Il m'est généralement difficile d'exprimer des sentiments publiquement. Ainsi, sachez que tous les mots de ces remerciements sont pensés avec suffisamment de ferveur pour avoir pu franchir la barrière de la pudeur.

Mes premiers remerciements vont à mon directeur de thèse, Nicolas Rougerie, qui a toujours su se rendre disponible pour répondre à mes interrogations et proposer des pistes de recherches pertinentes. La direction scientifique que tu m'as proposée a été à la fois passionnante et réalisable, ce qui, je trouve, m'a fourni un excellent contexte pour faire mes débuts dans le monde de la recherche. Durant nos séances de travail, j'ai été impressionné par ta culture, ta connaissance de la bibliographie de notre domaine et ta vision pour déceler les directions et schémas de preuves importants pour résoudre les problèmes présentés dans les pages qui suivront.

Que tous les membres de mon jury de soutenance soient remerciés, je tiens en grande estime vos travaux. Je suis honoré que cette thèse ait été rapportée par Chiara Saffirio et Nicolas Raymond. C'est aussi avec plaisir que je compte dans mon jury Laure Saint-Raymond, Thierry Champel, Mathieu Lewin. Merci aussi à Soeren Petrat, aux côtés de qui j'aurai l'opportunité de poursuivre ma carrière mathématique à Brême, de compléter cet exceptionnel jury. C'est avec hâte que j'attends cette collaboration.

En me remémorant le chemin qui m'a amené jusqu'ici, j'ai une pensée pour Sophie Schulz et Julien Esteban qui furent mes premiers professeurs de mathématiques dans les études supérieures. Ce sont eux qui m'ont permis de découvrir cette discipline et qui m'y ont donné goût. La découverte d'une vocation n'a pas de prix, je garderai toute ma vie des souvenirs, assis aux premiers rangs de leurs amphis bien sûr, leur bienveillance et pédagogie exemplaire. De l'autre côté de la connaissance, je suis reconnaissant envers tous les étudiants que j'ai pu avoir durant ces trois années pour m'avoir fait découvrir l'enseignement, activité que j'apprécie particulièrement.

À toutes les personnes de l'UMPA qui auront partagé ces trois années à mes côtés, les thésards du laboratoire, dont certains sont maintenant jeunes docteurs, vous avez contribué à rendre mon environnement de travail agréable. Notamment, Corentin et Raphaël qui ont successivement été mes colocataires du bureau. Raphaël, j'espère me souvenir des quelques enseignements d'algèbre que tu t'es aventuré à m'enseigner sur notre petit tableau. Bien que fort intéressants, cela ne m'aura pas permis de comprendre la troisième diapositive de ta belle soutenance. Merci Charlie, pour toutes tes questions sur la mécanique quantique, Basile, j'ai beaucoup apprécié préparer avec toi les TDs et DMs de Topologie, Antoine (Barrier), Jules, Léo et Ronan pour toutes les discussions.

Je souhaite dédier ce paragraphe à tous mes amis, dont les interactions me sont indispensables pour me changer les idées, pour tous les moments partagés. Merci Téo pour ton incroyable persistance à bien vouloir discuter de mes problèmes de mathématiques malgré leur aspect probablement rebutant. Je serai toujours étonné des bienfaits de l'exercice de vulgarisation qui, bien que complexe, permet de prendre un recul extrêmement bénéfique, de sortir la tête de ses équations, d'en faire le bilan pour enfin y voir un peu plus clair. Ta volonté et ton intuition font de toi un public remarquable pour que je puisse m'adonner à cette activité. Je n'oublierai pas Chris et Nathan pour votre participation assidue aux séminaires du discord, Antoine (Commaret) pour m'avoir rappelé l'existence de la formule sommatoire de Poisson une soirée de nouvel an, Robin et Emmanuel pour votre accueil récurrent (et la clim lors de la

canicule 2023), Mathieu, Julien, Martin, Cassandre, Aïlona et Billytis pour votre amitié.

Juliana, comment ne pas te remercier, pour avoir partagé ma vie et m'avoir supporté tout au long de la thèse. J'espère que tu retiendras que le Laplacien (-Laplacien pour être exact) représente l'énergie cinétique en mécanique quantique. Plus sérieusement, ta présence à mes côtés rend mon quotidien plus agréable et lumineux.

Enfin, je tiens à exprimer ma profonde gratitude envers ma famille qui m'a toujours chaleureusement entouré, tout particulièrement mes parents grâce à qui j'ai pu avoir la chance de me plonger sereinement dans de longues études. Je sais que vous ferez le trajet, le 09 Octobre, pour venir m'écouter prononcer mon charabia qui vous paraîtra certainement peu compréhensible. Toutefois, je serai très heureux de vous avoir à mes côtés en ce jour important pour moi qui viendra clore cette thèse.

Résumé

Cette thèse porte sur l'analyse mathématique de systèmes de fermions dans la limite où le nombre de particules tend vers l'infini. L'étude de ce genre de systèmes nécessite de calculer les énergies et densités des états physiques. Ce problème est extrêmement difficile à résoudre même numériquement en raison des interactions et des corrélations entre les particules. En effet avec un grand nombre de particules l'espace de Hilbert représentant l'espace des états possibles du système est immense. Cependant, l'objectif général est d'obtenir des théories effectives où l'énergie ne dépend que de la densité de particules (théorie fonctionnelle de la densité). Dans le cadre de la thèse, nous nous intéressons plus précisément à des fermions sans spin confinés dans un espace à deux dimensions soumis à un fort champ magnétique transverse et homogène. C'est à dire que l'amplitude du champ magnétique tend vers l'infini en même temps que le nombre de particules. Ce contexte physique est motivé par l'effet Hall quantique. Les principaux outils proviennent de l'analyse semi-classique mais aussi plus généralement de la théorie spectrale et de l'analyse fonctionnelle.

Après un premier chapitre introductif situant les résultats principaux par rapport à la bibliographie, le deuxième chapitre de la thèse reprend le contenu de la prépublication "Multiple Landau level filling for a mean field limit of 2D fermions". Dans ce travail, nous étudions la limite de champ moyen, couplée à une limite semi-classique et fort champ magnétique, pour l'état fondamental du système de fermions. En physique classique, les particules chargées soumises à un champ magnétique transverse et homogène décrivent des orbites. En mécanique quantique, l'énergie cinétique est quantifiée en niveaux d'énergie discrets appelés niveaux de Landau, séparés par un gap constant. Les résultats principaux de cet article étendent des travaux de Lieb-Solovej-Yngvason et Fournais-Madsen en considérant un régime où le gap entre niveaux de Landau est l'échelle dominante d'énergie par rapport aux interactions. De plus, les particules sont placées dans un domaine borné, permettant à la dégénérescence des niveaux de Landau d'être finie. Nous obtenons ainsi un modèle limite où un nombre arbitraire de niveaux de Landau sont remplis. Les travaux existants, utilisant un confinement par un potentiel de piégeage, ne peuvent décrire cette situation. Nous discutons aussi de la physique du modèle limite dans le dernier niveau de Landau partiellement rempli.

Le dernier chapitre présente un résultat sur la dynamique dans le même contexte que le chapitre précédent. Notre point de départ est l'équation de Hartree pour la première matrice densité. Il est bien connu que cette équation peut être obtenue par une approximation de champ moyen de la dynamique de Schrödinger à N corps pour des fermions en interaction. Nous étudions une limite fort champ magnétique, couplée avec une limite semi-classique et prouvons que la densité converge vers une solution d'une équation de transport gyro-cinétique. Dans un cadre de mécanique classique, des travaux de Golse-St-Raymond obtiennent des résultats similaires avec comme point de départ l'équation de Vlasov. Plus récemment un travail dans la limite semi-classique de Ben Porat traite le cas où le gap entre niveaux de Landau est très petit devant les interactions. La nouveauté principale de notre résultat est de considérer le régime quantique au départ où le gap entre niveaux de Landau est du même ordre de grandeur que l'énergie d'interaction. La preuve consiste à construire une densité semi-classique dont l'évolution en temps est obtenue puis comparée à l'équation cible.

Abstract

This thesis is about the mathematical analysis of fermionic systems in the large number of particles limit. The study of this kind of systems requires computing the energies and densities of the physical states. This problem is extremely difficult to solve, even numerically, due to the interactions and correlations between particles. Indeed, with a large number of particles, the Hilbert space representing the space of possible states of the system is very large. However, the general goal is to obtain effective theories where the energy only depends on the particle density (density functional theory). More precisely, we consider spinless fermions confined in a two-dimensional space subjected to a strong transverse and homogeneous magnetic field. That is to say the amplitude of the magnetic field goes to infinity simultaneously with the number of particles. This physical context is motivated by the quantum Hall effect. The main tools come from semi-classical analysis but also more generally from spectral theory and functional analysis.

After a first introductory chapter contextualising the main results in the bibliography, the second chapter is based on the content of the pre-publication "Multiple Landau level filling for a mean field limit of 2D fermions". In this work, we study the mean field limit, coupled with a semi-classical and strong magnetic field limit, of the fermionic system's ground state. In classical physics, charged particles subjected to a transverse and homogeneous magnetic field describe orbits. In quantum mechanics, kinetic energy is quantized into discrete energy levels called Landau levels, separated by a constant gap. The main results of this paper extend work by Lieb-Solovej-Yngvason and Fournais-Madsen by considering a regime where the gap between Landau levels is the dominant energy scale compared to the interactions. Moreover, the particles are placed in a bounded domain, allowing the degeneracy of the Landau levels to be finite. We thus obtain a limit model where an arbitrary number of Landau levels are filled. Existing works, using confinement by a trapping potential, cannot describe this situation. We also discuss the physics of the limit model in the last partially filled Landau level.

The last chapter presents a result on the dynamics in the same context as the previous chapter. Our starting point is the Hartree equation for the first density matrix. It is well known that this equation can be obtained by a mean-field approximation of the many-body Schrödinger dynamics for interacting fermions. We study a strong magnetic field limit, coupled with a semi-classical limit and prove that the density converges to a solution of a gyrokinetic transport equation. In a classical mechanics framework, Golse-St Raymond obtained similar results with the Vlasov equation as starting point. More recently, a work of Ben Porat in the semi-classical limit deals with the case where the gap between Landau levels is very small compared to the interactions. The main novelty of our result is to consider a quantum regime where the gap between Landau levels is of the same order as the interaction energy. The proof consists in constructing a semi-classical density for which the dynamic is obtained and then compared to the target equation. Le problème à N corps quantique correspond à l'étude d'un système de N particules en interaction régies par les lois de la mécanique quantique. On s'intéresse typiquement aux énergies possibles, aux densités et à la dynamique du système. En mécanique quantique, une particule placée dans un domaine Ω est décrite par un vecteur de l'espace de Hilbert L 2 pΩ, Cq. Un système de N particules est quant à lui décrit par un élément de L 2 pΩ, Cq bN -L 2 pΩ N , Cq L'énergie d'un système quantique est décrite par un opérateur auto-adjoint agissant sur l'espace de Hilbert des états, appelé Hamiltonien, noté H. Lorsque l'on mesure l'énergie d'un système quantique, le résultat est une valeur propre de l'Hamiltonien, puis, l'état se retrouve projeté sur le sous espace propre associé. Étant donné un état physique ψ, la densité de particules correspond au module carré de cet état, elle est ainsi intégrable. La dynamique est quant à elle régie par l'équation de Schrödinger ihB t ψ " Hψ h est la constante de Planck réduite, elle définit l'échelle des phénomènes quantiques. Dans un système d'unités pour lequel la masse des particules est choisie égale à 1{2, l'Hamiltonien d'un système de N particules sans spin prend la forme :

H N :" N ÿ i"1
`´h 2 ∆ i `V px i q ˘`ÿ iăj wpx i ´xj q (I.1.1)

L'opérateur ´∆ correspond à l'énergie cinétique du système. V : Ω Ñ R est le potentiel extérieur auquel sont soumises les particules. w représente le potentiel d'interaction entre une paire de particules. Les potentiels agissent multiplicativement sur les états physiques. Les notations ∆ i , V px i q signifient que ces opérateurs n'agissent que sur la i ième particule, autrement dit sur la i ième coordonnée de L 2 pΩ N , Cq. wpx i ´xj q agit quant à lui sur la paire de particules pi, jq, c'est-à-dire sur la i ième et la j ième coordonnée de L 2 pΩ N , Cq. L'Hamiltonien à N corps comprend alors la somme des énergies cinétique et potentielle de chaque particule ainsi que la somme des interactions entre chaque paire de particules.

Dans un état physique à N corps ψ N P L 2 pΩ N , Cq, les densités des particules sont corrélées si ψ N n'est pas factorisable sous la forme ψ N " u bN , avec u P L 2 pΩ, Cq La résolution du problème quantique à N corps, c'est-à-dire la diagonalisation de l'Hamiltonien à N corps est en général particulièrement difficile à cause des interactions et des corrélations entre les particules. Même numériquement, l'étude de ce problème devient vite impossible pour de grandes valeurs de N car l'espace de Hilbert des états devient immense. Le nombre d'électrons dans les systèmes physiques est typiquement de l'ordre de 10 10 électrons par mm 2 d'après [START_REF] Lieb | Quantum Dots[END_REF]. Comprendre comment les particules interagissent, se distribuent dans l'espace et évoluent dans le temps dans un système à N corps est essentiel pour de nombreux domaines de la physique, tels que la physique des particules, la chimie quantique et l'étude des matériaux quantiques. Dans un premier temps, on peut s'intéresser à la valeur propre la plus basse (l'énergie de l'état fondamental) qui est l'énergie du système sans excitation (à température nulle). Du côté des mathématiques, l'objectif général est l'obtention de théories efficaces où l'énergie ne dépend que de la densité physique des particules. La densité de l'état fondamental est ainsi donnée par le minimiseur et l'énergie fondamentale par le minimum de la fonctionnelle. Une telle théorie est appelée une théorie fonctionnelle de la densité. La minimisation de l'énergie se fait donc sur un espace fonctionnel des densités, plus simple que l'espace de Hilbert de départ. Cela vient cependant aux dépens d'une non-linéarité dans le terme d'interaction. Le modèle le plus simple de ce type est le modèle de Thomas Fermi (voir Subsection I.2.1). Les résultats mathématiques sur les théories fonctionnelles de la densité sont connus pour être utiles dans les méthodes numériques pour l'étalonnage des paramètres libres. Les théories fonctionnelles de la densité sont les principaux outils pour les calculs numériques quantiques, par exemple en chimie quantique où l'on veut calculer la densité électronique dans les systèmes moléculaires.

Par exemple, la théorie d'Hartree-Fock, qui sera détaillée en Subsection II.6.1, fait l'hypothèse de champ moyen. Cela revient à supposer que les particules ne sont pas corrélées, c'est-àdire que chaque particule ne voit que le potentiel moyen généré par les autres particules. Mathématiquement, cette hypothèse revient à supposer que les densités des particules sont indépendantes. Ainsi, une expression de l'énergie qui ne dépend que de la densité peut être obtenue. Certaines théories fonctionnelles de la densité ajoutent des termes pour prendre en compte les corrélations améliorant ainsi la théorie d'Hartree-Fock. Cela est possible grâce à différentes approximations dont la plus simple est l'approximation de densité locale. Cette approximation suppose que le terme d'échange-corrélation dans la fonctionnelle est localement égal à celui pouvant être obtenu analytiquement pour un gaz homogène d'électrons.

Mathématiquement, il est intéressant d'apporter une justification rigoureuse à ce genre d'approximations en montrant l'égalité du résultat apporté par les modèles approchés avec celui fourni par l'état fondamental de l'Hamiltonien quand N Ñ 8 au vu du nombre de particules dans les systèmes physiques concernés.

I.1.2 Fermions

Les particules physiques étant indiscernables, la densité physique de particules doit être invariante par permutation des coordonnées. Il y a deux manières canoniques d'imposer cette symétrie sur L 2 pΩ N , Cq permettant de modéliser des bosons ou des fermions. En effet, une fonction symétrique ou antisymétrique sous l'échange de deux coordonnées est respectivement appelée bosonique ou fermionique.

Notation I.1.1

On note H -L 2 pΩq l'espace des états modélisant une particule. On utilisera la notation bra-ket, aussi appelée formalisme de Dirac. Un vecteur ψ P H sera aussi noté |ψ⟩, et son dual, c'est-à-dire la projection sur ψ, ⟨ψ|. Le produit scalaire sur H sera quant à lui noté ⟨'|'⟩.

On considère l'action du groupe symétrique S N sur H bN , définie par

S N ˆHbN Ñ H bN σ, u :" Â N i"1 u i Þ Ñ U σ u :" Â N i"1 u σ ´1piq
On définit alors l'espace de Hilbert symétrique/antisymétrique à N corps comme

H b ˘N :" č σPS N
Ker pU σ ´ϵ˘p σqq où ϵ ˘pσq :" p˘1q |tiăj|σpiqąσpjqu| Dans la suite on utilise la convention suivante pour les listes :

x 1:Npx 1 , . . . , x N q Avec cette convention,

U σ upx 1:N q " N ź i"1 u σ ´1piq px i q " N ź i"1
u i px σpiq q " upx σpiq , . . . x σpN q q donc H b ˘N " ␣ ψ N P H bN |@σ P S N , ψ N px σpiq , . . . x σpN q q " ϵ ˘pσqψ N px 1:N q ( (I.1.2)

Si ψ N P H bN a une densité invariante par permutation des coordonnées, σ ¨|ψ N | 2 " |ψ N | 2 et donc σ ¨ψN et ψ N ne diffèrent que d'une phase : @x 1:N P Ω N , σ ¨ψN px 1:N q " e iϕ N pσ;x 1:N q ψ N px 1:N q Maintenant, en supposant l'invariance par translation du système, c'est-à-dire en supposant que la phase est la même après translation de ψ N , U σ ¨ψN px 1:N `y1:N q " e iϕ N pσ;x 1:N q ψ N px 1:N `y1:N q alors la phase est constante. Physiquement, l'échange de deux particules ne dépend pas de l'origine spatiale choisie. De plus, U στ " U σ U τ implique que U σ est unitaire d'inverse U σ ´1 et que e iϕ N pστ q " e iϕ N pσq e iϕ N pτ q

Cela revient à dire que la phase ϕ N est une représentation scalaire, donc unitaire, du groupe symétrique. Or, il existe seulement deux représentations scalaires irréductibles du groupe symétrique, données par le morphisme de signature et l'identité.

L'antisymétrie encode le principe de Pauli : on ne peut pas mettre deux particules dans le même état. En effet si ψ, χ P H, ψ " χ ùñ ψ ^χ " ψ b χ ´χ b ψ " 0 Il s'agit en fait d'un argument par l'absurde. Si l'on a un état antisymétrique à deux corps avec deux particules dans le même état, la densité à deux corps est nulle, donc il n'y a pas de particules, ce qui est absurde car l'état doit être normalisé. La classe de particules décrites par des états antisymétriques est appelée fermions, la classe de particules décrites par des états symétriques est appelée bosons.

Afin de mettre en perspective cet argument, rappelons que la classification de particules obtenue diffère si l'on essaye d'imposer la symétrie de permutation avant la quantification. En partant de l'espace de configuration classique pour N particules Ω N , puis en imposant la symétrie en identifiant les orbites sous l'action du groupe symétrique, un échange de particules est alors une boucle dans cet espace topologique. Comprendre les manières possibles d'échanger des particules revient alors à calculer son groupe fondamental. En dimension d ě 3, ce groupe fondamental est à nouveau S N , ainsi la quantification (recherche de représentations scalaires irréductibles) donne des bosons et des fermions. Mais en dimension d " 2, le groupe fondamental obtenu est le groupe des tresses B N qui a une structure plus riche et donne lieu à une autre classe de particules appelée anyons. Lors de la transposition de deux particules anyoniques, leur fonction d'onde change d'une phase e iπα , avec 0 ă α ă 1. Il s'agit en quelque sorte d'une interpolation entre les bosons et les fermions. Les anyons semblent apparaître comme des quasi-particules dans l'effet Hall quantique [START_REF] Kumar R.Bisognin | Fractional statistics in anyon collisions[END_REF].

On définit une projection conique sur l'espace des tenseurs symétriques/antisymétriques H b ˘N .

Proposition I.

1 ? N ! N â ȋ"1 u i " 1
Proof :

En utilisant ϵ ˘pσqϵ ˘pτ q " ϵ ˘pστ q et U σ U τ " U στ ,

P 2 ˘" 1 N ! 2 ÿ σ,τ PS N
ϵ ˘pστ qU στ " P ȃprès le changement de variable σ :" στ . Donc P ˘est une projection orthogonale sur Rg pP ˘q, et avec le même changement de variable,

U τ P ˘" 1 N ! ÿ σPS N
ϵ ˘pσqU τ σ " ϵ ˘pτ qP Donc Rg pP ˘q Ă H b ˘N . Pour l'autre inclusion,

P ˘|H b ˘N " 1 N ! ÿ σPS N ϵ ˘pσqU σ |H b ˘N " 1 N ! ÿ σPS N ϵ ˘pσq 2 " 1
Soit χ, ψ P H bN , avec le changement de variables σσ ´1,

⟨χ|P ˘ψ⟩ " 1 N ! ÿ σPS N ϵ ˘pσq ⟨χ|U σ ψ⟩ " 1 N ! ÿ σPS N ϵ ˘pσq ⟨U σ ´1 χ|ψ⟩ " ⟨P ˘χ|ψ⟩
Si u j " u k for j ‰ k, avec le changement de variables στ j,k σ,

u i ˇˇˇˇN â ȋ"i u i " ÿ σ,τ PS N ϵ ˘pστ q N â i"1 u σ piq ˇˇˇˇN â i"1 u τ piq " ÿ σ,τ PS N ϵ ˘pστ q N ź i"1 ⟨u σ piq|u τ piq⟩ " ÿ σ,τ PS N ϵ ˘pστ q N ź i"1 δ σpiq,τ piq " ÿ σ,τ PS N ϵ ˘pστ qδ σ,τ " ÿ σ ϵ ˘`σ 2 ˘" N ! I.1.

Régime de champ moyen et limite semi-classique

Étudier un système quantique de N fermions revient donc à diagonaliser l'Hamiltonien (I.1.1) agissant sur l'espace H b ´N défini en (I.1.2). Mais, pour avoir une limite intéressante quand N Ñ 8, il est nécessaire de s'assurer du fait que toutes les contributions dans l'énergie soient du même ordre de grandeur. Pour cela, remarquons que le nombre de termes d'interaction est de l'ordre OpN 2 q. En divisant le terme d'interaction par un facteur N on obtient l'Hamiltonien dans le scaling de champ moyen

H N :" N ÿ i"1
`´h 2 ∆ i `V px i q ˘`1 N ÿ iăj wpx i ´xj q (I. 1.3) Ainsi, les termes d'interaction et de potentiel extérieur sont tous les deux d'ordre N . Il reste à imposer la même chose pour le terme d'énergie cinétique. Dans le cas de bosons, pour l'état fondamental, ce terme est d'ordre N car toutes les particules peuvent minimiser leur énergie cinétique en se plaçant dans le même état. À cause du principe de Pauli, la situation est radicalement différente dans le cas des fermions. Toutes les particules ne pouvant pas minimiser leur énergie cinétique simultanément, l'ordre de grandeur de cette dernière se trouve être plus élevé que dans le cas bosonique. L'outil permettant de déterminer cet ordre de grandeur est l'inégalité de Lieb-Thirring.

Pour énoncer cette inégalité il faut introduire la notation de matrice densité. Dans la limite N Ñ 8, le nombre de variables des fonctions de L 2 pR d q b ´N diverge. Pour espérer obtenir des résultats de convergence, il est nécessaire de s'intéresser aux densités réduites qui appartiennent à un espace avec un nombre fini de variables.

Notation I.1.3: Matrices densités

On note L 1 l'espace des opérateurs à trace. Une matrice densité à N corps est un élément γ N P L 1 `Hb ˘N ˘positif de trace 1. Soit Tr I la trace partielle qui prend la trace sur les coordonnées dans I Ă 1, N de H bN , définie par

@A 1:N P L 1 pHq N , Tr I « N â i"1 A i ff -Tr « â iPI A i ff â iRI A i
puis étendue par linéarité sur L 1 pHq bN . On associe à γ N la k ième matrice densité réduite

γ pkq N -Tr k`1:N rγ N s
On associe à une matrice densité γ une densité

ρ γ pxq -γpx, xq
Ici, tout comme dans la suite, l'abus de notation consistant à noter de la même façon un opérateur et son noyau intégral est utilisé. ρ γ pkq N est appelée k ième densité réduite.

Dans le cas d'une projection

γ N " |ψ N ⟩ ⟨ψ N | , avec ψ N P L 2 `Hb ˘N ȏn retrouve bien ρ γ N " |ψ N | 2
De plus, si on identifie les opérateurs avec leurs noyaux intégraux, la k ième matrice densité réduite peut s'exprimer comme

γ pkq N px 1:k , y 1:k q - ż Ω pN ´kq γ N px 1:k , x k`1:N ; y 1:k , x k`1:N qdx k`1:N
Remarquons aussi qu'avec ces notations si µ pkq désigne la k ième marginale de µ, on a

ρ γ pkq N " ρ pkq γ N Notation I.1

.4: Convention pour les constantes

Tout au long du texte, on utilisera la notation C pour les constantes. C'est-à-dire qu'avec une famille de paramètres pλ i q i et deux fonctions positives f, g la relation f p', pλ i q i q " Opgp', pλ i q i qq sera notée f p', pλ i q i q ď C ppλ i q i q gp', pλ i q i q Nous sommes maintenant prêts pour l'énoncé de l'inégalité de Lieb-Thirring.

Theorem I.1.5: Inégalité de Lieb-Thirring

Soit ψ N P L 2 pR d q b ´N , et γ N -|ψ N ⟩ ⟨ψ N |, ψ N ˇˇˇˇN ÿ i"1 p´∆ i qψ N ě CN 1`2 d ż R d `ρp1q γ N ˘1`2 d (I.1.4)
Le livre [13, theorem 4.3] fournit une preuve de cette inégalité. La principale étape de cette preuve est l'obtention d'une borne inférieure pour la somme des valeurs propres négatives de l'Hamiltonien à un corps. Cette inégalité permet d'utiliser le contrôle sur l'énergie cinétique pour montrer que la première densité réduite n'est pas juste intégrable mais dans L 1`2 d . Une inégalité de ce genre sera démontrée dans le second chapitre en Section II.4 et adaptée au contexte du chapitre, c'est-à-dire avec un champ magnétique sur un domaine borné.

L'inégalité de Lieb-Thirring donne donc une borne inférieure pour l'ordre de grandeur du terme d'énergie cinétique dans (I.1.3). De plus, pour des électrons sans interaction : w " 0, confinés dans une boîte : V " 0 dans r0, Ls d et V " 8 en dehors, un calcul explicite de l'énergie fondamentale peut être réalisé. En effet, les vecteurs propres de l'énergie cinétique sont de la forme

ψ N ˇˇˇˇN ÿ i"1 p´h 2 ∆ i qψ N ě CN ż R d `ρp1q γ N ˘1`2 d
Ce régime est dit semi-classique car dans la limite h Ñ 0 un système quantique converge vers son équivalent en mécanique classique. Le régime de champ moyen est alors naturellement couplé à une limite semi-classique. Il est connu, voir [START_REF] Fournais | The semi-classical limit of large fermionic systems[END_REF], que l'énergie fondamentale de l'Hamiltonien à N corps converge alors vers le minimum de la fonctionelle de Thomas-Fermi :

E T F pρq - d d `2C T F ż R d ρpxq 1`d 2 dx `ż R d ρpxqV pxq `1 2 ij R d ˆRd ρpxqρpyqwpx ´yqdxdy avec comme constante C T F -4π 2 ˆd |S d´1 | ˙2 d
Ici, l'énergie cinétique est donnée par la même puissance de ρ dans dans l'inégalité de Lieb-Thirring. Cette intégrale peut être obtenue en regardant la limite de la somme des N plus petites valeurs propres h2 k 2 de l'énergie cinétique pour k dans le réseau réciproque (I.1.5).

Le modèle d'un atome neutre est un exemple de système physique respectant le régime champ moyen/semi-classique. Il prend en compte un noyau chargé positivement de charge Z -N où N est le numéro atomique de l'atome. Les fermions sont ici N électrons de charge ´1 interagissant entre eux avec un potentiel coulombien, placés dans un potentiel coulombien créé par le noyau. L'Hamiltonien est alors :

H N - N ÿ i"1 ˆ´∆ i ´Z |x i | ˙`ÿ 1ďiăjďN 1 |x j ´xi | (I.1.7) En dimension d " 2, H N N - N ÿ i"1 ˆ´´N ´1 2 ¯2 ∆ i ´1 |x i | ˙`1 N ÿ 1ďiăjďN 1 |x j ´xi | (I.1.8)
À un coefficient multiplicatif près, l'Hamiltonien de la forme (I.1.3) avec (I.1.6).

I.1.4 Effet Hall quantique

σ Hall " ν e 2 2πh
où e est la charge élémentaire, ν est appelé facteur de remplissage et correspond aux nombre de niveaux de Landau remplis dans le système. ν peut prendre des valeurs entières mais aussi certaines valeurs fractionnaires. L'effet Hall fractionnaire est dû aux interactions entre particules [START_REF] Jain | Composite fermions[END_REF]. La quantification de la conductivité de Hall a la propriété d'être extrêmement précise lors de mesures expérimentales. Ainsi l'effet Hall quantique trouve des applications en métrologie.

Le gap entre niveaux de Landau étant proportionnel à l'intensité du champ magnétique, le nombre de niveaux remplis dépend de l'intensité du champ magnétique. Lorsque le champ magnétique est très fort, le gap entre le plus bas niveau de Landau et le niveau suivant est très grand, ainsi toutes les particules se retrouvent dans le plus bas niveau afin de minimiser leur énergie cinétique. Dans cette thèse, les champ magnétiques seront d'une intensité permettant un mixage entre niveaux de Landau dans le but de fournir un contexte mathématique dans lequel le facteur de remplissage peut prendre des valeurs entières et fractionnaires. Cependant l'effet Hall fractionnaire reste hors de la portée des résultats présentés dans ce manuscrit car ce phénomène apparaît au delà des limites de champ moyen et semi-classiques qui sont ici étudiées.

I.2 Statique

I.2.1 Un théorème de Lieb, Solojev et Yngvason

On considère un système de N fermions sans spin dans un espace à deux dimensions. On se donne maintenant un champ magnétique uniforme d'intensité b, transverse au domaine des fermions. Pour un champ magnétique assez fort, les spins sont généralement considérés alignés avec le champ magnétique et donc ne sont pas à prendre en compte. Ce problème a initialement été étudié par Lieb Solovej et Yngvason in [START_REF] Lieb | Quantum Dots[END_REF], [START_REF] Lieb | Heavy atoms in the strong magnetic field of a neutron star[END_REF], [START_REF] Lieb | Asymptotics of Heavy Atoms in High Magnetic Fields I. Lowest Landau Band Region[END_REF], [START_REF] Lieb | Asymptotics of Heavy Atoms in High Magnetic Fields II. Semi-classical Regions[END_REF], [START_REF] Lieb | Ground states of large quantum dots in magnetic fields[END_REF] et plus récemment par Fournais, Lewin and Madsen in [START_REF] Fournais | The semi-classical limit of large fermionic systems[END_REF], [START_REF] Fournais | Semi-classical limit of confined fermionic systems in homogeneous magnetic fields[END_REF]. Dans ce qui suit, nous allons énoncer le résultat principal de [START_REF] Lieb | Ground states of large quantum dots in magnetic fields[END_REF]. On se place en jauge symétrique : En mécanique quantique la longueur magnétique est la distance sur laquelle les états propres de l'énergie cinétique sont localisés. L'état fondamental et l'énergie fondamentale de l'Hamiltonien sont

Apx, yq - 1
H N :" N ÿ i"1 `p´ih∇ i `bApx i qq 2 `V px i q ˘`1 N ÿ iăj wpx i ´xj q et agit
ψ 0 N -argmin ! ⟨ψ N |H N ψ N ⟩ , ψ N P L 2 `R2 ˘b´N tel que ∥Ψ N ∥ L 2 " 1 ) E 0 N -min ψ 0 N ˇˇH N ψ 0 N
On définit maintenant les modèles limite, à commencer par le modèle électrostatique :

E class pρq - ż R 2 ρpxqV pxqdx `1 2 ij R 2 ˆR2 ρpxqρpyq |x ´y| dxdy E 0 class -min ␣ E class pρq tel que ρ P L 1 `R2 ˘, ρ ě 0, ∥ρ∥ L 1 " 1 (
Ce modèle est purement classique, le premier terme correspond à l'énergie potentielle et le second à l'énergie d'interaction entre les particules. Ensuite, on considère le modèle de Thomas-Fermi :

E T F pρq - ż R 2 ρpxq 2 dx `Eclass pρq E 0 T F -min ␣ E T F pρq tel que ρ P L 1 `R2 ˘, ρ ě 0, ∥ρ∥ L 1 " 1 (
Finalement, on définit le modèle de Thomas-Fermi magnétique où le terme d'énergie cinétique est légèrement modifié par rapport au modèle de Thomas-Fermi :

E T F M pρq - ż R 2 j b pρpxqqdx `Eclass pρq E 0 T F M -min ␣ E T F M pρq tel que ρ P L 1 `R2 ˘, ρ ě 0, ∥ρ∥ L 1 " 1 ( avec j b P C 0 pR `q défini par j b p0q " 0 j 1 b pρq " 2hb ˆn `1 2 ˙si n 2πl 2 b ă ρ ă n `1 2πl 2 b
Dans ce contexte, l'énergie du niveau de Landau n P N est 2hb `n `1 2 ˘. Le gap entre deux niveaux voisins est donc 2hb. Il est bien connu, [START_REF] Jain | Composite fermions[END_REF] ou Subsection II.1.4, que la dégénérescence par unité d'aire à l'intérieur d'un niveau de Landau est d'ordre 1{p2πl 2 b q. Ceci justifie la définition de j b pρpxqq car les niveaux de Landau les plus bas sont remplis en priorité. Ainsi, pour x P R 2 tel que

n 2πl 2 b ă ρpxq ă n `1 2πl 2 b
n niveaux de Landau sont remplis localement en x et rajouter une particule en x coûte une énergie 2hb `n `1 2 ˘, c'est-à-dire l'énergie d'une particule dans le pn `1q ième niveau de Landau. Notons aussi que la fonction j b ainsi définie est une approximation de la fonction carré lorsque l b Ñ 8. Dans cette limite, le modèle de Thomas-Fermi magnétique se réduit à celui de Thomas-Fermi.

Le résultat de [26, Théorèmes 1.2 et 1.3] est alors le suivant : Theorem I.2.1: Ground states of large quantum dots in magnetic fields (LSY'95)

Si V P C 1,α loc pour un 0 ă α ď 1. En notant ρ N la première densité réduite de |ψ 0 N ⟩ ⟨ψ 0 N |, uniformément en h et b, E 0 N N Ñ N Ñ8 E 0 M T F ρ N N á N Ñ8 ρ 0 T F M au sens des mesures puis si l 2 b N Ñ 8, E 0 N N Ñ N Ñ8 E 0 T F ρ N N á N Ñ8 ρ 0 T F au sens des mesures et si l 2 b N Ñ 0, E 0 N N Ñ N Ñ8 E 0 class ρ N N á N Ñ8
ρ 0 class au sens des mesures En se plaçant dans un potentiel confinant, le système reste dans un volume borné. Ainsi, tout dépend de la limite du ratio

1 2πl 2 b N " 1 2πl 2
b N de la densité d'état dans un niveau de Landau divisé par le nombre de particules :

• Si l 2 b N Ñ 0, la densité d'état est suffisamment élevée pour permettre d'accueillir toutes les particules dans le plus bas niveau de Landau. Comme le gap entre niveaux de Landau est très grand, il est impossible pour les particules d'être excitées dans des niveaux supérieurs. L'énergie cinétique du système à N corps est alors triviale : N hb où hb est l'énergie du plus bas niveau de Landau. Et le modèle converge alors vers un modèle électrostatique décrit par E class .

• Si l 2 b N Ñ 8, tous les niveaux de Landau sont occupés, le gap entre niveaux est petit par rapport au nombre de particules. Le modèle limite, celui de Thomas-Fermi, est alors le même que l'on obtient en l'absence de champ magnétique.

• Si l 2 b N " Op1q, le nombre de niveau de Landau remplis varie en fonction de ρpxq. La limite obtenue est le modèle de Thomas Fermi magnétique. Le facteur de remplissage au point x étant donné par 2πl

I.2.2 Résultats

On souhaite traiter la situation intermédiaire par rapport à [START_REF] Lieb | Ground states of large quantum dots in magnetic fields[END_REF] où seul un nombre fini de niveaux de Landau sont complètement remplis. Précisément, notre résultat est une limite où les q premiers niveaux Landau sont entièrement remplis, le niveau Landau suivant est partiellement rempli avec un taux de remplissage r ă 1 et tous les niveaux Landau supérieurs sont vides. Cette configuration est physiquement motivée par l'effet Hall quantique qui se produit principalement dans un niveau Landau partiellement rempli tandis que les niveaux Landau inférieurs sont remplis et inertes, et les niveaux supérieurs sont vides, voir [START_REF] Jain | Composite fermions[END_REF].

Dans cette perspective, on fixe le rapport entre le nombre de particules et la dégénérescence des niveaux de Landau. Sur tout l'espace R 2 cette dégénérescence est infinie. Pour s'assurer de la finitude de la dégénérescence des niveaux de Landau, on travaille sur un domaine borné. Pour simplifier la situation, l'idéal serait de considérer des conditions aux bords périodiques. Mais, en présence d'un champ magnétique, les conditions aux bords périodiques doivent être modifiées. C'est un problème bien connu, par exemple voir [22, Section 3.9]. On définit alors des opérateurs de translation magnétique qui commutent avec le moment magnétique. Ces opérateurs de translations magnétiques définissent les conditions aux bords périodiques magnétiques. La conséquence de ce choix de domaine est qu'il faut adapter le formalisme semi-classique standard.

Notation I.2.2: Modèle

On se place sur Ω -r0, Ls 2 de taille L ą 0. On travaille en jauge de Coulomb :

∇ ¨A " 0 où A P C 8 pR 2 , R
H N - N ÿ j"1
´`ih∇ x j `bApx j q ˘2 `V px j q ¯`2 N ´1 ÿ 1ďjăkďN wpx j ´xk q agissant sur L 2 pΩq b ´N . Le potentiel d'interaction est supposé radial pour la métrique sur le tore :

wpx ´yqr wpdpx, yqq avec dpx, yqmin rPLZ 2 |x ´y `r|

Le domaine de H N est Dom pH N q -Dom pL h,b q b ´N et son énergie fondamentale E 0 N -inf txψ N |H N ψ N y , ψ N P DompH N q tel que ∥ψ N ∥ L 2 " 1u (I.2.3)
Le fait que L h,b soit auto-adjoint et l'existence de ses vecteur propres avec conditions aux bords magnétiques périodiques demande que le flux magnétique passant au travers du domaine, bL 2 , soit quantifié :

Dd Ă N tel que 2πd " b hL 2 " L 2 l 2 b
Ici, d se retrouve être la dégénérescence des niveaux de Landau. On peut donc fixer le nombre de niveaux remplis en choisissant un régime où N {d est fixé.

Notation I.2.3: Régime semi-classique

On choisit la constante de Planck hph N q N PN telle que

N ´1 2 ! h ! N ´1 4 (I.2.4) Soit q P N, r P r0, 1q, b -pb N q N PN tel que d - L 2 2πl 2 b Ă N et N d " N Ñ8 q `r `o ˆ1 hb ˙(I.2.5)
q donnera le nombre de niveaux Landau entièrement remplis et r le taux de remplissage du pq `1q ième niveau de Landau. Avec cette notation,

N d " 2πl 2 b N L 2 Ñ N Ñ8 q `r et 1 l 2 b " b h " N Ñ8 2πN pq `rqL 2
Dans ce régime, l'ordre de grandeur du champ magnétique est b " OphN q. Donc l'ordre de grandeur de l'énergie cinétique est hb " Oph 2 N q " 1 La contribution de l'énergie cinétique doit être de premier ordre par rapport aux termes potentiels si l'on veut imposer le nombre de niveaux de Landau remplis :

h2 N " 1 d'où la borne inférieure dans (I.2.4). La borne supérieure h ! N ´1 4 est une condition technique pour contrôler un terme d'erreur venant de l'énergie cinétique. La vitesse de convergence (I.2.5) est imposée pour la même raison.

On obtient un modèle semi-classique où l'énergie ne dépend plus de la fonction d'onde mais de la densité dans l'espace des phases N ˆΩ. Cela signifie que les particules ont deux degrés de liberté : le premier est n P N le nombre quantique représentant l'indice de niveau de Landau et x P Ω représentant la position des particules dans l'espace. En mécanique classique, x s'interprète comme le centre de l'orbite cyclotron. Ce modèle est semi-classique dans le sens où le principe de Pauli est toujours valable comme borne sur la densité. , .

-

E 0 sc,hb -inf mPDsc E sc,hb rms
On définit également le modèle électrostatique pour le niveau de Landau partiellement rempli qui ne dépend que de la densité.

Notation I.2.5: Modèle électrostatique pour le niveau de Landau partiellement rempli

E qLL rρs - ż Ω V ρ `ij Ω 2 wpx ´yqρpxqρpyqdxdy (I.2.7) D qLL - $ & % ρ P L 1 pΩq tel que ż Ω ρ " r q `r and 0 ď ρ ď 1 pq `rqL 2 , .
-

(I.2.8) E 0 qLL -inf D qLL E qLL Soit les énergies E q,r - q 2 `2qr `r q `r E q,r V - q q `r Ω V E q,r w - q 2 `2qr pq `rq 2 Ω 2 w et ρ P D qLL , m ρ pn, xq -1 năq 1 L 2 pq `rq
`1n"q ρpxq m ρ est une densité sur l'espace des phases construite avec les q plus bas niveaux de Landau maximisant la densité sous la contrainte du principe de Pauli dans (I.2.8) et une densité ρ dans le niveau de Landau suivant, partiellement rempli. La proportion de particules dans le niveau partiellement rempli est r q `r ce qui correspond à la constante de normalisation dans (I.2.8). Notons que le principe de Pauli est bien la borne correcte pour avoir

ż NˆΩ m ρ dη " 1
On démontrera dans la Proposition II.6.1 que E sc,hb rm ρ s " hbE q,r `Eq,r V `Eq,r w `EqLL rρs où • hbE q,r est l'énergie cinétique des q `1 plus bas niveaux de Landau

• E q,r V est l'énergie potentielle des q plus bas niveaux de Landau • E q,r w est l'énergie d'interaction contenant toutes les interactions exceptées celles à l'intérieur du niveau partiellement rempli.

On peut maintenant énoncer notre résultat principal sur la convergence de l'énergie fondamentale.

Theorem I.2.6: Convergence de l'énergie fondamentale Si V, w P L 2 pΩq,

E 0 N N " N Ñ8 hbE q,r `Eq,r V `Eq,r w `E0 qLL `op1q
Le premier ordre de l'énergie quantique fondamentale est l'énergie cinétique du système hbE q,r . Pour les termes d'ordre 1, les seules contributions non triviales à l'énergie fondamentale sont l'énergie potentielle et d'interaction à l'intérieur du niveau de Landau partiellement rempli. Les niveaux inférieurs sont totalement remplis, ainsi leurs contributions à l'énergie sont constante. L'interaction du niveau partiellement rempli avec tous les autres niveaux est également une constante. Pour les niveaux de Landau supérieurs, leur contribution à l'énergie est nulle car ils sont vides. La démonstration de la borne supérieure se base sur la théorie de Hartree-Fock. Cette dernière est obtenue en se restreignant aux déterminants de Slater (produit antisymétrique d'une famille orthonormale de L 2 pΩq). Pour ces états, de nombreux calculs sont simplifiés (voir le Theorem I.4.2 de Wick). La deuxième matrice densité réduite peut être reconstruite à partir de la première matrice densité réduite. Elle est ainsi donnée par un terme direct, ne comprenant aucune corrélation, et un terme d'échange qui encode les corrélations dues à l'antisymétrie des particules. La théorie de Hartree-Fock est donc une manière de supposer que toutes les autres corrélations sont négligeables. L'énergie de Hartree-Fock donne naturellement une borne supérieure pour l'énergie quantique à N corps puisque l'ensemble variationnel est réduit aux déterminants de Slater. Cette théorie peut être étendue à un opérateur à un corps (voir Notation II.6.3). En utilisant le principe variationnel de Lieb (Theorem II.6.5), on peut montrer que la théorie fournit toujours une borne supérieure approchée de l'énergie quantique à N corps. Il s'agit enfin prouver que l'énergie de Hartree-Fock est une approximation de l'énergie semi-classique (Proposition II.6.7).

Du côté de la borne inférieure, l'outil principal est le Theorem II.6.11 de De-Finetti. En effet, ce résultat permet d'écrire les k ième densité réduites comme des combinaisons convexes de densités à un corps tensorisées k fois. En ayant en tête le fait qu'une densité jointe de k particules ne possède aucune corrélation si elle peut s'écrire comme un produit d'une densité à un corps, il est clair que ce résultat joue un rôle important dans la justification de la décorrélation des densités.

I.2.4 Perspectives

Les hypothèses de régularité sur les potentiels ne sont pas optimales. Nous nous attendons à ce que ce résultat reste vrai si les potentiels ont une partie positive L 1 et une partie négative L 2 . Sous ces hypothèses, il faut prouver que les particules ne se concentreront pas dans les L 1 singularités positives des potentiels. Ceci a été fait dans [START_REF] Lieb | Ground states of large quantum dots in magnetic fields[END_REF] pour le potentiel de Coulomb.

Une autre piste intéressante à explorer serait de regarder le cas d'un champ magnétique non homogène (toujours transverse). En choisissant un champ magnétique suffisamment régulier, à l'échelle l b , le champ magnétique se comporte comme une constante. Ainsi l'énergie cinétique reste localement quantifiée en niveaux de Landau.

Le modèle pourrait aussi être complexifié en ajoutant les spins, ce qui revient à rajouter les matrices de Pauli dans le terme d'énergie cinétique de l'Hamiltonien à N corps.

Remplacer le Laplacien magnétique par l'opérateur de Dirac permettrait une description du graphène. En effet, dans le graphène, la relation de dispersion au niveau des cônes de Dirac dans la structure de bandes rend le transport des électrons relativiste. Le graphène est connu comme étant un bon matériau pour l'observation expérimentale de l'effet Hall quantique [START_REF] Zhang | Experimental observation of the quantum Hall effect and Berry's phase in graphene[END_REF]. Il est donc intéressant de considérer un modèle pour des fermions 2d dans un cadre de relativité restreinte. L'opérateur de Dirac n'étant pas positif, on ferait l'hypothèse de la mer de Dirac, supposant que dans le vide, tous les états d'énergie négative sont occupés. Une difficulté pour ce résultat serait de gérer des potentiels généraux : en effet, cela demanderait d'adapter une inégalité de Lieb-Thirring pour l'opérateur de Dirac avec condition aux bords magnétiques périodiques.

I.3 Dynamique I.3.1 Un petit tour de la bibliographie

Cette partie porte maintenant sur la dynamique donnée par l'Hamiltonien à N corps (I.1.3) et l'équation de Schrödinger

ihB t ψ N " H N ψ N (I.3.1) pour ψ N P L 8 ´R`, L 2 pΩq b ˘N ¯et R `-r0, 8r.
Il est bien connu que l'équation de Hartree peut être obtenue par une limite de champ moyen lorsque N Ñ 8. Si ψ N est une solution de (I.3.1), des résultats tels que [START_REF] Benedikter | Mean-Field Evolution of Fermionic Systems[END_REF] [39] prouvent que la première matrice densité réduite de |ψ N ⟩ ⟨ψ N | (voir Notation I.1.3), satisfait presque l'équation de Hartree :

ihB t γ " " ´h 2 ∆ `V `w ‹ ρ γ , γ ‰ (I.3.2) avec γ P L 8 pR `, L 1 pL 2 pΩqqq.
Dans la limite semi-classique h Ñ 0, en partant de l'équation de Hartree, on obtient l'équation de Vlasov [START_REF] Paul | Sur les mesures de Wigner[END_REF] [37] viendrons compléter la bibliographie. Certaines références choisissent aussi comme point de départ la dynamique de Newton [START_REF] Han-Kwan | From Newton's second law to Euler's equations of perfect fluids[END_REF].

Enfin, un résultat semi-classique [START_REF] Porat | Derivation of Euler's equations of perfect fluids from von Neumann's equation with magnetic field[END_REF] obtient l'équation d'Euler pour la vorticité depuis la dynamique de Schrödinger à N corps. Ce théorème concerne un régime dans lequel le gap entre les niveaux de Landau est petit comparé aux interactions. Ainsi l'espace des phases pour l'approximation semi-classique position-impulsion est toujours R d ˆRd .

I.3.2 Résultats

L'objet d'étude est toujours un système en deux dimensions de fermions soumis à un champ magnétique transverse et homogène. Notre point de départ est l'équation de Hartree (I.3.4), appropriée pour un système fermionique d'un grand nombre de particules. L'équation gyrocinétique (I.3.5) est obtenue dans une limite semi-classique couplée à une limite fort champ magnétique. Contrairement à [START_REF] Porat | Derivation of Euler's equations of perfect fluids from von Neumann's equation with magnetic field[END_REF], le régime est choisi pour que tous les termes de l'Hamiltonien pih∇ `bAq 2 `V `w ‹ ρ γ soient d'ordre 1. Cela entraîne un grand nombre de conséquences sur la méthode de preuve. En effet, vu que le gap entre niveaux de Landau hb doit être pris d'ordre 1, l'espace des phases pour l'approximation semi-classique est alors N ˆR2 . Dans cet espace des phases, n P N est le nombre quantique représentant le niveau de Landau et z P R 2 la position dans l'espace. Autrement dit, la quantification de l'énergie cinétique continue de jouer un rôle important dans la limite. Un fait marquant est que les fonctions de Wigner et la quantification de Weyl, utilisées par exemple dans [START_REF] Paul | Sur les mesures de Wigner[END_REF], n'ont pas d'équivalent sur cet espace des phases. Les travaux [START_REF] Liew | Combined mean-field and semiclassical limits of large fermionic systems[END_REF] [33] [START_REF] Liew | A mixed-norm estimate of the two-particle reduced density matrix of many-body Schrödinger dynamics for deriving the Vlasov equation[END_REF] utilisent plutôt les fonctions de Husimi, qui ont un équivalent sur N ˆR2 . Ces fonctions de Husimi sont construites à partir des états cohérents et seront introduites en Section II.5.

Dans notre méthode de preuve, une difficulté principale vient du fait que les états cohérents ne sont pas bien localisés pour les niveaux de Landau élevés. Il s'agit donc de considérer la dynamique dans les niveaux de Landau suffisamment bas et de contrôler le remplissage des niveaux élevés grâce aux moments de l'énergie cinétique. on définit la matrice densité remise à la bonne échelle de temps :

@t P R `, γ b ptq -γpbtq
Le choix de l'échelle de temps s'explique grâce à une étude classique Subsection III.1.4 qui nous montre que le mouvement du centre de l'orbite cyclotron s'effectue sur une échelle de temps d'ordre b. Ainsi Cela montre que tous les termes de l'Hamiltonien pih∇ `bAq 2 `V `w ‹ ρ γ sont bien d'ordre 1.

B t γ b " b ih " pih∇ `bAq 2 `V `w ‹ ρ γ b , γ b ‰ " 1 il 2 b " pih∇ `bAq 2 `V `w ‹ ρ γ b , γ b ‰ (I.
On énonce maintenant notre résultat principal. Si V, w P W 4,8 pR 2 q, alors @φ P C

I.3.4 Perspectives

Une première amélioration serait de réduire la régularité demandée sur les potentiels. Dans ce but, on pourrait utiliser des méthodes similaires à [START_REF] Laflèche | From Schrödinger to Hartree-Fock and Vlasov equations with Singular potentials[END_REF]. Pour cela, il faudrait établir l'équation pour la dynamique à l'intérieur d'un niveau de Landau, afin de pouvoir contrôler l'équation pour la fonction de Husimi directement et non la densité physique. Pour la dynamique de la fonction de Husimi, on s'attend à obtenir, en plus de l'équation gyro-cinétique, un terme de saut entre niveaux de Landau. Ainsi on obtiendrait un système d'équations couplées avec une équation pour chaque niveau de Landau. Cela permettrait de construire un objet quantique, analogue à la quantification de Weyl de la densité sur l'espace des phases position-impulsion R d ˆRd , à comparer à la matrice densité solution de l'équation de Hartree. Aussi, l'obtention de la propagation des moments de l'énergie cinétique d'ordre supérieur, faciliterait le contrôle des erreurs dans la dynamique de la première densité réduite. La propagation de certains I.4 Quelques outils

I.4.1 Matrices densités

En reprenant les Notation I.1.3, remarquons que les matrices densités étant positives, ce sont des opérateurs auto-adjoints. De plus, comme ce sont des opérateurs à trace, elles sont aussi compactes et ainsi diagonalisable par le théorème spectral. Voici quelques propriétés supplémentaires des matrices densités qui seront utilisées tout au long de la thèse:

Property I.4.1: Matrices densités Soit γ N P L 1 pH b ˘N q tel que Tr rγ N s " 1 et γ N ě 0, Tr " γ pkq N ı " 1, 0 ď γ pkq N Soit k P 1 : N , si A k P L 1 `Hbk ˘, alors Tr " γ N A k b Id bpN ´kq H ı " Tr " γ pkq N A k ı (I.4.1)
Proof :

On peut supposer γ N " Γ k b Γ N ´k, où Γ k P L 1 `Hbk ˘, Γ N ´k P L 1 `
γ N -|ψ N ⟩ ⟨ψ N |, avec ψ N " 1 ? N ! N â ȋ"1 ϕ j
où pϕ j q j est une famille orthonormale, alors

γ pkq N " pN ´kq! N ! ÿ 1ďi 1 ‰¨¨¨‰i k ďN ˇˇˇˇk â j"1 ϕ i j ÿ σPS k ϵ ˘pσq k â j"1 ϕ i σpjq ˇˇˇP roof : Premièrement, γ N " 1 N ! ÿ σ,τ PS N ϵ ˘pστ q ˇˇˇˇN â j"1 ϕ σpjq N â j"1 ϕ τ pjq ˇˇˇˇ" 1 N ! ÿ σ,τ PS N ϵ ˘pστ q N â j"1 ˇˇϕ σpjq ϕ τ pjq ˇĎonc γ pkq N " Tr k`1:N rγ N s " 1 N ! ÿ σ,τ PS N ϵ ˘pστ q ˜k â j"1 ˇˇϕ σpjq ϕ τ pjq ˇˇ¸N ź j"k`1 Tr "ˇˇϕ σpjq ϕ τ pjq ˇˇ‰ " 1 N ! ÿ σ,τ PS N ϵ ˘pστ q ˜k â j"1 ˇˇϕ σpjq ϕ τ pjq ˇˇ¸N ź j"k`1 δ σpjq,τ pjq
Avec le changement de variables τσ ´1τ on obtient

γ pkq N " 1 N ! ÿ σPS N ,τ PS N ϵ ˘pσστ q ˜k â j"1 ˇˇϕ σpjq ϕ στ pjq ˇˇ¸N ź j"k`1 δ σpjq,στ pjq " 1 N ! ÿ σPS N ,τ PS k ϵ ˘pτ q ˜k â j"1
ˇˇϕ σpjq ϕ στ pjq ˇˇ" Proof :

1 N ! ÿ 1ďi 1 ‰¨¨¨‰i k ďN ÿ σPS N ,τ PS k ϵ ˘pτ q ˜k â j"1 ˇˇϕ σpjq ϕ στ pjq ˇˇ¸k ź j"1 δ σpjq,i j " pN ´kq! N ! ÿ 1ďi 1 ‰¨¨¨‰i k ďN ÿ τ PS k ϵ ˘pτ q k â j"1 ˇˇϕ i j ϕ i τ pjq ˇ" pN ´kq! N ! ÿ 1ďi 1 ‰¨¨¨‰i k ďN ˇˇˇˇk â j"1 ϕ i j ÿ τ PS k ϵ ˘pτ q k â j"1 ϕ i τ pjq
On peut supposer γ N " |ψ N ⟩ ⟨ψ N |, avec ψ N " 1 ? N ! N â ȋ"1 ϕ j
où pϕ j q j est une famille orthonormale puis conclure par linéarité. Soit

ψ k - 1 ? k! k â j"1 ϕ j
En utilisant Theorem I.4.2, on calcule alors

ψ k ˇˇγ pkq N ψ k " pN ´kq! N !k! ÿ 1ďi 1 ‰¨¨¨‰i k ďN ÿ σ,τ,υPS k ϵ ˘pστ υq k â j"1 ϕ τ pjq ˇˇˇˇk â j"1 ϕ i j k â j"1 ϕ i σpjq ˇˇˇˇk â j"1 ϕ υpjq " pN ´kq! N !k! ÿ σ,τ,υ,ωPS k ϵ ˘pστ υq k â j"1 ϕ τ pjq ˇˇˇˇk â j"1 ϕ ωpjq k â j"1 ϕ ωσpjq ˇˇˇˇk â j"1 ϕ υpjq " pN ´kq! N !k! ÿ σ,τ,υ,ωPS k ϵ ˘pστ υqδ τ,ω δ ωσ,υ " pN ´kq! N !k! ÿ σ,ωPS k ϵ ˘pσωωσq " pN ´kq!k! N !
On peut construire une base de Hilbert de H en complétant pϕ j q 1ďjďN . Si ψ k est un produit d'éléments de la famille pϕ j q 1ďjďN , quitte à réordonner les ϕ j , on se ramène au calcul cidessus. On conclut alors en remarquant que si ψ k est un produit contenant des termes dans span ppϕ j q 1ďjďN q K , le résultat du calcul ci-dessus est 0.

I.4.2 États cohérents

Les états cohérents sont indispensables dans l'étude de limites semi-classiques.

⟨ψ α |N ψ α ⟩ " |α| 2 ∆N - b ⟨ψ α |N 2 ψ α ⟩ ´⟨ψ α |N ψ α ⟩ 2 " |α| 1 π ż C |ψ α ⟩ ⟨ψ α | dα " Id H Proof : Par un calcul direct, ⟨ψ α |ψ α ⟩ " e ´|α| 2 ÿ nPN αα n! " 1 et ra, Dpαqs "e ´|α| 2 2 "
a, e αa : ı e ´αa " e

´|α| 2 2 ˜ÿ nPN α n n! " a, a : n ‰ ¸e´αa "e ´|α| 2 2 ˜ÿ nPN ˚αn n! na : n´1 ¸e´αa " αDpαq donc aψ α " aDpαq |0⟩ " Dpαqa |0⟩ `αDpαq |0⟩ " αψ α ⟨ψ α |N ψ α ⟩ " ⟨aψ α |aψ α ⟩ " |α| 2 ψ α ˇˇN 2 ψ α " aψ α ˇˇaa : aψα " ⟨aaψ α |aaψ α ⟩ `⟨aψ α |aψ α ⟩ " |α| 4 `|α| 2 ∆N " b |α| 4 `|α| 2 ´|α| 4 " |α| Finalement, ż C |ψ α ⟩ ⟨ψ α | dα " ÿ n,mPN ż C e ´|α| 2 α n α m ? n!m! |n⟩ ⟨m| dα " ÿ nPN |ψ n ⟩ ⟨ψ n | ż C e ´|α| 2 |α| 2n n! dα "2π ÿ nPN |n⟩ ⟨n| 1 n! ż R `r2n`1 e ´r2 dr " π ÿ nPN |n⟩ ⟨n| 1 n! ż R `rn e ´rdr "π ÿ nPN |n⟩ ⟨n| " πId H Avec (I.4.
2), en interprétant a comme un opérateur de position dans C, ψ α est un état physique localisé autour α.

Chapter II Multiple Landau level filling for a large magnetic field limit of 2D fermions Abstract :

Motivated by the quantum hall effect, we study N two dimensional interacting fermions in a large magnetic field limit. We work in a bounded domain, ensuring finite degeneracy of the Landau levels. In our regime, several levels are fully filled and inert: the density in these levels is constant. We derive a limiting mean-field and semi classical description of the physics in the last, partially filled Landau level.

II.1 Context and result

II.1.1 Model

We consider a system of N interacting fermionic particles in two dimensions. They are placed under a homogeneous magnetic field perpendicular to the domain. In this context the kinetic energy of the particles is quantized into discrete energy levels called Landau levels, separated by a finite energy gap. This problem has initially been studied by Lieb Solovej and Yngvason in [START_REF] Lieb | Quantum Dots[END_REF], [START_REF] Lieb | Heavy atoms in the strong magnetic field of a neutron star[END_REF], [START_REF] Lieb | Asymptotics of Heavy Atoms in High Magnetic Fields I. Lowest Landau Band Region[END_REF], [START_REF] Lieb | Asymptotics of Heavy Atoms in High Magnetic Fields II. Semi-classical Regions[END_REF], [START_REF] Lieb | Ground states of large quantum dots in magnetic fields[END_REF] and more recently by Fournais, Lewin and Madsen in [START_REF] Fournais | The semi-classical limit of large fermionic systems[END_REF], [START_REF] Fournais | Semi-classical limit of confined fermionic systems in homogeneous magnetic fields[END_REF].

Our goal is to study the mean field and semi-classical limit under high magnetic field so the Landau level quantization plays an important role. This setup is related to that of [START_REF] Lieb | Ground states of large quantum dots in magnetic fields[END_REF] where three regimes are studied. In the first one, the energy gap is small with respect to the potential contributions in the energy so particles occupy all Landau levels and a standard Thomas-Fermi model is obtained in the limit. In the second one, the energy gap is comparable to the potential energy terms, particles optimise both their Landau level and their position in the potentials and the limit is a magnetic Thomas-Fermi model. For the last scaling, the gap is large compared to the potential energies so all particles occupy the lowest Landau level and the limit is described with a classical continuum electrostatic theory in this level. We want to deal with the intermediate situation where only a finite number of Landau levels are completely filled. Precisely, our result is a limit where the q first Landau Level are fully filled, the next Landau level is partially filled with filling ratio r ă 1 and all higher Landau levels are empty. We also provide a model for the physics in the partially filled Landau level. This setup is physically motivated by the quantum Hall effect which mostly takes place in a partially filled Landau level while lower Landau levels are filled and inert, and higher levels are empty, see [START_REF] Jain | Composite fermions[END_REF].

In this perspective we want to fix the limit ratio of the number of particles to the degeneracy of Landau levels. On the whole space R 2 this degeneracy is infinite. To ensure finiteness of the degeneracy of the Landau levels (see Proposition II.2.11), we work on a bounded domain. For simplicity, we would like to consider a torus with periodic boundary conditions. But, in the presence of a magnetic field the periodic boundary conditions must be modified. This is a well known issue, for example see [START_REF] Jain | Composite fermions[END_REF]Section 3.9]. As explained in Subsection II.2.1, we define magnetic translation operators to ensure commutation with the magnetic momentum. These magnetic translations operators define the so called magnetic periodic boundary conditions.

Notation II.1.1: Model

We work on the domain Ω -r0, Ls 2 of fixed size L ą 0. The one body kinetic energy operator, also called magnetic Laplacian, is

L h,b -pih∇ `bAq 2 (II.1.1)
We work in the Coulomb gauge:

∇ ¨A " 0 where A P C 8 pR 2 , R 2 q is the vector potential generating the constant magnetic field: identifying R 2 with R 2 ˆt0u Ă R 3 , we assume

∇ ^A " p0, 0, 1q (II.1.2)
b is the magnetic field amplitude with associated magnetic length

l b - c h b
We identify R 2 with C and use complex notation for the variables px, yq P R 2 namely px, yq " x `iy P C Let z 0 P C, by (II.1.2)

∇ ^pA ´Ap' ´z0 qq " 0 so we can choose φ z 0 P C 8 pR 2 , Rq such that

A ´Ap' ´z0 q -l 2 b ∇φ z 0 (II.1.3)
For some usual expressions see (II.2.3) and (II.2.4). As detailed in Subsection II.2.1, for a wave-function ψ P L 2 pΩq the magnetic periodic boundary conditions are @t P r0, Ls, # ψpL `itq " e iφ L pL`itq ψpitq ψpt `iLq " e iφ iL pt`iLq ψptq (II. 1.4) and the domain of the magnetic Laplacian is Dom pL h,b q -␣ ψ P H 2 pΩq such that (II.1.4) holds ( (II.1.5)

Now, the N -body Hamiltonian is

H N - N ÿ j"1
´`ih∇ x j `bApx j q ˘2 `V px j q ¯`2 N ´1 ÿ 1ďjăkďN wpx j ´xk q (II. 1.6) acting on the space of N -body fermionic wave-functions

L 2 ´`Ω N ˘-N ľ L 2 pΩq.
We denote T -R 2 {LZ 2 . V P L 2 pTq is the external potential and w P L 2 pTq the interaction potential assumed to be radial for the metric on the torus:

wpx ´yq -r wpdpx, yqq with dpx, yq -min rPLZ 2 |x ´y `r|
The domain of the N -body Hamiltonian (II.1.6) is

Dom pH N q - N ľ Dom pL h,b q
We define the N -body ground state energy

E 0 N -inf txψ N |H N ψ N y , ψ N P DompH N q such that ∥ψ N ∥ L 2 " 1u (II.1.7)
When we change the gauge, wave-functions change by a phase thus the energy is independent of the choice of gauge. There are N pN ´1q{2 interacting pairs of fermions. Thus, we divide the interactions term by pN ´1q{2 so that the order of the contribution coming from interactions is OpN q and comparable to the contribution coming from the external potential.

As we will see in Subsection II.2.2, the self adjointness of the magnetic Laplacian and the existence of its eigenvectors require the magnetic field flux bL 2 going through the domain to be quantized in multiples of 2πh:

Dd Ă N such that 2πd " b h L 2 " L 2 l 2 b
We will prove in Proposition II.2.11 that d is the degeneracy of Landau levels. Now, we can fix the number of filled Landau levels by choosing a scaling for which the ratio N {d is fixed.

Notation II.1.2: Scaling

We take Planck's constant to be a sequence hph N q N PN such that

N ´1 2 ! h ! N ´1 4 (II.1.8) Let q P N, r P r0, 1q, b -pb N q N PN be such that d - L 2 2πl 2 b Ă N ˚(II.1.9)
and

N d " N Ñ8 q `r `o ˆ1 hb ˙(II.1.10)
where E ˚-Ezt0u for E Ă R.

q will give the number of fully filled Landau levels and r the filling ratio of the pq `1q th Landau level. Note that the lowest Landau level index is 0 in our convention. With this notation,

N d " 2πl 2 b N L 2 Ñ N Ñ8 q `r and 1 l 2 b " b h " N Ñ8 2πN pq `rqL 2 (II.1.11)
With this scaling, we find that the order of the magnetic field is b " OphN q. It is known (II.2.18), that the order or the kinetic energy is hb " Oph 2 N q " 1 (II.1.12)

The kinetic energy contribution needs to be of leading order compared to the potential terms if we want to impose the number of filled Landau level and this is true if and only if h2 N " 1 hence the lower bound in (II. 1.8). The upper bound h ! N ´1 4 is necessary in our approach to control some error terms coming from the kinetic energy. This is also the reason why we impose the convergence rate in (II. 1.10). This scaling is a semi-classical limit because Planck's constant goes to 0.

To satisfy (II.1.10), one can take for example

d - Z N q `r ŝo N q `r ´1 ď d ď N q `r ùñ q `r ď N d ď N N q `r ´1 " q `r 1 ´q `r N " N Ñ8 q `r `O ˆ1 N
Ṅote that if r is rational one can take sequences such that there is no error in (II.1.10), and if r is irrational (see [START_REF] Durand | Topics in metric number theory[END_REF]Proposition 1.4]) it is always possible to have

N d " N Ñ8 q `r `O ˆ1 N 2 İI.1.2
Semi-classical limit model

In the limit, we obtain a semi-classical model where the energy no longer depends on the wavefunction but on the density in phase space. This comes with a non linearity in the interaction term. The phase space is N ˆΩ. This means that particles have two degrees of freedom: the first one is n P N the quantum number representing the Landau Level index and R P Ω representing the position of particles in space. In classical mechanics, one can think of R as the center of the cyclotron orbit of the particles and n as the index of the quantized angular velocity of the cyclotron orbit. This model is semi-classical in the sense that the Pauli principle still holds as a bound on the density.

Notation II.1.3: Semi-classical functional

We consider the measure on phase space

η - ˜ÿ nPN δ n ¸b λ Ω
where λ Ω is the Lebesgue measure on Ω. For a phase space density m P L 1 pN ˆΩ, R `q, the semi-classical energy is

E sc,hb rms - ż NˆΩ E n mpn, Rqdηpn, Rq `ż NˆΩ V mdη `ż pNˆΩq 2 wm b2 dη b2 (II.1.13)
where, as we will see in Section II.2,

E n -2hb ˆn `1 2 ˙(II.1.14)
is the energy of the n th Landau level. Define the semi-classical domain

D sc - $ & % m P L 1 pN ˆΩq such that ż NˆΩ mdη " 1 and 0 ď m ď 1 pq `rqL 2
, .

-(II. 1.15) and the semi-classical ground state energy

E 0 sc,hb -inf mPDsc E sc,hb rms
We also define the electrostatic model for the partially filled Landau level that only depends on the density.

Notation II.1.4: Electrostatic model for the partially filled level

Define

E qLL rρs - ż Ω V ρ `ij Ω 2 wpx ´yqρpxqρpyqdxdy (II.1.16)
with domain

D qLL - $ & % ρ P L 1 pΩq such that ż Ω ρ " r q `r and 0 ď ρ ď 1 pq `rqL 2 , . - (II.1.17)
The associated ground state energy is

E 0 qLL -inf D qLL E qLL
We define the following energies:

E q,r - q 2 `2qr `r q `r (II.1.18) E q,r V - q q `r Ω V (II.1.19) E q,r w - q 2 `2qr pq `rq 2 Ω 2 w (II.1.20) Let ρ P D qLL , define m ρ pn, xq -1 năq 1 L 2 pq `rq `1n"q ρpxq (II.1.21)
m ρ is a phase space density constructed with the qLL lowest Landau levels saturating the Pauli principle in (II.1.15) and (II.1.17) and with the density ρ in the partially filled Landau level. The ratio of particles in the partially filled Landau level is r q `r This corresponds to the normalization constraint in (II. 1.17). With this we see that the Pauli principle is indeed the correct bound on the densities to have

ż NˆΩ m ρ dη " 1
We will see in Proposition II.6.1 by a direct computation that E sc,hb rm ρ s " hbE q,r `Eq,r V `Eq,r w `EqLL rρs where • hbE q,r is the kinetic energy contribution from the q `1 lowest Landau levels

• E q,r
V is the external potential energy contribution from the q lowest Landau levels • E q,r w is the energy contribution from interactions between the q lowest Landau levels and the interactions between the q lowest Landau levels and the pq `1q th Landau level. in other words, it contains all the interactions except the ones inside the partially filled level.

The particles in the partially filled Landau level try to optimise their localisation with respect to the self consistent potential V `w ˚ρ:

E qLL rρs " ż Ω pV `w ˚ρq ρ
If w has a non-negative fourier transform, this functional is convex and thus has a unique minimizer. For example, with V " 0, the minimizer is

ρ 0 - r pq `rqL 2

II.1.3 Main results

We can now state our main result:

E 0 N N " N Ñ8 hbE q,r `Eq,r V `Eq,r w `E0 qLL `op1q
This means that in the limit, the first order in the quantum many body energy per particle is the trivial energy hbE q,r . Then for terms of order 1, the only non trivial contribution to the energy are the external potential term and the interaction term inside the partially filled Landau level. The lower Landau levels are totally filled and therefore their contribution to the energy is constant. The interaction of the partially filled level with all other level will also be a constant. For higher Landau levels, their contribution to the energy is null because they are totally empty.

The regularity assumptions on the potentials are not minimal, we expect this result to hold true if potentials have an L 1 positive part and an L 2 negative part. Under these assumptions, one needs to prove that the particles will not concentrate in the L 1 positive singularities of the potentials. This has been done in [START_REF] Lieb | Ground states of large quantum dots in magnetic fields[END_REF] for the repulsive 1{ |x| Coulomb potential. We will not deal with this issue in this paper.

The number of variables of the densities is going to infinity in our limit. As usual for a large number of particles, obtaining a convergence of densities requires to work in a space with a finite number of variables and therefore look at reduced densities.

Notation II.1.6: Reduced densities

We denote L p the set of p-Schatten class operators along with ∥'∥ L p the p-Schatten norm. Let γ N P L 1 `L2

´`Ω N ˘˘a positive operator (thus self adjoint) of trace 1. We call such an operator an N -body density matrix. By the spectral theorem, γ N is diagonalizable in a Hilbert basis of L 2

´`Ω N ˘:

γ N " ÿ iPN λ i |u i ⟩ ⟨u i | with 0 ď λ i ď 1 and ÿ iPN λ i " 1
We will denote in the same way operators and their integral kernel. We introduce compact notation for lists:

1 : n -p1, . . . , nq x 1:n -px 1 , . . . , x n q
The density associated to γ N is

ρ γ N px 1:N q -γ N px 1:N , x 1:N q
Let Tr I be the partial trace that traces out coordinates in I Ă 1, N of L 2 pΩq bN , it is defined as the linear operator on L 1 `L2 `ΩN ˘˘satisfying

@A 1:N P L 1 `L2 pΩq ˘, Tr I « N â i"1 A i ff -Tr « â iPI A i ff â iRI A i Let 1 ď k ă N , we define the k th reduced density matrix associated to γ N by γ pkq N " Tr k`1:N rγ N s (II.1.22)
with the convention that γ pN q N -γ N . For an N variables symmetric density ρ N we denote ρ pkq N its k th marginal. If one starts from a wave-function ψ N P L 2

´`Ω N ˘we use the notation

γ ψ N -|ψ N ⟩ ⟨ψ N | ρ ψ N -ρ γ ψ N " |ψ N | 2 (II.1.23)
Note that with this notation

ρ γ pkq N " ρ pkq γ N (II.1.24)
In (II.1.22), we have integrated the last N ´k variables but the result does not depend of the choice of these variables. Indeed, a permutation of coordinates brings a sign ˘in front of each |u i ⟩ and this keeps γ N invariant. The domain Ω is a locally compact metric space, the set of Radon measures on it is the dual of continuous and compactly supported functions

MpΩq " C 0 c pΩq
In our case, since Ω is compact this is just the dual of continuous functions. We denote M `pΩq the set of positive Radon measures. Let PpΩq be the set of probabilities on Ω:

PpΩq -tµ P M `pΩq such that µpΩq " 1u
On this space the weak star topology is metrizable using a Wasserstein metric [10, Section 7.12]. Moreover PpΩq is also locally compact [3, section 17.E], thus it is possible to iterate and define the space of probability measures on PpΩq namely P pPpΩqq. Now, we have the following theorem for the convergence of reduced densities:

Theorem II.1.7: Densities convergence with magnetic periodic conditions With the definitions introduced in Notation II.1.1, Notation II.1.2, Notation II.1.3, Notation II.1.4, Notation II.1.6, if pψ N q is a sequence of minimizers of (II.1.7), then Dµ P P pD qLL q such that • µ only charges minimizers of the limit energy functional (II.1.16)

• @k P N ˚, in the sense of Radon measures

ρ pkq ψ N á N Ñ8 ż D qLL ˆq L 2 pq `rq `ρ˙b k dµpρq (II.1.25)
The density of particles converge to a convex combination of densities of the form q L 2 pq `rq `ρ

From the Pauli principle in (II.1.17) we see that the constant term in this expression corresponds to particles in the q lowest and fully filled Landau levels. Then the density of particles in the partially filled Landau level is given by a minimizer ρ of the limit functional (II.1.16).

II.1.4 Scaling

Another way to obtain the scaling in Notation II.1.2 is to observe that we have two characteristic length-scales:

• L ? N , measuring the mean distance between particles

• l b , the magnetic length, which, in classical mechanics corresponds to the radius of a cyclotron orbit. Due to the Pauli principle, l b will be the order of the minimum distance between particles inside a Landau level. More precisely the Pauli principle takes the form of an upper bound on the density in phase space.

The square ratio of these length is

L 2 N l 2 b " bL 2 hN (II.1.26)
If this ratio goes to zero, the mean distance between particles is very small compared to the minimal length-scale between two particles in a fixed Landau level. This implies that the particles must fill many Landau levels and this corresponds to the scaling in [START_REF] Lieb | Ground states of large quantum dots in magnetic fields[END_REF] where the energy gap between Landau level is small compared to the potential terms.

If this ratio goes to infinity, the mean distance between particles is very large compared to the minimal length-scale between two particles in a fixed Landau Level. As a consequence, all particles can be placed in the lowest Landau level and this corresponds to the regime in [START_REF] Lieb | Ground states of large quantum dots in magnetic fields[END_REF] where particles only occupy the lowest Landau level and do not feel the Pauli principle.

In the limit we study, we see from (II. 1.11) that the ratio (II.1.26) has been fixed to be

L 2 N l 2 b Ñ N Ñ8 2π q `r (II.1.27)
in order to fill a finite number of Landau levels. In our limit we fixed L and took l b going to zero, but one can also ensure (II.1.27) by fixing a magnetic length r l b ą 0 and taking a domain length r L going to infinity as

r L - r l b l b L (II.1.28)
In this limit the density of particles in the domain Ω is fixed:

r L 2 N Ñ N Ñ8 r l b 2 2π q `r (II.1.29)
The two scalings can be unified by saying that we use a small dimensionless parameter

α - l b L " r l b r L (II.1.30)
Indeed, taking a small magnetic length as in the first scaling or a large domain length in the second scaling implies that α goes to zero as in (II.1.30). One can go from a limit to the other by unitary transformation of the Hamiltonian.

Proposition II.1.8: Re-scaling Define the unitary operator

U τ : L 2 pr0, Ls 2 q Ñ L 2 ˆ"0, r L ı 2 u Þ Ñ u τ -τ u pτ 'q with τ - L r L
then the N -body Hamiltonian (II.1.6) is unitarily equivalent to

U bN τ H N U τ bN " 1 τ H N,τ
where

H N,τ - N ÿ j"1 `pih τ ∇ j `bτ A τ px j qq 2 `Vτ px j q ˘`2 N ´1 ÿ 1ďjăkďN w τ px j ´xk q (II.1.31) and hτ -h ? τ A τ - 1 τ A pτ 'q b τ -τ 3 2 b V τ -τ V pτ 'q w τ -τ w pτ 'q Proof :
First, we check unitarity. Let u P L 2 pΩq,

∥u τ ∥ 2 " τ 2 ż r0, r Ls 2 u pτ ¨xq 2 dx " ż r0,Ls 2 upxq 2 dx " ∥u∥ 2
With the same change of variable we see that the adjoint of U τ is

U τ : L 2 ˆ"0, r L ı 2 ˙Ñ L 2 pr0, Ls 2 q u Þ Ñ 1 τ u ˆ1 τ ' Ṫhe transformation of the kinetic momentum is U τ pih∇ `bAq U τ u " 1 τ U τ pih∇ `bAq u ˆ1 τ ' " 1 τ U τ ˆih 1 τ ∇u ˆ1 τ ' ˙`bAp'qu ˆ1 τ ' ˙" ih 1 τ ∇up'q `bA pτ 'q up'q " 1 ? τ pih τ ∇ `bτ A τ q u
The action of U τ on the potentials is

U τ V pxqU τ " V pτ ¨xq U b2 τ wpxqU τ b2 " w pτ ¨xq
We conclude by putting together this two computations and factorising by 1 τ We confirm that the new magnetic length is

c hτ b τ " r Ll b L " r l b
Moreover if one chooses a linear A (like (II.2.3) or (II.2.4)) and

V pxq " ϱ ˚1 |x| wpxq " 1 |x|
then the vector potential and the interaction potential are not rescaled :

A τ " A w τ " w
If we assume that the external potential is generated by a background charge density ϱ P L 1 pΩq it transforms as

V τ pxq " ż Ω τ ϱpτ x ´yq 1 |y| dy " ż r0, r Ls 2 τ 2 ϱpτ py ´xqq 1 |y| dy -ϱ τ ˚1 |x|
The re-scaling preserves the total charge

ż r0, r Ls 2 ϱ τ dx " ż Ω ϱdx and H N,τ " N ÿ j"1 ˆpih τ ∇ j `bτ Apx j qq 2 `ρτ ˚1 |x| ˙`2 N ´1 ÿ 1ďjăkďN wpx j ´xk q
We conclude that our initial scaling is equivalent to a thermodynamic limit.

II.1.5 Organisation of the paper

The next two sections contain preparations and necessary tools. Section II.2 is about the diagonalisation of the magnetic Laplacian (II.1.1). In Section II.3 we define the orthogonal projection on Landau levels and localise it in space, this will be the central object in the definition of the semi-classical densities. Then we prove a Lieb-Thirring inequality in Section II.4 to deal with L 2 potentials. The last two sections contain the proof of Theorem II.1.5 and Theorem II.1.7. In Section II.5 we justify the semi-classical approximation and express the energy in terms of semi-classical densities. Finally, in Section II.6 we prove the mean-field approximation giving an upper and a lower energy bound.

II.2 Quantization

In this Section, we recall the diagonalization of the magnetic Laplacian (II.1.1). We construct an orthonormal basis of L 2 pΩq adapted to the Landau levels in terms of magnetic periodic eigenstates of L h,b . This result is stated in Proposition II.2.16. This fact is already well known in the literature, see [START_REF] Serfaty | Lowest Landau level approach in superconductivity for the Abrikosov lattice close to HC2[END_REF], [START_REF] Almog | Abrikosov Lattices in Finite Domains[END_REF] or in [22, section 3.9]. Thus the reader may go directly to Proposition II.2.16 and accept its statement.

To prove Proposition II.2.16 we will see that on a finite domain, the degeneracy of Landau levels is finite in Proposition II.2.11. We use the fact that the Landau levels are isomorphic and we study the lowest Landau level for which we prove the following properties:

• the wave-functions have a finite number of zeros inside the domain (Proposition II.2.10)

• the degeneracy is the number of zeros of the wave-functions (Proposition II.2.11) Then, we will prove another expression for the eigenfunctions in Proposition II.2.18 using the Poisson summation formula.

II.2.1 Magnetic translation operators

In this subsection we explain the definition of the boundary conditions (II.1.4). If T z 0 is the translation operator by z 0 and we try to commute the magnetic momentum with a translation we get rih∇ `bA, T z 0 s " b rA, T z 0 s " bpA ´Ap' ´z0 qqT z 0 Thus we cannot impose periodic boundary conditions, which would mean finding joint eigenfunctions of L h,b and of the translation operators with eigenvalue 1. The remedy is to compose the translation operator with a change of phase chosen to ensure commutation with P h,b . Thus, L h,b and the magnetic translations can be diagonalized jointly. This means that we can Notation II.2.1 Let z 0 P C, define the translation operator on u P L 2 loc pR 2 q by T z 0 u -up' ´z0 q

We define the magnetic translation operators as

τ z 0 -e iφz 0 T z 0 (II.2.1)
They define the conditions (II.1.4) on BΩ. Let k ě 1, we define the magnetic periodic Sobolev spaces as

H k mp pΩq - ␣ ψ P H k pΩq such that (II.1.

4) hold (

We will use similar notation for other usual functional spaces where the subscript mp stands for magnetic periodic and p for periodic. The domain of the magnetic momentum

P h,b -ih∇ `bA is DompP h,b q -H 1 mp pΩq
In Coulomb gauge, there exists ϕ P C 8 pR 2 , Rq such that the vector potential satisfies

A " ∇ K ϕ - ˆ´B y ϕ B x ϕ ˙(II.2.2)
For k ą 1, H k pΩq ãÑ C 0 pΩq, so the conditions (II.1.4) are well defined. For k " 1 they are defined with the trace operator T and ψ |Ω -T ψ.

For some examples of Coulomb gauges, one can take the symmetric gauge:

ϕ sym - |z| 2 4 A Sym - 1 2 px, yq K - 1 2 p´y, xq φ z 0 ,sym - x 0 y ´y0 x 2l 2 b (II.2.3)
or the Landau gauge: In the next proposition we also emphasize the importance of the flux quantization (II.1.9). The magnetic translations in the two directions L and iL defining the lattice commute if and only if the flux is quantized. Therefore when the flux is quantized, we are able to impose magnetic periodic boundary conditions in both directions L and iL.

ϕ Lan - y 2 2 A Lan -p´y, 0q φ z 0 ,Lan - ´y0 x l 2 b (II.2.
Proposition II.2.2: Commutation between magnetic Laplacian and magnetic translations

We have the commutation relation

rτ L , τ iL s " 0 ðñ L 2 l 2 b P 2πZ
and in this case,

@k ě 1, H k mp pΩq " ␣ ψ |Ω , ψ P H k loc `R2 ˘such that τ L ψ " τ iL ψ " ψ ( (II.2.8)
moreover, the magnetic Laplacian (II.1.1) commutes with the magnetic translations defined in (II.1.3) and (II.2.1):

rP h,b , τ z 0 s " 0 on H 1 mp pΩq and rL h,b , τ z 0 s " 0 on H 2 mp pΩq
In view of (II.2.8) we will identify ψ P H k mp pΩq and its extension on R 2 from now on. Proof of Proposition II.2.2:

We compute rτ L , τ iL s " e iφ L T L e iφ iL T iL ´eiφ iL T iL e iφ L T L " `eipφ L `TL φ iL q ´eipφ iL `TiL φ L q ˘TL T iL (II.2.9) So rτ L , τ iL s " 0 ðñ Dd P Z such that φ L `TL φ iL ´φiL ´TiL φ L " 2πd
and it is sufficient to prove

φ L `TL φ iL ´φiL ´TiL φ L " L 2 l 2 b (II.2.10)
With the Stokes theorem:

ż BΩ A.dl " ż Ω dS " L 2 (II.2.11)
Using (II.1.3) we get another computation for this integral

ż BΩ A.dl " L ż 0 pApuq ´Apu `iLqq ¨p1, 0qdu `i L ż 0 pApL `iuq ´Apiuqq ¨p0, 1qdu " l 2 b L ż 0 p´B x φ iL pu `iLq `iB y φ L pL `iuqq du " l 2 b r´φ iL pu `iLq `φL pL `iuqs L 0 " l 2 b pφ L `TL φ iL ´φiL ´TiL φ L q pL `
iLq but because of (II.2.11) this quantity is constant. Dividing by l 2 b gives (II.2.10). The right to left inclusion in (II.2.8) is just the evaluation of τ L ψ " τ iL ψ " ψ on BΩ. The left to right inclusion is obtained by extending ψ with the magnetic translations τ L , τ iL and the extension is well defined due to the commutation relation rτ L , τ iL s " 0. Next we compute P h,b e iφz 0 " e iφz 0 pih∇ `bA ´h∇φ z 0 q so e iφz 0 T z 0 P h,b " pih∇ `bAp' ´z0 q `h∇φ z 0 qe iφz 0 T z 0 With the definitions (II.1.3), (II.2.1) this ensure that rP h,b , τ z 0 s " 0.

To emphasize even more the importance of the flux quantisation, we state that

␣ ψ |Ω , ψ P H k loc `R2 ˘such that τ L ψ " τ iL ψ " ψ ( is trivial without it. Property II.2.3 ␣ ψ |Ω , ψ P H k loc pR 2 q such that τ L ψ " τ iL ψ " ψ ( is dense in L 2 pΩq if and only if (II.1.9) holds. Otherwise ␣ ψ |Ω , ψ P H k loc `R2 ˘such that τ L ψ " τ iL ψ " ψ ( " t0u
Proof :

If (II.1.9) holds, by Proposition II.2.2,

C 8 c pΩq Ă H k mp pΩq " ␣ ψ |Ω , ψ P H k loc `R2 ˘such that τ L ψ " τ iL ψ " ψ (
hence the density in L 2 pΩq. If (II.1.9) does not hold, let ψ P H k loc pR 2 q such that τ L ψ " τ iL ψ " ψ, then rτ L , τ iL sψ " 0 but using (II.2.9) and (II.2.10)

rτ L , τ iL sψ "e ipφ iL `TiL φ L q `eipφ L `TL φ iL q´φ iL ´TiL φ L ´1˘T L T iL ψ "e ipφ iL `TiL φ L q ˆei L 2 l 2 b ´1˙T L T iL
and since the flux is not quantized,

L 2 l 2 b R 2πZ
so T L T iL ψ " 0 and ψ " 0.

The next lemma proves that

␣ u P L 2 pΩq such that L h,b u P L 2 pΩq ( " H 2 pΩq Indeed, Lemma II.2.4
Let ψ P L 2 pΩq, assume L h,b ψ P L 2 pΩq, then ψ P H 2 pΩq and

∥ψ∥ H 2 ď Cpb, hq ∥L h,b ψ∥
Note that the converse embedding is also true because A and its gradient are bounded.

Proof of Lemma II.2.4:

∥L h,b ψ∥ ě h2 ∥∆ψ∥ ´ pb 2 A 2 `ihb∇ ¨Aqψ ´∥2hbA ¨∇ψ∥ ě h2 ∥∆ψ∥ ´Cpbq ∥ψ∥ ´2h ∥A∥ 8 ∥∇ψ∥
But the gradient term can be controlled by the Laplacian. Let D ą 0, using that

@k P 1 L Z 2 , k 2 ď

II.2.2 Landau Level quantization

In this subsection, we set up the usual formalism for the description of the magnetic Laplacian in term of annihilation and creation operators. More details about these operators and the properties of Landau levels can be found in [START_REF] Rougerie | Holomorphic quantum hall states in higher landau levels[END_REF].

Notation II.2.5

We denote by π x , π y the coordinates of the magnetic momentum: and the number of excitation operator Na : a.

P h,b - ˆihB x `
The quantization of the magnetic Laplacian comes from the following commutation relations:

rπ x , π y s " ihb (II.2.16) "
a, a : ‰ " Id (canonical commutation relation)

rτ z 0 , as " " τ z 0 , a : ‰ " 0 (II.2.17)
and the magnetic Laplacian is diagonal in terms of creation and annihilation operators:

L h,b " 2hb ˆN `Id 2
˙(II.2.18)
In the next lemma we prove that the magnetic Laplacian L h,b defines a Sobolev space whose norm turns out to be equivalent to the H 1 pΩq norm. Proof :

First, we prove a Green formula for the magnetic momentum. Let χ P H 1 pΩq, ⃗ ψ P H 1 pΩ, C 2 q, we use the Stokes theorem:

ż BΩ χ ⃗ ψ K ¨⃗ npxqdx " ż Ω ∇ ¨´χ ⃗ ψ ¯" ż Ω ∇χ ¨⃗ ψ `ż Ω χ∇ ¨⃗ ψ
where ⃗ npxq is the outer normal vector of Ω when x P BΩ. So

xχ|P h,b ¨⃗ ψy " ż Ω χpxq pih∇ `bAq ¨⃗ ψpxqdx " ż Ω ⃗ ψpxq ¨pih∇ `bAq χpxqdx `ih ż BΩ χpxq ⃗ ψ K pxq ¨⃗ npxqdx "x ⃗ ψ|P h,b χy `ih ż BΩ χ ⃗ ψ K ¨⃗ n
Further assume χ, ⃗ ψ are magnetic periodic, then χ ⃗ ψ is periodic so the boundary term vanishes. Thus P h,b is symmetric. The symmetry of L h,b follows from

P h,b H 2 mp pΩq Ă H 1 mp pΩ, C 2 q
Indeed, if ψ P H 2 mp pΩq, P h,b ψ P H 1 pΩ, C 2 q and P h,b ψ is magnetic periodic since the magnetic translations commute with P h,b . We deduce that π x and π y are symmetric on H Finally to prove that pDom pL h,b q , x'y L q is a Hilbert space we need to prove that it is closed in H 2 pΩq. Let ψ n P Dom pL h,b q such that ψ n Ñ ψ in H 2 pΩq, the limit also satisfies magnetic periodic boundary conditions because

τ z 0 ψ n " ψ n ùñ ∥τ z 0 ψ ´ψ∥ L 2 ď ∥τ z 0 ψ ´τz 0 ψ n ∥ L 2 `∥ψ n ´ψ∥ L 2 " 2 ∥ψ n ´ψ∥ L 2 Ñ nÑ8 0 ùñ τ z 0 ψ " ψ Continuity and coercivity are trivial, if χ, ψ P Dom pL h,b q, |xχ|ψy| L " |xP h,b χ|P h,b ψy| ď ∥P h,b χ∥ L 2 ∥P h,b ψ∥ L 2 " ∥χ∥ L ∥ψ∥ L (II.2.22) xψ|ψy L " ∥ψ∥ 2 L ě ∥ψ∥ 2 L
The proposition implies spectral properties of L h,b .

Corollary II.2.7: Spectral analysis of the magnetic Laplacian L h,b is a closed positive self-adjoint operator and the embedding Dom pL h,b q ãÑ L 2 pΩq is continuous and compact.

Proof :

The

`Dom pL h,b q ãÑ L 2 pΩq ˘" `H1 pΩq ãÑ L 2 pΩq ˘˝`D om pL h,b q ãÑ H 1 pΩq
The boundary of Ω is Lipschitz so the left embedding is compact due to the Rellich-Kondrachov theorem and the right one is continuous from Lemma II.2.6. Thus, the composition is compact.

H 2
mp pΩq contains the smooth and compactly supported functions, so it is dense in L 2 pΩq. We can conclude using the Lax-Milgram theorem [START_REF] Cheverry | A guide to spectral theory applications and exercises[END_REF]Corollary 4.26] that the resolvent of L h,b is well defined and compact. Applying the spectral theorem to the resolvent of L h,b proves that its spectrum is punctual and L 2 pΩq is a Hilbertian direct sum of the eigenspaces of L h,b . The same conclusions also holds for the N -body Hamiltonian (II.1.6) since the magnetic Laplacian is of dominant order in it.

N inherits the properties of L h,b in Corollary II.2.7 and it is well known that sppN q " N Notation II.2.8: Landau levels

We define the n th Landau level as the eigenspace associated to n P N:

nLL -tψ P Dom pL h,b q such that N ψ " nψu
The ground level, denoted LLL for Lowest Landau Level has energy E 0 " hb.

It is well known that the Landau levels are isomorphic, and that the operator a : { ? n `1 is a unitary mapping from nLL to (n+1)LL of inverse a{ ? n `1. Using the creation operator, if we find a basis of the lowest Landau level we will be able to generate a basis of any Landau level. This is why, in the next session we start with a study of the lowest level.

II.2.3 Lowest Landau level

We start with the following characterisation: Proposition II.2.9: Lowest Landau level Denote by OpΩq the set of holomorphic functions, then

LLL Ă kerpaq Ă OpΩqe ´ϕ l 2 b
where ϕ is defined in (II.2.2).

Proof :

Take ψ P LLL, then aψ " 0, using (II.2.6) pπ y ´iπ x qψ " pihB y `hB x `bA y ´ibA x qψ " p2hB z ´ibAqψ " 2phB z `bB z ϕqψ " 0

So Df P OpΩq such that ψ " f e ´ϕ l 2
b . This proves that the zeros of a wave-function of LLL are given by the zeros of a holomorphic function. Since zeros of a holomorphic function must be isolated, the compactness of the domain implies that wave-functions have a finite number of zeros. Actually, the next proposition says that this number of zeros is d defined in (II.1.9), and therefore independent of the choice of wave-function. One can see [51, b P LLL and n 0 be the number of zeros of ψ which can be computed with the logarithmic derivative through

n 0 " 1 2iπ ż BΩ B z lnpf qdz " 1 2iπ ż BΩ B z f f dz (II.2.23)
With Definitions (II.2.2) and (II.1.3)

A ´Tz 0 A " ˆ´B y ϕ `Tz 0 B y ϕ B x ϕ ´Tz 0 B x ϕ ˙" l 2 b ∇φ z 0 (II.2.24) If τ z 0 ψ " ψ, the boundary condition on f is f " e ϕ l 2 b ψ " e ϕ l 2 b `iφz 0 T z 0 ψ " e ϕ´Tz 0 ϕ l 2 b `iφz 0 T z 0 f (II.2.25)
Using equation (II.2.24), we get

l 2 b B z φ z 0 " l 2 b B x φ z 0 ´iB y φ z 0 2 " ´iB x ´By 2 ϕ `Tz 0 iB x `By 2 ϕ " ´iB z ϕ `iT z 0 B z ϕ (II.2.26)
With equations (II.2.25) and (II.2.26), the boundary condition on B z f takes the form

l 2 b B z f " e ϕ´Tz 0 ϕ l 2 b `iφz 0 `Tz 0 l 2 b B z f `2 pB z ϕ ´Tz 0 B z ϕq T z 0 f Ȃs for l 2 b B z f f , we have l 2 b B z f f " T z 0 l 2 b B z f f `2B z pϕ ´Tz 0 ϕq
Finally, we can compute the integral in (II.2.23):

2iπl 2 b n 0 " l 2 b L ż 0 ˆBz f f ptq `i B z f f pL `itq ´Bz f f pt `iLq ´i B z f f pitq ˙dt " ´L ż 0 2B z pϕ ´TiL ϕqpt `iLqdt `i L ż 0 2B z pϕ ´TL ϕqpL `itqdt " L ż 0 `2B z ϕptq ´2B z ϕpt `iLq `2iB z ϕpL `itq ´2iB z ϕpitq ˘dt " ż BΩ 2B z ϕdz " i ż BΩ Apzqdz (II.2.27)
where the last equality comes from Equation (II.2.6) which implies 2B z φ " iA. But the integral of A over a complex loop can be related to the integral of A over a loop in R 2 . Let γγ x `iγ y : r0, 1s Ñ BΩ be a parametrization of BΩ:

ż BΩ Apzqdz " 1 ż 0 Apγpuqqγ 1 puqdu " 1 ż 0 Apγpuqq ¨γ1 puqdu `i 1 ż 0 `Ax pγpuqqγ 1 y puq ´Ay pγpuqqγ 1 x puq ˘du " ż BΩ A ¨dl `i 1 ż 0 `Ax pγpuqqγ 1 y puq ´Ay pγpuqqγ 1 x puq ˘du " ż BΩ A ¨dl `i ż BΩ A K ¨dl " ż BΩ A ¨dl `i ż Ω ∇ ¨A " ż BΩ A ¨dl
Combining this with (II.2.27) and (II.2.11) we get

2πn 0 " l 2 b L 2
so with (II.1.9), we conclude that n 0 " d.

An elliptic function can be expressed as a rational function in terms of the Weierstrass elliptic function and its derivative. In the case of magnetic periodic boundary conditions we will see that we have a decomposition in terms of theta functions from which we construct our basis of LLL. A similar proof of the following proposition can be found in [START_REF] Almog | Abrikosov Lattices in Finite Domains[END_REF] or [ Since all Landau levels are isomorphic, a proof for the lowest Landau level is sufficient. The Landau level dimension is independent of the gauge, for simplicity we use the Landau gauge in this proof. In Landau gauge (II.2.4), the boundary condition on f in Equation (II.2.25) becomes

f pzq " e y 2 ´py´y 0 q 2 2l 2 b ´i y 0 x l 2 b f pz ´z0 q " e ´y2 0 2l 2 b ´i y 0 z l 2
b f pz ´z0 q using equation (II.1.9), this can be rewritten as

f pz ´Lq " f pzq f pz ´iLq " f pzqe L 2 2l 2 b `i Lz l 2 b " f pzqe πd`2iπd z L (II.2.28)
The periodicity along the real axis allows us to expand in Fourier series:

f pzq " ÿ kPZ c k pyqe 2iπk x L (II.2.29)
The holomorphy of f implies that

2B z f " 0 " ÿ k ˆ2iπk L c k pyq `ic 1 k pyq ˙e2iπk x L
Solving the EDO in y that we obtain by identifying the Fourier coefficients gives

c k pyq " c k p0qe ´2πk y L Plugging this in (II.2.29) leads to f pzq " ÿ k c k p0qe 2iπk z L (II.2.30)
Finally we impose the pseudo-periodicity along the imaginary axis (II.2.28):

ÿ k c k p0qe 2iπk z L `2πk " ÿ k c k p0qe πd`2iπ z L pk`dq
Identifying the Fourier coefficients c k e 2πk " c k´d e πd (II.2.31) implies that they are only d independent Fourier coefficients.

We will prove in the next Section that the above relation between Fourier coefficients gives a decomposition of f in terms of theta functions.

II.2.4 Magnetic periodic eigenfunctions

This Section contains computations of the eigenfunctions of L h,b with magnetic periodic boundary conditions.

Notation II.2.12: Theta functions Let τ be a complex parameter in the upper half plane, we define

θpz, τ q - ÿ kPZ e iπτ k 2 `2iπkz
Theta functions are pseudo-periodic: θpz `1, τ q " θpz, τ q θpz `τ, τ q " θpz, τ qe ´iπpτ `2zq

We complete the computation of Proposition II.2.11 and express the wave-functions of the magnetic Laplacian (II.1.1) in term of theta functions.

Proposition II.2.13: LLL wave-functions

The following family, indexed by l P rr0, d ´1ss, is an orthonormal basis of the lowest Landau level in Landau gauge:

ψ 0l pzq - π ´1
e 2iπl z L ÿ kPZ c l`kd e 2iπkd z L " d´1 ÿ l"0 c l e 2iπl z L ÿ kPZ e ´2πkl´πdk 2 `2iπkd z L " d´1 ÿ l"0 c l e 2iπl z L ÿ kPZ e iπpidqk 2 `2iπkpd z L `ilq " d´1 ÿ l"0 c l e 2iπl z L θ ´d z L `il, id
We found a family of the lowest Landau level indexed by l P rr0, d ´1ss with expression in Landau gauge. We need to normalise this family and to verify that the wave-functions are orthogonal. We start by proving that

e ´y2 2l 2 b `2iπl z L θ ´d z L `il, id ¯" e πl 2 d `2iπl x L ÿ kPZ e 2iπkd x L ´1 2l 2 b py`kL`l L d q 2
Using (II.1.9),

e ´y2 2l 2 b `2iπl z L θ ´d z L `il, id ¯" e ´y2 2l 2 b `2iπl z L ÿ kPZ e ´πdk 2 ´2πkl`2iπkd z L " e 2iπl x L ÿ kPZ e ´y2 2l 2 b ´2πl y L ´πdk 2 ´2πkl´2πkd y L `2iπkd x L " e 2iπl x L ÿ kPZ e 2iπkd x L ´1 2l 2 b ´y2 `2ly L d `L2 k 2 `2kl L 2 d `2kLy " e πl 2 d `2iπl x L ÿ kPZ e 2iπkd x L ´1 2l 2 b py`kL`l L d q 2
Finally we check the orthonormality. Let 0 ď l ď q ă d,

xψ 0l |ψ 0q y " 1 ? πLl b ÿ k,pPZ ż Ω e 2iπpq´lq x L `2iπdpp´kq x L ´1 2l 2 b py`kL`l L d q 2 ´1 2l 2 b py`pL`q L d q 2 dxdy " 1 ? πLl b ÿ k,pPZ L ż 0 e 2iπpq´l`drp´ksq x L dx L ż 0 e ´1 2l 2 b py`kL`l L d q 2 ´1 2l 2 b py`pL`q L d q 2 dy
Since 0 ď q ´l ă d we have a simplification:

L ż 0 e 2iπpq´l`drp´ksq x L dx " Lδ lq δ kp Therefore xψ 0l |ψ 0q y " δ lq 1 ? πl b ÿ kPZ L ż 0 e ´1 l 2 b py`kL`l L d q 2
dy " δ lq One can check that the above wave-functions satisfy the boundary conditions (II.2.5). Using (II.2.32) we observe the L-periodicity along the real axis. Along the imaginary axis we increment the index k by 1: We recall some basis properties of Hermite polynomials that will be useful: H 0 " 1 and for all n P N we have the relations

ψ 0l pz `iLq " π ´1 4 ? Ll b e 2iπl x L ÿ kPZ e 2iπkd x L ´1 2l 2 b py`pk`1qL`l L d q 2 " e ´2iπd x
H n`1 " 2xH n ´H1 n (II.2.37)
H n p´xq " p´1q n H n pxq (II.2.38)

H 1 n " 2nH n´1 (II.2.39)
Using this, we can give expressions for the full basis.

Proposition II.2.16: nLL wave-functions

The following family indexed by pn, lq P N ˆrr0, d ´1ss is a Hilbert basis of eigenfunctions of L h,b in Landau gauge:

ψ nl - a : n ? n! ´τ´i L d ¯l ψ 00 " c n ? Ll b e 2iπl x L ÿ kPZ H n ˆ1 l b " y `kL `l L d ȷ˙e 2iπkd x L ´1 2l 2 b py`kL`l L d q 2 (II.2.40)
with the normalization factor

c n - 1 π 1 4 ? n! ˆ´i ? 2 ˙n
Proof :

Due to (II.2.17) the order of application of the magnetic translations and the creation operators does not matter. Also, due to Corollary II.2.14, it is enough to deal with l " 0. In order to lighten the computations we define the dimensionless variable

y k " y `kL l b
We proceed by induction in n. The initialisation is given by H 0 " 1 and (II.2.32):

ψ 00 pzq " c 0 ? Ll b ÿ kPZ H 0 py k qe 2iπkd x L ´y2 k 2
In complex notation a : (II.2.15) becomes in Landau gauge

a : " ´2hB z ´iby ? 2hb " ´?2l b B z ´iy ? 2l b " ´i ? 2 ˆ´l b piB x `By q `y l b ṡo a : n`1 a pn `1q! ψ 00 pzq " a : ? n `1ψ n,0 " c n`1 ? Ll b ÿ kPZ ˆ´l b piB x `By q `y l b ˙Hn py k qe 2iπkd x L ´y2 k 2 " c n`1 ? Ll b ÿ kPZ ˆ"2πk dl b L `yk `y l b ȷ H n py k q ´H1 n py k q ˙e2iπkd x L ´y2 k 2 " c n`1 ? Ll b ÿ kPZ r2y k H n py k q ´H1 n py k qs e 2iπkd x L ´y2 k 2 " c n`1 ? Ll b ÿ kPZ H n`1 py k qe 2iπkd x L ´y2 k 2
where the last equality uses (II.2.37).

As expected with our boundary conditions the modulus of the wave-functions:

|ψ nl | " ˇˇˇˇcn ? Ll b ÿ kPZ H n ˆ1 l b " y `kL `l L d ȷ˙e 2iπkd x L ´1 2l 2 b py`kL`l L d q 2 ˇˇˇˇ( II.2.41)
is periodic on the lattice LZ 2 , but the periodicity along the real axis is even shorter. Indeed we see in (II.2.41) that |ψ nl | is L{d-periodic in x.

We can write another useful form of equation (II.2.40) using the Poisson summation formula. The advantage of the expression in Proposition II.2.18 is the fact that the index l is decoupled from the polynomials and the Gaussian factors which is not the case in (II.2.40). This will simplify the computation of the Landau level's projector when we will sum over l in (II.

ψ nl pzq " r c n ? l b L 3 2 e ´i xy l 2 b ÿ kPZ H n ˆ1 l b " x `k L d ȷ˙e ´2iπkp y L `l d q´1 2l 2 b px`k L d q 2
with the normalization factor

r c n - π 1 4 p´1q n 2 1´n 2 ? n! Proof :
We start from (II.2.40) expressed in terms of h n :

ψ nl pzq " c n ? Ll b e 2iπl x L ÿ kPZ h n ˆ1 l b " y `kL `l L d ȷ˙e 2iπkd x L Define gpuq -h n ˆ1 l b " y `uL `l L d ȷ˙e 2iπdu
x L so we have

ψ nl pzq " c n ? Ll b e 2iπl x L ÿ kPZ gpkq (II.2.46)
in order to apply the Poisson summation formula to g. To do so, we compute ĝ with a change of variable and equations (II.1.9) and (II.2.44):

ĝpνq " 1 ? 2π ż R h n ˆ1 l b " y `uL `l L d ȷ˙e ´iupν´2πd x L q du " l b L ? 2π e ip y L `l d qpν´2πd x L q ż R h n puqe ´i ul b L pν´2πd x L q du " l b L ? 2π e ip y L `l d qpν´2πd x L q x h n ˆlb L ν ´x l b ˙" p´iq n l b L e ip y L `l d qpν´2πd x L q h n ˆlb L ν ´x l b
ṡo by using (II.1.9) again:

ĝp2πkq " p´iq n l b L e ip y L `l d qp2πk´2πd x L q h n ˆ2πk l b L ´x l b " p´iq n l b L e ´i xy l 2 b ´2iπl x L e 2iπkp y L `l d q h n ˆ1 l b " k L d ´xȷṪ
o conclude the computation we insert this after applying the Poisson summation formula (II.2.43) to (II.2.46):

ψ nl pzq " c n ? Ll b e 2iπl x L ? 2π ÿ kPZ ĝ p2πkq " c n ? Ll b ¨?2πp´iq n l b L e ´i xy l 2 b ÿ kPZ H n ˆ1 l b " k L d ´xȷ˙e 2iπkp y L `l d q´1 2l 2 b pk L d ´xq 2 " r c n ? l b L 3 2 e ´i xy l 2 b ÿ kPZ H n ˆ1 l b " x `k L d ȷ˙e ´2iπkp y L `l d q´1 2l 2 b px`k L d q 2
by changing the sum index k to ´k, using the parity of Hermite polynomials and the relation

c n ? 2πp´iq n " r c n

II.3 Projectors on Landau levels

From the construction of an L 2 pΩq basis adapted to Landau levels, we define the projectors on Landau levels in Notation II.3.1. Since the phase space is N ˆΩ we also want to localise the projectors in space. Then we prove some properties of the projector that will be needed for the semi-classical analysis. In Proposition II.3.4 we give an equivalent for the diagonal of the projector's integral kernel, and in Corollary II.3.6 an equivalent for its trace.

II.3.1 nLL projectors

Notation II.3.1: Projectors

The orthogonal projector on nLL is

Π n - d´1 ÿ l"0 |ψ nl ⟩ ⟨ψ nl |
Let g P C 8 pR 2 , R `q radial with support included in the ball Bp0, L{2q such that ∥g∥ L 2 " 1.

Let λ ě 1, define the localizer g λ P C 8 pTq defined by

g λ pxq - # λgpλxq if x P B `0, L 2λ 0 else Note that ∥g λ ∥ L 2 " 1
@Xpn, Rq P N ˆΩ, define the localised projector

Π X -Π n,R -g λ p' ´RqΠ n g λ p' ´Rq (II.3.1)
We assume the following scaling for λpλ N q N :

1 ! λ ! N ´1 2 h2 (II.3.2)
This localised projector was introduced by Lieb, Solovej and Yngvason in [START_REF] Lieb | Quantum Coherent Operators: A Generalisation of Coherent States[END_REF] and [START_REF] Yngvason | Thomas-Fermi Theory for Matter in a Magnetic Field as a Limit of Quantum Mechanics[END_REF] where it has been called coherent operator. We take the bounds (II.3.2) in order to have g 2 λ á δ so the projector is well localised and

h2 l b λ " hbλl b ! 1
This is necessary because hbλl b is the order of some error terms coming from the kinetic energy (for example in Proposition II.6.7).

The projectors satisfy the following resolutions of identity:

Property II.3.2 @Xpn, Rq P N ˆΩ, Π n and Π X are positive and

ÿ nPN Π n " Id, ż NˆΩ Π X dηpXq " Id (II.3.3)
Proof :

Let ψ P L 2 pΩq
xψ|Π n,R ψy " xgp' ´Rqψ|Π n gp' ´Rqψy ě 0 because Π n is a projector. The first resolution of the identity is a consequence of the completeness of the basis and implies the second one:

ż NˆΩ Π X dηpXq " ż Ω g λ p¨´Rq ˜ÿ nPN Π n ¸gλ p¨´RqdR " ż Ω g λ p¨´Rq 2 dR " Id

II.3.2 Integral kernels of the projectors

The computations of Section II.2 lead to the following expression of the nLL-projector's kernel.

Proposition II.3.3: Kernel of the nLL-projector

With notation (II.2.42) and xx 1 `ix 2 , y " y 1 `iy 2 ,

Π n px, yq " 1 ∥h n ∥ 2 L 2 Ll b e i y 1 y 2 ´x1 x 2 l 2 b ÿ k,qPZ H n ˆ1 l b " x 1 `k L d ȷ˙H n ˆ1 l b " y 1 `qL `k L d ȷė 2iπk y 2 ´x2 L `2iπdq y 2 L ´1 2l 2 b px1`k L d q 2 ´1 2l 2 b py1`qL`k L d q 2 (II.3.4) Proof :
From Proposition II.2.18:

Π n px, yq "r c 2 n l b L 3 e i y 1 y 2 ´x1 x 2 l 2 b d´1 ÿ l"0 ÿ k,pPZ H n ˆ1 l b " x 1 `k L d ȷ˙H n ˆ1 l b " y 1 `p L d ȷė 2iπl p´k d `2iπ py 2 ´kx 2 L ´1 2l 2 b px1`k L d q 2 ´1 2l 2 b py1`p L d q 2
Then, we use to conclude. The computation of the normalization factor can be performed with (II.2.45):

r c 2 n l b L 3 ¨d " r c 2 n 2π ¨1 Ll b " 1 ? π2 n n! ¨1 Ll b " 1 ∥h n ∥ 2 L 2 Ll b
The above simplification for the sum in l is the reason why we used the Poisson formula on wave-functions. The argument does not work on the expression in (II.2.40) since the Gaussian terms depend on l.

If we consider the same setup on the whole space R 2 instead of Ω, the expression of the projector in Landau gauge becomes (see [START_REF] Jain | Composite fermions[END_REF]Section 3.2]):

Π 8 0 px, yq - 1 2πl 2 b e ´|x´y| 2l 2 b `i Imrxys 4l 2 b `i y 1 y 2 ´x2 x 1 2l 2 b
The next proposition states that the diagonal of the projector's kernel on Ω converges to that of the projector on the whole space. This is expected since the limit is equivalent to a scaling where the size of the domain goes to infinity. This result will be important to estimate the trace of Π n,R .

Proposition II. Moreover with notation (II.2.42),

pP h,b Π n qpz, zq ´b l b ¨1 2π ∥h n ∥ 2 L 2 ż R ˆih 1 n puq uh n puq ˙hn puqe ´u2 du L 8 ď Cpnqb (II.3.7)
The proof needs the following technical lemma.

Lemma II.3.5

Let m P N, c ą 0, the following series are uniformly bounded in α, a, b:

@α P R `, a
P n ˆ1 l b " x `k L d ȷ˙Q n ˆ1 l b " x `qL `k L d ȷ˙e 2iπqd y L ´1 2l 2 b px`k L d q 2 ´1 2l 2 b px`qL`k L d q 2 is of order 1 l b
and can be uniformly approximated as

Ξpzq ´L 2πl b ż R P n puqQ n puqe ´u2 du L 8
ď Cpnq (II.3.10)

Proof of Proposition II.3.4:

We start from (II.3.4):

Π n pz, zq " 1 ∥h n ∥ 2 L 2 Ll b ÿ k,qPZ h n ˆ1 l b " x `k L d ȷ˙h n ˆ1 l b " x `qL `k L d ȷ˙e 2iπqd y L
We apply Lemma II.3.5 and thus compute

1 ∥h n ∥ 2 L 2 Ll b ż R h n ˆx l b `2πu l b L ˙2 du " 1 Ll b ¨L 2πl b " 1 2πl 2 b
and obtain (II.3.6). Starting again from (II.3.4) and using notation (II.3.11), we compute in Landau gauge

pP h,b Π n qpx, yq " ˆihB x 1 ´bx 2 ihB x 2 ˙1 ∥h n ∥ 2 L 2 Ll b e i y 1 y 2 ´x1 x 2 l 2 b ÿ k,qPZ h n pk b,x 1 q h n pk b,y 1 `qL q ¨e2iπk y 2 ´x2 L `2iπdq y 2 L " 1 ∥h n ∥ 2 L 2 Ll b e i y 1 y 2 ´x1 x 2 l 2 b ÿ k,qPZ h l b ˆih 1 n pk b,x 1 q k b,x 1 h n pk b,x 1 q ˙hn pk b,y 1 `qL q ¨e2iπk y 2 ´x2 L `2iπdq y 2 L So pP h,b Π n qpz, zq " b ∥h n ∥ 2 L 2 L ÿ k,qPZ
ˆih 1 n pk b,x q k b,x h n pk b,x q ˙hn pk b,x`qL q e 2iπdq y L and with Lemma II.3.5,

pP h,b Π n qpz, zq ´L 2πl b ¨b ∥h n ∥ 2 L 2 L ż R ˆih 1 n puq uh n puq ˙hn puqe ´u2 du L 8 ď Cpnqb
Finally, we compare the trace of Π n,R to the trace of the projector on the whole space.

Corollary II.3.6: Approximation of the projector's trace

ˇˇˇT r rΠ n,R s ´1 2πl 2 b ˇˇˇď Cpnq l b Proof :
This is a direct consequence of Proposition II.3.4 after integrating on z P Ω:

Tr rΠ n,R s " ż Ω Π n,R pz, zqdz " 1 2πl 2 b ż Ω g λ pz ´Rq 2 dz `O ˆ1 l b ˙" 1 2πl 2 b `O ˆ1 l b
Ẇe end this section with the proof of the technical Lemma.

Proof of Lemma II.3.5:

Step 1: Proof of (II.3.8) and (II.3.9)

Let α P R `, a, b P r´1, 1s. If q ě 2 then q ď 2pq ´1q so @u P rq ´1, qq, |a `b `αq| m e ´cpa`αqq 2 ď p2 `2αuq m e ´cpa`αuq For (II.3.9), let α P r0, 1s, a P R, b P r´1, 1s. We see that the series is α-periodic in a so we can assume 0 ď a ď 1 and use (II.3.8) and for k " 0:

α |a `b| m e ´ca 2 ď 2 m
Step 2: Approximation of Ξ Due to the Gaussian factor in Ξ, all terms for which q ‰ 0 have a fair chance to vanish when l b Ñ 0. Thus, we focus first on the term indexed by q " 0. To simplify notation we introduce

u b,x - 1 l b ˆx `uL d ˙" x l b `2πu l b L (II.3.11)
ξpuq -P n pu b,x q Q n pu b,x q e ´ub,x 2 Ξ |q‰0 pzq -Ξpzq ´ÿ kPZ ξpkq so ÿ kPZ ξpkq "

ÿ kPZ P n ˆ1 l b " x `k L d ȷ˙Q n ˆ1 l b " x `k L d ȷ˙e ´1 l 2 b px`k L d q 2
is the term for q " 0 and Ξ |q‰0 pzq contains the other terms. Note that Ξ is L{d-periodic in x so we can choose x P r0, L{ds and

x l b ď 2π l b L Ñ N Ñ8 0 (II.3.12)
For q " 0, if we replace the sum in k by the associated integral we obtain:

ż R ξpuqdu " L 2πl b ż R P n puqQ n puqe ´u2 du
which is the approximation in (II.3.10). For the convergence of the Riemann sum, we compute the derivative of the integrand. There exists R n a polynomial of degree 2n `1 such that

ξ 1 puq " 2π l b L R n pu b,x q e ´ub,x 2
Now, use the mean value theorem:

ˇˇˇˇˇÿ kPZ ξpkq ´ż R ξpuqdu ˇˇˇˇˇď ÿ kPZ k`1 ż k |ξpkq ´ξpuq| du ď 2π l b L ÿ kPZ sup kďuďk`1
|R n pu b,x q| e ´ub,x 2 (II. 3.13) To control this we only need to control monomials.

If k ď u ď k `1, |u b,x | m e ´ub,x 2 ď |k b,x | m e ´kb,x 2 `ˇˇp k `1q b,x ˇˇm e ´pk`1q b,x 2 `|k b,x | m e ´pk`1q b,x 2 `ˇˇp k `1q b,x ˇˇm e ´kb,x 2 " |k b,x | m e ´kb,x 2 `ˇˇp k `1q b,x ˇˇm e ´pk`1q b,x 2 `ˇˇp k `1q b,x´L d ˇˇm e ´pk`1q b,x 2 `ˇˇk b,x`L d ˇˇm e ´kb,x 2
Thus after some change of indices,

2π l b L ÿ kPZ sup kďuďk`1 |u b,x | m e ´ub,x 2 ď 2π l b L ÿ kPZ ´2 |k b,x | m `ˇˇk b,x´L d ˇˇm `ˇˇk b,x`L d ˇˇm ¯e´k b,x 2
Using (II.3.9) with α " 2π

l b L Ñ 0, a " x l b , b P " 0, 2π l b L , ´2π l b L * , c " 1: ˇˇˇˇˇÿ kPZ ξpkq ´ż R ξpuqdu ˇˇˇˇˇď Cpnq
Step 3: Control of Ξ |q‰0

Let ϵ ą 0, with Young's inequality:

´ˆx l b `2πk l b L ˙¨q L l b ď ϵ 2 ˆx l b `2πk l b L ˙2 `1 2ϵ ˆq L l b ˙2 so e ´1 2 ´x l b `2πk l b L ¯2´1 2 ´x l b `2πk l b L `q L l b ¯2 ď e ´p1´ϵ 2 q ´x l b `2πk l b L ¯2´p 1 2 ´1 2ϵ q ´q L l b

¯2

We take ϵ " 3{2. As in (II.3.13), we need to deal with monomial terms of the form

ÿ q‰0,k ˇˇˇx l b `2πk l b L ˇˇˇm ˇˇˇq L l b ˇˇˇr m e ´1 4 ´x l b `2πk l b L ¯2´1 6 ´q L l b
¯2 by using

• (II.3.8) for the sum in q with α " L l b , a " 0, b " 0, c " 1 6

• (II.3.9) for the sum in k with α " 2π

l b L Ñ 0, a " x l b Ñ 0, b " 0, c " 1 4
We conclude that

ˇˇΞ |q‰0 ˇˇď Cpnq L l b ¨lb L " Cpnq

II.4 A Lieb-Thirring inequality

In this section we prove a Lieb-Thirring inequality for the magnetic Laplacian with magnetic periodic boundary conditions:

Theorem II.4.1: Kinetic energy inequality Let γ P L 1 pL 2 pΩqq a positive operator, then for large enough b,

ż Ω ρ 2 γ ď C ∥γ∥ L 8 h2 Tr rL h,b γs (II.4.1) Moreover if ψ N P L 2 ´pΩ N q with ∥ψ N ∥ L 2 " 1, ρ p1q ψ N 2 L 2 ď C hb Tr " L h,b γ p1q ψ N ı and ˇˇˇˇˇż Ω V ρ p1q ψ N ˇˇˇˇˇď C hb Tr " L h,b γ p1q ψ N ı ∥V ∥ L 2 (II.4.2) ż Ω 2 wρ p2q ψ N ď C hb Tr " L h,b γ p1q ψ N ı ∥w∥ L 2 (II.4.3)
This result is well known in the absence of magnetic field, see [START_REF] Lieb | A positive density analogue of the Lieb-Thirring inequality[END_REF]Theorem 5.2]. Here, we adapt the proof of [START_REF] Lieb | The stability of matter in quantum mechanics[END_REF]Chapter 4] to magnetic periodic boundary conditions. To achieve this we prove the following sequence of inequalities: a Kato inequality (Lemma II.4.2), a diamagnetic inequality for Green functions (Proposition II.4.5), a Lieb-Thirring inequality (Theorem II.4.6) from which we deduce the inequality on the kinetic energy (Theorem II.4.1). The reader already familiar with Lieb-Thirring inequalities might jump to Section II.5.

II.4.1 Reduced densities

We give some usual properties of the reduced density matrices, see Notation II.1.6. Let γ N be an N -body density matrix, since the Hamiltonian only contains one-body and two-body terms, the quantum N -body energy in the state γ N only depends on the two first reduced densities:

Tr rH N γ N s N " Tr " pL h,b `V q γ p1q N ı `Tr " wγ p2q N ı (II.4.4)
moreover, reduced densities inherit trace and Pauli principle from γ N :

Tr

" γ pkq N ı " 1, 0 ď γ pkq N ď k! pN ´kq! N ! (II.4.5)
We can also express the reduced density matrices in term of integral kernels:

γ pkq N px 1:k , y 1:k q - ż Ω N ´k γ N px 1:k , x k`1:N ; y 1:k , x k`1:N qdx k`1:N (II.4.6)
The reduced density matrices are symmetric under permutation of coordinates: @σ P S k , γ pkq N `xσp1:kq , y σp1:kq ˘" γ pkq N px 1:k , y 1:k q and consistent: @q P 1 : k , γ pqq N px 1:q , y 1:q q " ż Ω k´q γ N px 1:q , x q`1:k ; y 1:q , x q`1:k qdx q`1:k Note that the reduced densities ρ pkq γ N inherit the symmetry and the consistency from the reduced density matrices.

II.4.2 A Kato inequality with periodic boundary conditions

One can look at [6, Theorem X.27] for a proof of the Kato inequality in the non magnetic case.

Lemma II.4.2: Periodic Kato inequality Define the complex sign spuq - So ∇u ϵ is bounded in L 2 pΩ, R 2 q and converges weakly to v P L 2 pΩ, R 2 q up to sequence of ϵ.

$ & % u |u| if u ‰ 0 0 if u " 0 Let u P C 8 pΩq
Let φ P C 8 c pintpΩq, R 2 q, since φ P L 2 pΩ, R 2 q and 0 ď u ϵ ´|u| ď ϵ ż

Ω v ¨φ " lim ż Ω ∇u ϵ ¨φ " ´lim ż Ω u ϵ ∇ ¨φ "
´ż Ω |u| ∇ ¨φ so v " ∇ |u| and the bound (II.4.10) passes to the limit and we obtain (II.4.7).

Step 2: Proof of (II.4.8)

We use polar coordinates:

u -|u| e iθ Let x P Ω, if |u| pxq ‰ 0, |u| is smooth on a neighbourhood V x of

II.4.3 Diamagnetic inequality

The main lemma for the Lieb-Thirring inequality in the magnetic case is the diamagnetic inequality in term of Green functions because it allows to restrict ourselves to the non magnetic case.

Notation II.4.3: Green functions

The resolvents of ´h 2 ∆ with periodic boundary conditions and L h,b are well defined for λ ą 0:

G bA,λ -pL h,b `λq ´1 G λ -p´h 2 ∆ `λq ´1
Their integral kernels define the corresponding Green functions.

They have the following properties:

Property II.4.4

Let x P Ω, then G bA,λ px, 'q, G λ px, 'q P L 2 pΩq and

G λ px, yq " G λ px ´yq " G λ py ´xq " 1 L 2 ÿ kP 2πh L Z 2 1 k 2 `λe ik¨px´yq ě 0 Proof :
The periodic Laplacian is diagonalizable in the plane wave basis indexed by k P 2π L Z 2 :

e k pxq - 1 L e ik¨x Indeed ´h 2 ∆ `λ " ÿ kP 2π L Z 2 `h 2 k 2 `λ˘| e k ⟩ ⟨e k | so G λ px, yq " 1 L 2 ÿ kP 2π L Z 2 1 h2 k 2 `λe ik¨px´yq A change of index k -´k gives G λ px, yq P R. Let f P L 2 pΩq, since G λ f solves p´h 2 ∆ `λqu " f, u P H 2 p pΩq
by the Lax-Milgram theorem, G λ f is the unique minimizer of the following functional

J puq - ż Ω `h 2 |∇u| 2 `λ |u| 2 ´f u ˘dx
Assuming f ě 0, we see that J puq ě J p|u|q and conclude that G λ f ě 0. This implies G λ px, yq ě 0. Finally,

´h 2 ∆ `λ ě λ and L h,b `λ ě λ so ∥G λ ∥ L 8 ď 1 λ and ∥G bA,λ ∥ L 8 ď 1 λ
and G bA,λ px, 'q, G λ p', yq P L 2 pΩq.

Now we prove a diamagnetic inequality:

Proposition II.

II.4.4 Lieb-Thirring inequality

We would like to prove a kinetic inequality of the form

Tr rγL h,b s ě C ∥γ∥ L 8 ż Ω ρ 2 γ
with γ P L 1 pL 2 pΩqq a positive operator. We will deal with the magnetic field with the diamagnetic inequality and use Lieb-Thirring inequality for the Laplacian. But the previous inequality for the Laplacian is false if we take γ -|e 0 ⟩ ⟨e 0 |,

Tr r´∆γs " 0 ă C L 2 " C ∥γ∥ L 8 ż Ω ρ 2 γ
To avoid this, we add 1 to the Laplacian so that the constant mode has a non-zero energy.

Theorem II.4.6: Lieb-Thirring Inequality

Let V P L 2 pΩ, R `q, ´Tr " 1 pLh,b`1´Vqď0 pL h,b `1 ´Vq ı ď C LT h2 ż Ω Vpxq 2 dx (II.4.20)
Proof :

Step λ px, yq So taking m -3{2 and using an inequality on the traces of powers (see [13, Theorem 4.5]), Step 2: Estimate of G m λ p0q

N λ " B λ ď Tr rK m λ s ď Tr " V m 2 K m λ V m 2 ‰ ď ż Ω Vpxq m ˇˇG m A,λ`1 px, xq ˇˇdx ď ż Ω Vpxq m G m λ`1 px, xqdx
The kernel of G m λ is

G m λ pxq " 1 L 2 ÿ kP 2πh L Z 2 1 pk 2 `λq m e i k¨x h
We use the integral bound for the sum

ÿ kPZ 1 pk 2 `λq m ď λ ´m `ż R 1 pu 2 `λq m du so G m λ p0q " 1 L 2 ÿ k,qPZ 1 ´`2πh L ˘2 pk 2 `q2 q `λ¯m " 1 L 2 ˆL 2πh ˙2m ÿ k,qPZ 1 ´k2 `q2 ``L 2πh ˘2 λ ¯m ď 1 L 2 ˆL 2πh ˙2m ÿ kPZ ¨1 ´k2 ``L 2πh ˘2 λ ¯m `ż R 1 ´k2 `u2 ``L 2πh ˘2 λ ¯m du ' We estimate the integral ż R 1 ´k2 `u2 ``L 2πh ˘2 λ ¯m du " ˜k2 `ˆL 2πh ˙2 λ ¸´m ż R 1 ˆu2 k 2 `p L 2πh q 2 λ `1˙m du " Ipmq ´k2 ``L 2πh ˘2 λ ¯m´1 2 with m ą 1 2 ùñ Ipmq - ż R 1 p1 `u2 q m du ă 8 Similarly G m λ p0q ď 1 L 2 ˆL 2πh ˙2m ÿ kPZ ¨1 ´k2 ``L 2πh ˘2 λ ¯m `Ipmq ´k2 ``L 2πh ˘2 λ ¯m´1 2 ‹ ' ď λ ´m L 2 `Ipmq 2πhL λ ´m`1 2 `1 L 2 ˆL 2πh ˙2m ¨ż R 1 ´u2 ``L 2πh ˘2 λ ¯m du `ż R Ipmq ´u2 ``L 2πh ˘2 λ ¯m´1 2 du ‹ ' ď λ ´m L 2 `Ipmq πhL λ ´m`1 2 `1 p2πhq 2 IpmqI ˆm ´1 2 ˙λ´m`1 ď Cpmq h2 λ ´m`1
since λ ě 1. We need m ą 1 for the integrals to converge.

Step 3: Conclusion

We use the same trick as [START_REF] Lieb | The stability of matter in quantum mechanics[END_REF] changing the potential to

W λ pxq -max ˆV ´λ 2 , 0
Ċombining this with (II.4.21) and the change of variable

µ - 2V pxq λ , dλ " ´2V pxq µ 2 dµ we obtain ´Tr " 1 pLh,b`1´Vqď0 pL h,b `1 ´Vq ı ď Cpmq h2 8 ż 1 ż Ω λ ´m`1 max ˆV pxq ´λ 2 , 0 ˙m dλdx ď Cpmq h2 ż Ω ¨2V pxq ż 0 λ ˆ2V pxq λ ´1˙m dλ 'dx " Cpmq h2 ż Ω ¨8 ż 1 2V pxq µ pµ ´1q m ¨2V pxq µ 2 dµ 'dx " Cpmq h2 ż Ω V pxq 2 ¨8 ż 1 pµ ´1q m µ 3 dµ 'dx
The integral in µ converges if 3 ´m ą 1. To conclude we need 1 ă m ă 2 hence the choice m " 3{2 is convenient.

This leads to proof of the Fundamental inequality of kinetic energy:

Proof of Theorem II.4.1:

Step 1: Inequality for ρ ż

Ω V 2 `ż Ω Vρ γ Then choose V -C N 1 ργ ďc ρ γ : Tr rpL h,b `1q γs ě C N ˆ1 ´CN C LT ∥γ∥ L 8 h2 ˙ż ργ ďc ρ γ 2
The constant preceding the integral is maximal when With this and monotone convergence we take the limit c Ñ 8 in inequality (II.4.22) and obtain (II.4.1). Applying this to (II.4.5), we have

C N - h2 2C LT ∥γ∥ L 8
1 hb Tr " L h,b γ p1q ψ N ı ě C l 2 b γ p1q ψ N L 8 ρ p1q ψ N 2 L 2 ě CN l 2 b ρ p1q ψ N 2 L 2 ě C ρ p1q ψ N 2 L 2
Step 2: Inequality for ρ

p2q ψ N
For the second reduced density, by symmetry

N ´Tr " L h,b γ p1q ψ N ı ´Tr " wγ p2q ψ N ı¯" ψ N ˇˇˇˇ˜N ÿ i"1 L h,b px i q ´2 N ´1 ÿ iăj wpx i ´wj q ¸ψN " ψ N ˇˇˇˇ˜N ÿ i"1 L h,b px i q ´N N ´1 N ÿ j"2 wpx 1 ´wj q ¸ψN ě ψ N ˇˇˇˇN ÿ j"2 ˆLh,b px i q ´N N ´1wpx 1 ´xj q ˙ψN " ż Ω ψ N px, 'q ˇˇˇˇN ÿ j"2
ˆLh,b px i q ´N N ´1wpx ´xj q ˙ψN px, 'q dx

" ż Ω ¨ ψ N px, 'q ˇˇˇˇN ÿ j"2 L h,b px i qψ N px, 'q ´N ż Ω wpx ´yqρ p1q ψ N px,'q dy 'dx
Then using (II.4.2) and then Young's inequality,

Tr " L h,b γ p1q ψ N ı ´Tr " wγ p2q ψ N ı ě 1 N ż Ω ¨ChbpN ´1q ρ p1q ψ N px,'q 2
L 2 ´N ż Ω wpx ´yqρ p1q ψ N px,'q dy 'dx ě ż Ω ¨Chb ρ p1q ψ N px,'q 2
L 2 ´ż Ω wpx ´yqρ p1q ψ N px,'q dy 'dx ě ż Ω ˆChb ρ p1q ψ N px,'q 2
L 2 ´1 2 ˆ1 2Chb ∥w∥ 2 L 2 `2Chb ρ p1q ψ N px,'q 2
L 2 ˙˙dx ě ´C hb ∥w∥ 2 L 2
Changing w to ϵw, dividing by ϵ and using (II.4.4) gives

ż Ω 2 wρ p2q ψ N ď 1 ϵ Tr " L h,b γ p1q ψ N ı `C hb ϵ ∥w∥ 2 L 2
To optimise in ϵ, we choose ϵhb ∥w∥ L 2 , we get

ż Ω 2 wρ p2q ψ N ď ˆ1 hb Tr " L h,b γ p1q ψ N ı `C˙∥ w∥ L 2 ď C hb Tr " L h,b γ p1q ψ N ı ∥w∥ L 2 because L h,b ě hb.
Similarly with Young's inequality and (II.4.2),

ˇˇˇˇˇż Ω V ρ p1q ψ N ˇˇˇˇˇď C hb Tr " L h,b γ p1q ψ N ı ∥V ∥ L 2

II.5 Semi-classical limit

In this section we introduce the Husimi functions representing densities in the phase space. The fundamentals properties of these functions can be found in Property II.5.3. Then we prove that the N -body quantum energy can be approximated by a semi-classical functional depending only on Husimi functions in Proposition II.5.4.

II.5.1 Husimi functions

Notation II.5.1

Let k P N ˚, γ k P L 1 pL 2 pΩ k qq, recalling Notation II.3.1 and (II.1.23) we define the associated Husimi functions or lower symbol as

m γ k pX 1:k q -Tr « γ k k â i"1 Π X i ff with X 1:k P pN ˆΩq k
Conversely, if m k P L 1 `pN ˆΩq k ˘, define the associated density matrix

γ m k -p2πl 2 b q k ż pNˆΩq k m k pX 1:k q k â i"1 Π X i dη bk pX 1:k q
We call m k the upper symbol of γ m k . We also associate a density to m k :

ρ m k - ÿ n 1:k PN k m k pn 1:k ; 'q
we extend the definition (II.1.13) to Husimi functions, if k ě 2:

E sc,hb rm k s - ż NˆΩ E n m p1q k pn, Rqdηpn, Rq `ż NˆΩ V m p1q k dη `ż pNˆΩq 2 wm p2q k dη b2 (II.5.1)
and we also extend (II.1.16) to ρ k P L 1 `Ωk ˘:

E qLL rρ k s " ż Ω V ρ p1q k `ż Ω 2 wρ p2q k (II.5.2)
If one starts from a wave-function ψ N P L 2

´`Ω N ˘we use the notation

m ψ N -m γ ψ N
For another discussion and further references about lower and upper symbols one can look at [START_REF] Rougerie | Scaling limits of bosonic ground states from many-body to nonlinear Schrödinger[END_REF]Definition 3.13].

The k-body Husimi function is the joint probability distribution for k particles in phase space. Similarly as for (II.1.24), we have

m pkq γ N " m γ pkq N and ρ pkq mγ N " ρ m pkq γ N
The next lemma provides a translation between Husimi functions and density matrices.

Lemma II.5.2: Relations between Husimi functions and reduced densities Let γ k P L 1 `L2 `Ωk ˘˘be a positive operator, then m

γ k P L 1 `pN ˆΩq k ˘and 0 ď m γ k ď ∥γ k ∥ L 8 p2πl 2 b q k p1 `Opl b qq ż pNˆΩq k m γ k dη bk " Tr rγ k s
Conversely if m k P L 1 `pN ˆΩq k ˘is positive, then γ m k P L 1 pL 2 pΩ k qq and

0 ď γ m k ď `2πl 2 b ˘k ∥m k ∥ L 8 Tr rγ m k s " ∥m k ∥ L 1 `Opl b q Moreover if γ N P L 1 `L2 ´`Ω k ˘˘and 1 ď k ď N , then m pkq γ N ď pN ´kq! p2πl 2 b q k N ! Tr rγ N s p1 `O pl b qq Proof :
m γ k is positive because @X P N ˆΩ, Π X and γ k are positive. With Corollary II.3.6,

m γ k pX 1:k q ď ∥γ k ∥ L 8 k ź i"1 Tr rΠ X i s " ∥γ k ∥ L 8 ˆ1 2πl 2 b `O ˆ1 l b ˙˙k " ∥γ k ∥ L 8 p2πl 2 b q k p1 `
γ m k ď `2πl 2 b ˘k ∥m k ∥ L 8 ż pNˆΩq k k â i"1 Π X i dη bk pX 1:k q " `2πl 2 b ˘k ∥m k ∥ L 8
Finally, using Corollary II.3.6,

Tr rγ m k s " p2πl 2 b q k ż pNˆΩq k m k pX 1:k qTr « k â i"1 Π X i ff dη bk pX 1:k q " ż pNˆΩq k m k dη bk `Opl b q " ∥m k ∥ L 1 `Opl b q
Π X P L 1 pL 2 pΩqq is positive, thus it can be diagonalized:

Π X " ÿ iPN λ i,X |ψ i,X ⟩ ⟨ψ i,X | with λ i,X ě 0 and ÿ iPN λ i,X " Tr rΠ X s
We have

m γ pkq N pX 1:k q " ÿ i 1:k PN k ˜k ź j"1 λ i j ,X j ¸Tr « γ pkq N ˇˇˇˇk â j"1 ψ i j ,X j k â j"1 ψ i j ,X j ˇˇˇˇff " ÿ i 1:k PN k ˜k ź j"1 λ i j ,X j ¸Tr « γ N ˇˇˇˇk â j"1 ψ i j ,X j k â j"1 ψ i j ,X j ˇˇˇˇb Id L 2 pΩ N ´k q ff (II.5.3)
Let ψ 1:N P L 2 pΩq be an orthonormal family, we claim that

ˇˇˇˇk â i"1 ψ i k â i"1 ψ i ˇˇˇˇb Id L 2 pΩ N ´k q ď pN ´kq! N ! on L 1 `L2
´`Ω N ˘˘(II.5.4)

Indeed, if we consider the Slater state

χ N - 1 ? N ! ÿ σPS N ϵpσq N â j"1 ψ σpjq then χ N ˇˇˇˇ˜ˇˇˇˇk â i"1 ψ i k â i"1 ψ i ˇˇˇˇb Id L 2 pΩ N ´k q ¸χN " 1 N ! ÿ σ,τ PS N ϵpστ q k â i"1 ψ i ˇˇˇˇk â i"1 ψ τ piq N â i"1 ψ σpiq ˇˇˇˇ˜k â i"1 ψ i ¸b N â i"k`1 ψ τ piq " 1 N ! ÿ σ,τ PS N ϵpστ q ˜k ź i"1 δ σpiq,i δ τ piq,i ¸N ź i"k`1 δ σpiq,τ piq " 1 N ! ÿ σPS N k ź i"1 δ σpiq,i " pN ´kq! N !
If the Slater determinant does not contain the ψ 1:k then the result of this computation is 0, thus we obtain (II.5.4). Then with (II.5.3) and Corollary II.3.6,

m γ pkq N pX 1:k q ď pN ´kq! N ! ÿ i 1:k PN k ˜k ź j"1 λ i j ,X j ¸Tr rγ N s " pN ´kq! N ! Tr rγ N s k ź j"1 Tr " Π X j ‰ " pN ´kq! p2πl 2 b q k N ! Tr rγ N s p1 `O pl b qq
We have the following properties for the Husimi functions coming from reduced density matrices.

Property II.5.3: Husimi functions Let γ N be an N -body density matrix, then m pkq γ N are symmetric, consistent and satisfy

0 ď m pkq γ N ď pN ´kq! p2πl 2 b q k N ! `Opl b q (II.5.5) ż pNˆΩq k m pkq γ N dη bk " m pkq γ N L 1 " 1 (II.5.6)
ρ pkq mγ N " pg 2 λ q bk ˚ρpkq γ N (II.5.7)

Proof :

Let k ą q ě 1 and X 1:q P pN ˆΩq q . Recalling the results of Subsection II.4.1, we prove that the N -body Husimi functions are consistent marginals using (II.3.3):

ż pNˆΩq k´q m pkq γ N pX 1:k qdηpX q`1:k q " Tr » - - ż pNˆΩq k´q γ pkq N k â i"1 Π X i dηpX q`1:k q fi ffi fl " Tr « γ pkq N q â i"1 Π X i b Id bpk´qq NˆΩ ff " Tr « γ pqq N q â i"1 Π X i ff " m pqq γ N pX 1:q q (II.5.8)
Let σ P S k , the symmetry of the Husimi measures follows from the symmetry of the reduced density matrices: For the last point we perform a straightforward computation:

σ ´1 ¨mpkq γ N pX 1:k q " Tr « γ pkq N q â i"1 Π X σpiq ff " Tr « ´σ ¨γpkq N ¯q â i"1 Π X i ff " Tr « γ pkq N q â i"1 Π X i ff " m pkq γ N pX 1:k q (II.
ÿ n 1:k ě0 m pkq γ N pn 1:k ; R 1:k q " Tr « γ pkq N ÿ n 1:k ě0 k â i"1 Π n i ,R i ff " ż Ω k γ pkq N px 1:k , x 1:k q k ź i"1
g λ px i ´Ri q 2 dx 1:k " pg 2 λ q bk ˚ρpkq γ N pR 1:k q Equation (II.5.7) tells us that summing the densities inside every Landau level approximate the total density.

II.5.2 Semi-classical energy

We now prove that the quantum energy can be approximated by the following semi-classical energy, only depending on the one body and two body Husimi functions.

Proposition II.5.4: Semi-classical approximation

Let ψ N P L 2 ´pΩ N q, ∥ψ N ∥ L 2 " 1, the quantum energy can be approximated with the semiclassical energy (II.5.1)

⟨ψ N |H N ψ N ⟩ N " N Ñ8 E sc,hb rm ψ N s `O ˆf pλq hb Tr " L h,b γ p1q N ı ˙`O `phλq 2 ˘(II.5.9)
where

f pλq -max ` g 2 λ ˚V ´V L 2 , pg 2 λ q b2 ˚w ´w L 2 ˘Ñ λÑ8 0 (II.5.10)
The kinetic energy

1 hb Tr " L h,b γ p1q ψ N ı
will be bounded when we will take a sequence of minimizers of the N -body quantum energy. Recalling (II.1.8) and (II. 1.11),

bl b " O phN l b q " O ´hN 1 2 ¯" 1 so with (II.3.2) phλq 2 ! hλ ! hbλl b ! 1 (II.5.11)
Moreover, λ Ñ 8 so the error terms in (II.5.9) will be small.

Proof of Proposition II.5.4:

Step (II.5.15)

Step 2: Potential energy

If we take a k variable potential

V k P L 1 `Ωk ˘, Tr " V k γ pkq ψ N ı " ż Ω k γ pkq ψ N px 1:k ; x 1:k q V k px 1:k qdx 1:k " ż Ω k ρ pkq ψ N V k
To express this in terms of Husimi functions we use (II.5.7):

Tr

" V k γ pkq ψ N ı " ż Ω k ρ pkq mγ N V k `ż Ω k ´ρpkq ψ N ´pg 2 λ q bk ˚ρpkq ψ N ¯Vk " ż Ω k ρ pkq mγ N V k `ż Ω k ρ pkq ψ N `Vk ´pg 2 λ q bk ˚Vk
Step

3: Conclusion

Thus applying (II.4.4) and using (II.5.15),

⟨ψ N |H N ψ N ⟩ N " Tr " pL h,b `V qγ p1q ψ N ı `Tr " wγ p2q ψ N ı " ż NˆΩ E n m p1q ψ N pn, Rqdηpn, Rq `ż Ω ρ p1q ψ N V `ż Ω 2 ρ p2q ψ N w ´phλq 2 ∥∇g∥ 2 L 2 " E sc,hb rm ψ N s `ż Ω ρ p1q ψ N " V ´g2 λ ˚V ‰ `ż Ω 2 ρ p2q ψ N " w ´pg 2 λ q b2 ˚w‰ ´phλq 2 ∥∇g∥ 2 L 2
Using V, w P L 2 pΩq and the fact that w and thus

pg 2 λ q b2 ˚wpx, yq " ij Ω 2
g 2 λ pzqg 2 λ ptqwpx ´y `t ´zqdzdt only depends on x ´y we can use the kinetic energy inequalities (II.4.2) and (II.4.3) to control the errors terms:

ˇˇˇ⟨ ψ N |H N ψ N ⟩ N ´Esc,hb rm ψ N s ˇˇˇď ˇˇˇˇˇż ω ρ p1q ψ N " V ´g2 λ ˚V ‰ ˇˇˇˇˇ`ˇˇˇˇˇż Ω 2 ρ p2q ψ N " w ´pg 2 λ q b2 ˚w‰ ˇˇˇˇp hλq 2 ∥∇g∥ 2 L 2 ď C hb Tr " L h,b γ p1q N ı f pλq `phλq 2 ∥∇g∥ 2 L 2
and we have

f pλq Ñ λÑ8 0

II.6 Mean field limit

In Section II.5, we went from the quantum N -body energy to the semi-classical energy (II.5.1) (Proposition II.5.4). The last step needed to obtain the limit models (II.1.13) and (II.1.16) out of (II.5.1) and (II.5.2) is to remove correlations. Indeed we see that for m P L 1 pN ˆΩq and ρ P L 1 pΩq

E sc,hb " m b2 ‰ " E sc,hb rms E qLL " ρ b2 ‰ " E qLL rρs
For fermionic states there are always some correlations due to anti-symmetry. Therefore the objective of this section is to prove that all other correlations are negligible. Neglecting correlations except those coming from the anti-symmetry is called the mean field approximation. We prove that this approximation holds in the mean field limit using Lieb's variational principle (Theorem II.6.5) for the energy upper bound in Subsection II.6.1 and the De Finetti Theorem II.6.11 for the lower bound in Subsection II.6.2.

The next proposition is a computation of the semi-classical energy when the low Landau levels are saturated. In this case the semi-classical energy is a sum of constant energies and of the semi-classical functional (II. 1.16) for particles in qLL.

Proposition II.6.1: Saturated low Landau level energy Let ρ P D qLL , using Notation II.1.4 and Notation II.5.1 E sc,hb rm ρ s " hbE q,r `Eq,r V `Eq,r w `EqLL rρs " hbE q,r `EqLL rρ mρ s

Proof :

With a straightforward computation:

E sc,hb rm ρ s " ÿ nPN E n ż Ω m ρ pn, xqdx `ÿ nPN ż Ω V pxqm ρ pn, xqdx `ÿ n,r nPN ż Ω 2
wpx ´yqm ρ pn, xqm ρ pr n, yqdxdy

" 1 q `r q´1 ÿ n"0 E n `r q `r E q `q pq `rq Ω V `ż Ω V ρ `q2 pq `rq 2 Ω 2 w `2q L 2 pq `rq ij Ω 2 wpx ´yqρpxqdxdy `ż Ω 2 wpx ´yqρpxqρpyqdxdy " E qLL rρs `2hb q `r ¨q 2 ¨ˆq ´1 `1 2 `1 2 ˙`r2hb q `r ¨ˆq `1 2 q q `r Ω V `q2 pq `rq 2 Ω w `2qr pq `rq 2 Ω w " E qLL rρs `q2 `2qr `r q `r hb `q q `r Ω V `q2 `2qr pq `rq 2 Ω
w " E qLL rρs `hbE q,r `Eq,r V `Eq,r w

We obtain the second equality with

E qLL " ρ mρ ‰ " ÿ nPN ż Ω V pxqm ρ pn, xqdx `ÿ n,r nPN ż Ω 2
wpx ´yqm ρ pn, xqm ρ pr n, yqdxdy " E qLL rρs `Eq,r V `Eq,r w

II.6.1 Energy upper bound

In this part we prove the energy upper bound:

Proposition II.6.2: Upper energy bound

E 0 N N ď hbE q,r `Eq,r V `Eq,r w `EqLL rρs `hbO ˆ1 ´dpq `rq N ˙`O pf pλqq `O phbλl b q
The main tool for this proof is the Hartree-Fock theory obtained when one only consider Slater trial states. For Slater states, many energy computation are simplified (see Wick's Theorem II.6.4): the second reduced density matrix can be reconstructed from the first reduced density matrix. The second reduced density matrix is given in term of a perfectly uncorrelated term and an exchange term that will reduce the energy in the case of repulsive interactions. The exchange term contains the correlations due to anti-symmetry, these are the minimal correlations fermionic states can have. Thus Hartree Fock theory is a way to assume that all other correlations are negligible. The Hartree-Fock energy gives a canonical upper bound for the N -body quantum energy since the variational ensemble is restricted to Slater sates. Hartree-Fock theory can be extended to one body operator (see Notation II.6.3), and using Lieb's variational principle (Theorem II.6.5) one can show that the theory still provides an approximate upper bound for the N -body quantum energy. Then we show that the Hartree-Fock energy is an approximation for the semi-classical energy (Proposition II.6.7). Theorem II.6.4: Wick's theorem

Let ψ N " 1 ? N ! Ź N j"1 ϕ j P L 2
´pΩ N q with pϕ j q j an orthonormal family, then

γ p1q ψ N " 1 N N ÿ i"1 |ϕ i ⟩ ⟨ϕ i | and γ p2q ψ N " N N ´1 p1 ´Exq ´γp1q ψ N ¯b2 " 1 N pN ´1q N ÿ i,j"1 |ϕ i b ϕ j ⟩ ⟨ϕ i b ϕ j ´ϕj b ϕ i | Thus for a Slater state γ ψ N ´γp1q ψ N ¯2 " γ p2q ψ N
and the Hartree-Fock energy is exactly what we obtain for the quantum N -body energy:

E HF " γ p1q ψ N ı " Tr " pL h,b `V q γ p1q ψ N ı `Tr " wγ p2q ψ N ı
Lieb's theorem [START_REF] Lieb | Variational Principle for Many-Fermion Systems[END_REF] extends the usual variational principle for operators of the form (II.6.2).

Theorem II.6.5: Lieb's variational principle

Let γ P L 1 pL 2 pΩqq satisfying Tr rγs " 1 0 ď γ ď 1 N
there exits an N -body density matrix γ N and a positive operator L 2 such that

γ p1q N " γ γ p2q N " γ 2 ´L2
We start with Lieb's variational principle to get an energy upper bound in term of the operator γ 2 . An important remark here is that we don't assume that the interaction potential is repulsive to get the upper bound as it is usually done when dealing with Lieb's variational principle. The reason why we were able to relax the assumption w ě 0 is independent of our specific model and comes from the fact that L 2 looks like an exchange term in the mean field limit since (see (II.6.8)) Tr rL 2 s ď 1 N ´1 Hence the contributions coming from this term in the computation (II.6.6) will be treated as error terms. Lieb's variational principle has also been recently generalised in [START_REF] Bach | Hartree-Fock Theory, Lieb's Variational Principle and their Generalizations[END_REF].

Proposition II.6.6

Let γ P L 1 pL 2 pΩqq such that Tr rγs " 1 and 0 ď γ ď 1 N , then

E 0 N N ďE HF rγs `Tr rL h,b γs hb O pl b q Proof :
Step 1: Estimate for the interaction term

First we prove a lower bound for the interaction term. Using The Gagliardo-Nirenberg inequality for ψ P L 2 pΩq,

∥ψ∥ 2 L 4 ď C GN ´b∥ψ∥ L 2 ∥∇ψ∥ L 2 `∥ψ∥ L 2
ālong with Hölder's, Young's and Kato's (II.4.7) inequalities,

|⟨ψ|Vψ⟩| ď ∥ψ∥ L 4 ∥Vψ∥ L 4 3 ď ∥V∥ L 2 ∥ψ∥ 2 L 4 ď C GN ∥V∥ L 2 `∥ψ∥ L 2 ∥∇ |ψ|∥ L 2 `∥ψ∥ 2 L 2 ď C GN ∥V∥ L 2 ˆ1 h ∥ψ∥ L 2 ∥P h,b ψ∥ L 2 `∥ψ∥ 2 L 2 ď C GN ∥V∥ L 2 ˆϵ ∥P h,b ψ∥ 2 L 2 `ˆ1 `1 4ϵh 2 ˙∥ψ∥ 2 L 2 Ṡo for ψ 2 P L 2 pΩq b Dom pL h,b q, |⟨ψ 2 |wψ 2 ⟩| ď ż Ω 2 |wpx ´yq| |ψ 2 px, yq| 2 dxdy ď ∥w∥ L 2 ż Ω ∥ψ 2 px, 'q∥ 2 L 4 ď C GN ∥w∥ L 2 ż Ω ˆϵ ∥P h,b ψ 2 px, 'q∥ 2 L 2 `ˆ1 `1 4ϵh 2 ˙∥ψ 2 px, 'q∥ 2 L 2 ˙dx (II.6.4) " C GN ∥w∥ L 2 ˆϵ ∥1 b P h,b ψ 2 ∥ 2 L 2 `ˆ1 `1 4ϵh 2 ˙∥ψ 2 ∥ 2 L 2 Ṫhus ψ 2 ˇˇ`C GN ∥w∥ L 2 ϵ `Id L 2 pΩq b L h,b ˘`w ˘ψ2 "C GN ∥w∥ L 2 ϵ ∥1 b P h,b ψ 2 ∥ 2 L 2 `⟨ψ 2 |wψ 2 ⟩ ě ´CGN ∥w∥ L 2 ˆ1 `1 4ϵh 2 ˙∥ψ 2 ∥ 2 L 2
and

ϵC GN ∥w∥ L 2 `Id L 2 pΩq b L h,b ˘`w ě ´CGN ∥w∥ L 2 ˆ1 `1 4ϵh 2 ˙(II.6.5)
Step 2: Lieb's variational principle Let γ N and L 2 be the operators in Theorem II.6.5. Now we use (II.4.4), and (II.6.5):

E 0 N N ď Tr rH N γ N s N " Tr " pL h,b `V q γ p1q N ı `Tr " wγ p2q N
ı "Tr rpL h,b `V q γs `Tr rw pγ 2 ´L2 qs " E HF rγs ´Tr rwL 2 s ďE HF rγs `CGN ∥w∥ L 2 ˆˆ1 `1 4ϵh 2 ˙Tr rL 2 s `ϵTr

"`I d L 2 pΩq b L h,b ˘L2 ‰ ˙(II.6.6)
To conclude we need to estimate the error terms.

Step 3: First error term

If A is an operator on L 2 pΩq it follows from (II.6.1) that

Tr "`I d L 2 pΩq b A ˘Exγ b2 ‰ " Tr " Aγ 2 ‰ (II.6.7)
Indeed, if we decompose γ in an orthonormal family:

γ - ÿ iPN λ i |u i ⟩ ⟨u i | then Tr "`I d L 2 pΩq b A ˘Exγ b2 ‰ " ÿ i,jPN λ i λ j Tr " Id L 2 pΩq b A |u i b u j ⟩ ⟨u j b u i | ‰ " ÿ i,
" γ p2q N ı " N N ´1 Tr " γ b2 ´Exγ b2 ‰ ´1 " N N ´1 ´N N ´1Tr " γ 2 ‰ ´1 ď 1 N ´1 (II.6.8)
If ϵ Ñ 0, we can control the first error term in (II.6.6) with 0 ď ˆ1 `1 4ϵh 2 ˙Tr rL 2 s ď C N ϵh 2 (II.6.9)

Step 4: Second error term

For the second error term, using Theorem II.6.5, (II.6.2) and (II.6.7) for A -L h,b ,

0 ďTr "`I d L 2 pΩq b L h,b ˘L2 ‰ " Tr " `Id L 2 pΩq b L h,b ˘´γ 2 ´γp2q N ¯ı " N N ´1Tr "`I d L 2 pΩq b L h,b ˘p1 ´Exq γ b2 ‰ ´Tr " `Id L 2 pΩq b L h,b ˘γp2q N ı " N N ´1Tr rL h,b γs ´N N ´1Tr "`I d L 2 pΩq b L h,b ˘Exγ b2 ‰ ´Tr rL h,b γs " 1 N ´1Tr rL h,b γs ´N N ´1Tr " L h,b γ 2 ‰ ď 1 N ´1Tr rL h,b γs
When the kinetic energy is minimised Tr rL h,b γs is of order hb so we estimate the second error term in (II.6.6) with:

0 ď ϵTr "`I d L 2 pΩq b L h,b ˘L2 ‰ ď C ϵhb N ¨Tr rL h,b γs hb (II.6.10)
Step 5: Conclusion

We optimise in ϵ so the bounds in (II.6.9) and (II.6.10) are of the same order: Recalling definitions (II.1.13) and (II.5.10), we now go from the Hartree-Fock energy to the semi-classical energy.

1 N ϵh 2 " ϵhb N ùñ ϵ " 1 ? h3 b " N 2δ´1 2 " op1q ùñ 1 N ϵh 2 " ϵhb N " 1 l b N " Opl b q (II.
Proposition II.6.7: Semi-classical approximation of Hartree-Fock energy Let n 0 P N, m P L 1 pN ˆΩq such that @n ą n 0 , mpn, 'q " 0 and

0 ď m ď 1 2πl 2 b N (II.6.12) then E HF rγ m s " E sc,hb rms `O pf pλqq `O phbλl b q Proof :
Step 1: Kinetic term

We start by proving that we recover the semi-classical functional from the direct terms. We compute the kinetic term using the commutation relation (II.5.13) and Corollary II. Step 2: Direct term

Let k P N ˚and W k P L 2 `Ωk ˘, with the Fubini theorem Tr " W k γ bk m ‰ "p2πl 2 b q k ż pNˆΩq k m bk pX 1:k qTr « W k k â i"1 Π X i ff dη bk pX 1:k q " `2πl 2 b ˘k ż pNˆΩq k m bk pX 1:k q ż Ω k W k px 1:k q ˜k â i"1 Π X i ¸px 1:k , x 1:k qdx 1:k dη bk pX 1:k q " ż Ω k W k px 1:k q ¨k ź i"1 2πl 2 b ż pNˆΩq mpXqΠ X px i , x i qdη pXq ‹ 'dx1:k " ż Ω k W k px 1:k q ¨k ź i"1 ż pNˆΩq mpn, Rqg 2 λ px i ´Rq p1 `lb Epn, x i qq dη pn, Rq ‹ 'dx1:k " ż Ω k W k `ρbk m ˚pg 2 λ q bk ˘dx `lb ż Ω k W k px 1:k q ¨k ź i"1 ż Ω g 2 λ px i ´Rq n 0 ÿ n"0 mpn, RqEpn, x i qdR 'dx 1:k
But m has finitely many filled Landau level so with the Pauli principle (II.6.12), ρ m P L 8 pΩq and

Tr " W k γ bk m ‰ " ż Ω k W k ρ bk m `O ` W k ´Wk ˚pg 2 λ q bk L 1 ˘`O pl b q
(II.6.17)

Step 3: Exchange term Now we need to control the exchange term. It follows from (II.6.1) that Exγ b2 m px, y; z, tq " γ m px, tqγ m py, zq so with (II.6.4) for γ m P L 2 pΩq b Dom pL h,b q as an integral kernel, ˇˇTr " wExγ b2 m ‰ˇˇ" ˇˇˇˇˇż

Ω 2
wpx ´yq |γ m px, yq| 2 dxdy ˇˇˇˇď

C GN ∥w∥ L 2 ż Ω ˆϵ ∥P h,b γ m px, 'q∥ 2 L 2 `ˆ1 `1 4ϵh 2 ˙∥γ m px, 'q∥ 2 L 2 ˙dx (II.6.18)
With an integration by part,

ż Ω ∥P h,b γ m px, 'q∥ 2 L 2 dx " ż Ω 2 P h,b γ m px, 'qpyq ¨Ph,b γ m px, 'qpyqdxdy " ż Ω 2 L h,b γ m px, 'qpyqγ m px, yqdxdy " ż Ω 2 γ m px, yqL h,b γ m p', xqpyqdxdy " ż Ω 2 γ m px, yq pL h,b γ m q py, xqdxdy " Tr rγ m L h,b γ m s
Inserting this in (II.6.18), using the cyclicity of the trace we get Step 4: Conclusion

ˇˇTr " wExγ b2 m ‰ˇˇ" ˇˇTr " wExγ m b2 ‰ˇˇď C GN ∥w∥ L 2 ˆϵTr " L h,b γ 2 m ‰ `ˆ1 `1 4ϵh 2 ˙Tr " γ 2 m ‰ ďC GN ∥w∥ L 2 N ˆϵTr
With (II.6.3) and (II.6.2) then (II.6.16) and (II.6.17) applied to V and w,

E HF rγ m s "Tr rL h,b γ m s `Tr rV γ m s `N N ´1 Tr " wγ b2 m ‰ `N N ´1Tr " wExγ b2 m ‰ " ż NˆΩ E n mpXqdηpXq `ż Ω V ρ m `N N ´1 ż Ω 2 wρ b2 m `O ` V ´V ˚pg 2 λ q L 1 N N ´1 O ` w ´w ˚pg 2 λ q b2 L 1 ˘`O pl b q `O phbλl b q `O `phλq 2 Ȓecalling
(II.5.10), the semi-classical energy expression (II.1.13), (II.5.11) and hbλ " 1,

E HF rγ m s " E sc,h rms `f pλq `O phbλl b q
With the notation of equation (II.1.21), we would like to define a one body operator with saturated low Landau levels:

γ ρ - L 2 pq `rq N ż ΩˆN m ρ pXqΠ X dηpXq
We need to prove that the direct term gives the limit model for qLL and to control the exchange terms. But we cannot apply directly Lieb's principle because with Lemma II.5.2 we have an error on the trace Tr rγ ρ s " 1 `op1q and 0 ď γ ρ ď 1 N To cure this we modify m ρ slightly.

Proposition II.6.8: Corrected Husimi function Let n 0 P N, m P L 1 pN ˆΩq such that @n ą n 0 , mpn, 'q " 0, ∥m∥ L 1 " 1 `op1q and

0 ď m ď 1 2πl 2 b N
then there exist r m P L 1 pN ˆΩq, n 1 P N such that @n ą n 1 , r mpn, 'q " 0, Step 1: Tr rγ m s ă 1

If Tr rγ m s ă 1 we add some mass to m where it is possible without breaking the Pauli principle. Let n 1 P N and

0 ď τ ď 1 2πl 2 b N we define r mpτ, n 1 q -m `min ˆτ, 1 2πl 2 b N ´m˙1 nďn 1
By construction

0 ď m ď r mpτ, n 1 q ď 1 2πl 2 b N
and τ 1 nďn 1 ď r mpτ, n 1 q (II.6.20)

We choose n 1 ą n 0 and remark that

Tr rγ r m s ˆ1 2πl 2 B N , n 1 ˙" 1 2πl 2 B N ż NˆΩ 1 nďn 1 p1 `lb EpXqq dηpXq ě L 2 2πl 2 b N n 1 ´lb Cpn 1 q
Since Dn 1 P N such that Thus we can estimate

ÿ nďn 1 ż Ω min ˆτ, 1 2πl 2 b N ´mpn, xq ˙dx " ż NˆΩ p r m ´mq dη " 1 ´lb ż NˆΩ r mEdη ´ż NˆΩ mdη " Opl b q `O p1 ´∥m∥ L 1 q so τ " 1 L 2 ż Ω min ˆτ, 1 2πl 2 b N ´mpn 1 , xq ˙dx ď Opl b q `O p1 ´∥m∥ L 1 q (II.6.21)
Step 2: Tr rγ m s ą 1

Now if Tr rγ m s ą 1 we remove some mass to m: r mpτ q -max p0, m ´τ q " m ´minpm, τ q

ż NˆΩ r mEdη " Opl b q `O p1 ´∥m∥ L 1 q " ż măτ mdη `ż τ ďm τ dη " ∥m∥ L 1 `ż τ ďm pτ ´mqdη So ∥m∥ L 1 `Opl b q `O p1 ´∥m∥ L 1 q " ż τ ďm pm ´τ qdη ď 1 πl 2 b N |1 τ ďm | and τ ď 1 |1 τ ďm | ż NˆΩ minpm, τ qdη " 1 |1 τ ďm | pOpl b q `O p1 ´∥m∥ L 1 qq ď 1 πl 2 b N ¨Opl b q `O p1 ´∥m∥ L 1 q ∥m∥ L 1 `Opl b q `O p1 ´∥m∥ L 1 q " Opl b q `O p1 ´∥m∥ L 1 q (II.6.23)
Step

E n `pn 1 `1q ∥V ∥ L 1 `L2 πl 2 b N ∥w∥ L 1 pn 1 `1q 2 ∥m ´r m∥ L 8
We conclude with (II. 6.24).

Putting all of this together we obtain the upper bound.

Proof of Proposition II.6.2:

Recalling (II.1.21), let ρ P D qLL and define

m ρ,N - dpq `rq N m ρ (II.6.25) then 0 ď m ρ,N ď d L 2 N " 1 2πl 2 b N ż NˆΩ m ρ,N dη " dpq `rq N " 1 `op1q
We consider r m ρ,N the corrected Husimi function in Proposition II.

E 0 N N ďE HF " γ m ρ,N ‰ `O pl b q " E sc,hb r r m ρ,N s `O pf pλqq `O phbλl b q "E sc,hb rm ρ,N s `hbO ˆ1 ´dpq `rq N ˙`O pf pλqq `O phbλl b q "hbE q,r `Eq,r V `Eq,r w `EqLL " dpq `rq N ρ ȷ `hbO ˆ1 ´dpq `rq N ˙`O pf pλqq `O phbλl b q "hbE q,r `Eq,r V `Eq,r w `EqLL rρs `hbO ˆ1 ´dpq `rq N ˙`O pf pλqq `O phbλl b q
For the last equality we use the estimate

ˇˇˇE qLL " dpq `rq N ρ ȷ ´EqLL rρs ˇˇď ˇˇˇ1 ´dpq `rq N ˇˇˇ∥ V ∥ L 2 ∥ρ∥ L 2 `˜1 ´ˆdpq `rq N ˙2¸∥ w∥ L 2 ∥ρ∥ 2 L 2 and ˇˇˇˇ˜1 ´ˆdpq `rq N ˙2¸ˇˇˇˇˇď C ˇˇˇ1 ´dpq `rq N ˇˇǏ I.6.

Energy lower bound

In this part we prove the Energy lower bound :

Proposition II.6.9: Lower bound Let pψ N q N be a sequence of minimizers of (II.1.7), E sc,hb rm ψ N s ě hbE q,r `Eq,r

V `Eq,r w `E0 qLL `op1q
The main tool for the lower bound is the De Finetti Theorem II.6.11. Husimi functions are symmetric and consistent measures. The De Finetti theorem states that such measures are indeed reduced to trivial measure of this kind, namely tensorized products of one body measures and their convex combinations. This result plays an important role in the justification of the decorrelation of densities for the lower bound. We start by extracting some limit Husimi functions and give their fundamental properties. Similar arguments can be found in [START_REF] Fournais | The semi-classical limit of large fermionic systems[END_REF]Section 2]. With Notation II.5.1, Proposition II.6.10

Let pψ N q N be a sequence of minimizers of (II.1.7), up to a subsequence a) there exists limit Husimi functions M pkq P L 8 `pN ˆΩq k ˘such that

m pkq ψ N á N Ñ8
M pkq in the weak star topology on L 8 ´pN ˆΩq k ¯(II.6.27)

0 ď M pkq
E qLL " ρ m ψ N ı Ñ N Ñ8
E qLL rρ M s (II.6.31)

Proof :

a) From inequality (II.5.5) the Husimi functions are uniformly bounded, with a diagonal extraction we obtain (II.6.27) and the bound (II.5.5) with (II.1.11) induce (II.6.28) in the limit.

b) Now since we took a minimizer of the energy, with the upper bound Proposition II.6.2 and the Kinetic energy inequalities (II.4.2) and (II.4.3),

E 0 N N "Tr " L h,b γ p1q ψ N ı `ż Ω V ρ p1q ψ N `ż Ω 2 wρ p2q ψ N " Tr " L h,b γ p1q ψ N ı ˆ1 `O ˆ1 hb ˙ďE sc,hb rm ρ s `hbO ˆ1 ´dpq `rq N ˙`O pf pλqq `O phbλl b q
so by Proposition II.6.1 we know that

Tr " L h,b γ p1q ψ N ı " O phbq (II.6.32)
Since the contribution of the potential are bounded, the only thing we have to look at are the kinetic terms. Let m ρ be the Husimi function with saturated low Landau levels defined here (II.1.21). We denote

c N,n - ż Ω ´mp1q ψ N pn, .q ´mρ pn, .q
By definition of m ρ and Lemma II.5.2 we have

ÿ nPN c N,n " ż NˆΩ m p1q ψ N ´ż NˆΩ m ρ " 1 ´1 " 0 n ă q ùñ c N,n ď L 2 2πl 2 b N `O pl b q ´1 q `r " O pl b q `O ˆ1 ´dpq `rq N ṅ ą q ùñ c N,n " m p1q ψ N pn, 'q L 1 ě 0 Since pE n q n is increasing ÿ nPN E n c N,n ě q ÿ n"0 E n c N,n `Eq ÿ nąq c N,n " ´q´1 ÿ n"0 pE q ´En qc N,n
ěO phbl b q `hbO ˆ1 ´dpq `rq N ˙(II.6.33)

Now we compute E sc,hb rm ψ N s ´Esc,hb rm ρ s "

ÿ nPN E n c N,n `ż NˆΩ V ´mp1q ψ N ´mρ ¯dη `ż pNˆΩq 2 w ´mp2q ψ N ´mb2 ρ ¯dη b2 (II.6.34)
From the semi-classical approximation (Proposition II.5.4), (II.6.32) and the upper bound (Proposition II.6.2),

E 0 N N " ⟨ψ N |H N ψ N ⟩ N " E sc,hb rm ψ N s `O pf pλqq `O `phλq 2 ďE sc,hb rm ρ s `hbO ˆ1 ´dpq `rq N ˙`O pf pλqq `O phbλl b q
so with (II.5.11),

E sc,hb rm ψ N s ´Esc,hb rm ρ s ď hbO ˆ1 ´dpq `rq N ˙`O pf pλqq `O phbλl b q (II.6.35)
All the potential terms in (II.6.34) are of order 1, therefore the sum in (II.6.33) is bounded and we have

O phbl b q `hbO ˆ1 ´dpq `rq N ˙ď ´q´1 ÿ n"0 pE q ´En qc N,n ď ÿ nPN E n c N,n ď C So q´1 ÿ n"0 E n ´Eq hb c N,n " O ˆ1 hb ˙(II.6.36)
With a similar inequality as (II.6.33) but with E q`1 instead of E q we deduce

C ě ÿ nPN E n c N,n ě q ÿ n"0 E n c N,n `Eq`1 ÿ nąq c N,n " q ÿ n"0 pE n ´Eq`1 qc N,n ě q ÿ n"0 E n c N,n `Eq ÿ nąq c N,n ě O phbl b q `hbO ˆ1 ´dpq `rq N ˙(II.6.37)
and therefore (II.6.36) implies

c N,q " O ˆ1 hb Ṫhen ÿ nąq E n hb c N,n " ÿ nPN E n hb c N,n ´q ÿ n"0 E n hb c N,n " O ˆ1 hb ˙ě ÿ nąq ż Ω m p1q ψ N pn, 'q and c N,q " ż Ω m p1q N pq, RqdR ´ż Ω ρpRqdR " m p1q N pq, 'q L 1 ´r q `r " O ˆ1 hb ˙(II.6.38)
From the consistency of m pkq ψ N in Property II.5.3,

m p1q ψ N pn 1 , 'q L 1 " ż Ω ¨ż pNˆΩq k´1 m pkq ψ N pn 1 , x 1 ; X 2:k qdη bpk´1q pX 2:k q ‹ 'dx1 " ÿ n 2:k PN k´1 m pkq ψ N pn 1:k , 'q L 1 (II.6.39) Since N k z 0 : q k " k ğ i"1
N i´1 ˆpNz 0 : q q ˆNk´i by the symmetry of m pkq ψ N , (II.6.39) and (II.6.36),

ÿ n 1:k PN k z 0:q k m pkq ψ N pn 1:k , 'q L 1 " k ÿ n 1 ąq m p1q ψ N pn 1 , 'q L 1 " O ˆ1 hb ˙(II.6.40)
Ω is bounded, thus testing (II.6.27) against 1 tn 1:

k uˆΩ P L 1 ´pN ˆΩq k ¯, m pkq ψ N pn 1:k ; 'q L 1 Ñ N Ñ8
M pkq pn 1:k ; 'q L 1 So (II. 6.38) gives

M p1q pq, 'q L 1 " r q `r
and with (II.6.40), if n 1:k P N k z 0 : q k , then M pkq pn 1:k , 'q " 0 and we see that the norm (II.5.6) passes to the limit:

M pkq L 1 " ÿ n 1:k P 0:q k M pkq pn 1:k , 'q L 1 " lim N Ñ8 ÿ n 1:k P 0:q k m pkq ψ N pn 1:k , 'q L 1 " lim N Ñ8 ¨ÿ n 1:k P 0:q k m pkq ψ N pn 1:k , 'q L 1 `ÿ n 1:k PN k z 0:q k m pkq ψ N pn 1:k , 'q L 1 ' " lim N Ñ8 m pkq ψ N L 1 " 1
If n ă 0, by (II.6.36),

m p1q ψ N pn, 'q ´1 L 2 pq `rq L 1 ď m p1q ψ N pn, 'q ´1 2πl 2 b N L 1 `O ˆ1 ´dpq `rq N " ż Ω ˆ1 2πl 2 b N ´mp1q ψ N pn, 'q ˙`O ˆ1 ´dpq `rq N " ż Ω ˆ1 L 2 pq `rq ´mp1q ψ N pn, 'q ˙`O ˆ1 ´dpq `rq N " ´CN,n `O ˆ1 ´dpq `rq N ˙" O ˆ1 hb ˙`O ˆ1 ´dpq `rq N ṡo M p1q pn, 'q " 1 L 2 pq `rq .
c) Testing (II.5.8) against φ q P C 0 c ppN ˆΩq q q, we have

ż pNˆΩq q φ q m pqq ψ N dη bq " ż pNˆΩq k
φ q pX 1:q qm pkq ψ N pX 1:k qdη bk pX 1:k q (II.6.41)

Since φ q P L 1 ´pN ˆΩq k ¯, with (II.6.27),

ż pNˆΩq q φ q m pqq ψ N dη bq Ñ N Ñ8 ż pNˆΩq q
φ q M pqq dη bq (II.6.42)

In order to pass to the limit in the right term of (II.6.41), for the low Landau levels we use (II.6.27) on

1 p 0:q ˆNq k ´φq b Id pNˆΩq k´q ¯P L 1 ´pN ˆΩq k ānd
for the high Landau levels we use (II.6.40) and φ q P L 8 ´pN ˆΩq k ¯:

ż pNˆΩq k φ q pX 1:q qm pkq ψ N pX 1:k qdη bk pX 1:k q " ż Ω k 1 p 0:q ˆNq k ´φq b Id pNˆΩq k´q ¯mpkq ψ N dη bk `ÿ n 1:k PN k z 0:q k ż Ω k φ q pn 1:q , x 1:q qm pkq ψ N pn 1:k , x 1:k qdx 1:k Ñ N Ñ8 ż Ω k 1 p 0:q ˆNq k ´φq b Id pNˆΩq k´q ¯M pkq dη bk " ż pNˆΩq k
φ q pX 1:q qM pkq pX 1:k qdη bk pX 1:k q (II.6.43)

Thus passing to the limit in (II.6.41) and inserting (II.6.42) and(II.6.43) we obtain @φ q P C 0 c ppN ˆΩq q q , ż pNˆΩq q φ q M pqq dη bq " ż pNˆΩq k φ q pX 1:q qM pkq pX 1:k qdη bk pX 1:k q and this proves that the limit Husimi functions are also consistent. Testing against φ q , we also obtain that the symmetry of Husimi functions passes to the limit. Then we can conclude with the Kolmogorov extension theorem that there exists M a symmetric measure on pN ˆΩq N whose marginals are pM pkq q k . d) Let φ k P C 0 pΩ k q, φ k is bounded and

1 0:q k b φ k P L 1 ´pN ˆΩq k
so using (II.6.27) and (II.6.40)

ż Ω k φ k ρ pkq m ψ N " ż pNˆΩq k ´1 0:q k b φ k ¯mpkq ψ N dη bk `ÿ n 1:k PN k z 0:q k ż Ω k φ k m pkq ψ N Ñ N Ñ8 ż pNˆΩq k ´1 0:q k b φ k ¯M pkq dη bk " ż Ω k φ k ρ pkq M
e) Let V k P L 2 pΩ k q, and pV k,n q n Ă C 8 pΩ k q a sequence regularised with a convolution to a regular function so that

∥V k ´Vk,n ∥ L 2 Ñ nÑ8 0 we have ż Ω k V k ´ρpkq m ψ N ´ρpkq M ¯" ż Ω k V k,n ´ρpkq m ψ N ´ρpkq M ¯`ż Ω k ρ pkq m ψ N pV k ´Vk,n q `ż Ω k ρ pkq M pV k,n ´Vk q
For a fixed n, since V k,n P C 0 pΩ k q by (II.6.30) the first term goes to 0 when N Ñ 8. For the second term we use (

II.4.2) if V 1 " V , (II.4.3) if V 2 "
w and (II.5.7)

ˇˇˇˇˇż

Ω k ρ pkq m ψ N pV k ´Vk,n q ˇˇˇˇˇ" ˇˇˇˇˇż Ω 2 ´`g k λ ˘bk ˚ρpkq N ¯pV k ´Vk,n q ˇˇˇˇˇď C pV k ´Vk,n q ˚`g 2 λ ˘bk L 2 ďC ∥pV k ´Vk,n q∥ L 2
For the third term we use Hölder's inequality since ρ pkq M P L 8 `Ωk ˘so we have

lim N Ñ8 ˇˇˇˇˇż Ω k V k ´ρpkq m ψ N ´ρpkq M ¯ˇˇˇˇˇď C ∥V k ´Vn,k ∥ L 2 Ñ nÑ8 0
Now we want to apply the De Finetti theorem to M :

Theorem II.6.11: De Finetti or Hewitt-Savage

Let X be a metric space, µ P P s pX N q be a symmetric probability measure with marginals `µpnq ˘ně1 . DP µ P PpPpXqq such that:

@n P N ˚, µ pnq " ż PpΩq ρ bn dP µ pρq (II.6.44)
For a proof of this via the the Diaconis-Freedman theorem see [41, Section 2.1.] and for some further context one can look at [48, Section 2.2.].

Recalling the definition of the semi-classical domain (II.1.15), we obtain: Proposition II.6.12: Low Landau level filling of the limit factorised densities There exists P M P P pD sc q such that @k P N ˚, M pkq " ż Dsc m bk dP M pmq (II.6.45)

Let µ be the push-forward measure of P M by the application

L 1 pN ˆΩq Ñ L 1 pΩq m Þ Ñ
E qLL rρ M s " ż D qLL E qLL " q L 2 pq `rq `ρȷ dµpρq " E q,r V `Eq,r w `ż D qLL E qLL rρs dµpρq (II.6.47)
Proof :

Step 1: µ P P pD qLL q Applying Theorem II.6.11 to M obtained in Proposition II.6.10 gives the existence of P M P P pP pN ˆΩqq such that

@k P N ˚, M pkq " ż PpNˆΩq m bk dP M pmq (II.6.48)
Let φ P C 0 c pN ˆΩ, R `q, φ ‰ 0, ϵ ą 0, k P N ˚, and

A ϵ pφq - $ & % m P P pN ˆΩq | ż NˆΩ φdm ě 1 `ϵ L 2 pq `rq ż NˆΩ φ , .
-

If m P A ϵ pφq, then 1 ď L 2 pq `rq p1 `ϵq ∥φ∥ L 1 pηq ż NˆΩ φdm ď ¨L2 pq `rq p1 `ϵq ∥φ∥ L 1 pηq
ż NˆΩ φdm 'k so with (II.6.28),

P M pA ϵ pφqq " ż PpNˆΩq 1 Aϵpφq dP M ď ż PpNˆΩq ¨L2 pq `rq p1 `ϵq ∥φ∥ L 1 pηq ż NˆΩ φdm 'k dP M pmq " ˜L2 pq `rq p1 `ϵq ∥φ∥ L 1 pηq ¸k ż PpNˆΩq ¨ż pNˆΩq k φ bk dm bk ‹ 'dPM pmq " ˜L2 pq `rq p1 `ϵq ∥φ∥ L 1 pηq ¸k ż pNˆΩq k φ bk dM pkq ď ˆ1 1 `ϵ˙k Ñ kÑ8 0
we proved that P M pA ϵ pφqq " 0 and therefore

P M ¨č φPC 0 c pNˆΩ,R `q ϵą0 P pN ˆΩq zA ϵ pφq ‹ '" 1 ´PM ¨ď φPC 0 c pNˆΩ,R `q ϵą0
A ϵ pφq ‹ '" 1 therefore for P m almost every m P P pN ˆΩq, @φ P C 0 c pN ˆΩ, R `q , ϵ ą 0,

ż NˆΩ φdm ă 1 `ϵ L 2 pq `rq ż NˆΩ φ (II.6.49)
So for P m almost every m P P pN ˆΩq, m is the density of a probability measure thus a positive function such that ∥m∥ L 1 " 1 and by (II.6.49), m P L 8 pN ˆΩq and m ď 1 L 2 pq `rq (II.6.50)

We have shown P M P P pD sc q, therefore (II.6.48) implies (II.6.45). Moreover if n ă q by (II.6.29),

ż Ω 1 L 2 pq `rq dx " ż NˆΩ 1 tnuˆΩ dM p1q " ż PpNˆΩq ¨ż Ω mpn, xqdx 'dP M pmq so ż
PpNˆΩq ¨ż Ω ˆ1 L 2 pq `rq ´mpn, xq ˙dx 'dP M pmq " 0 By (II.6.50) the integrand is positive thus null P M almost everywhere, we conclude that for P M almost every m n ă q ùñ mpn, 'q " 1 L 2 pq `rq (II.6.51)

If n ą q by (II.6.29),

0 " ż NˆΩ 1 tnuˆΩ dM p1q " ż PpNˆΩq ¨ż Ω mpn, xqdx 'dP M pmq
Once again by (II.6.50) the right integrand is positive and thus null so for P M almost every m n ą q ùñ mpn, 'q " 0 (II.6.52)

Finally if n " q, since m P P pN ˆΩq we conclude using (II.6.52) and (II. Gathering (II.6.50), (II.6.51), (II.6.52) and (II.6.53), we now know that for P M almost every m we have mpq, 'q P D qLL . This means that µ P P pD qLL q.

Step 2: Computation of E qLL Finally we compute

ρ pkq M " ÿ n 1:k M pkq
E qLL rρ M s " ż D qLL E qLL " q L 2 pq `rq `ρȷ dµpρq " ż D qLL pE q,r V `Eq,r w `EqLL rρsq dµpρq "E q,r V `Eq,r w `ż D qLL E qLL rρs dµpρq
Now we are ready for the proof of the lower bound.

Proof of Proposition II.6.9:

Let ρ P D qLL , starting from (II.6.34), using inequality (II.6.37) and Proposition II.6.1 we have

E sc,hb rm ψ N s ě E sc,hb rm ρ s `EqLL " ρ m ψ N ı ´EqLL " ρ mρ ‰ `O phbl b q `hbO ˆ1 ´dpq `rq N "hbE q,r `EqLL " ρ m ψ N ı `O phbl b q `hbO ˆ1 ´dpq `rq N
Ẇe conclude with (II.6.31) and (II.6.47) and that

E sc,hb rm ψ N s ěhbE q,r `EqLL " ρ m ψ N ı `O phbl b q `hbO ˆ1 ´dpq `rq N
"hbE q,r `EqLL rρ M s `op1q " hbE q,r `Eq,r V `Eq,r w `ż D qLL E qLL rρs dµpρq `op1q

(II.6.54) ěhbE q,r `Eq,r V `Eq,r w `E0 qLL `op1q

II.6.3 Conclusion

Proof of Theorem II.1.5:

Let pψ N q N be a sequence of minimizers of (II.1.7), by (II.5.9)

EpN q N " ⟨ψ N |H N ψ N ⟩ N " E sc,hb rm ψ N s `op1q
Since the lower bound is true up to a subsequence for which the have Proposition II.6.10, for every adherence value of EpN q{N we conclude by gathering Proposition II.6.2 and Proposition II.6.9.

Proof of Theorem II.1.7:

With (II.6.46) and (II.6.30) we get

ρ pkq m ψ N á N Ñ8 ż D qLL ˆq L 2 pq `rq `ρ˙b k dµpρq Let φ P C 8 `Ωk ˘with (II.5.7), ż Ω k φ ´ρpkq m ψ N ´ρpkq ψ N ¯" ż Ω k φ ´pg 2 λ q bk ˚ρpkq ψ N ´ρpkq ψ N ¯" ż Ω k ρ pkq ψ N `pg 2 λ q bk ˚φ ´φ˘Ñ N Ñ8
0 (II.6.55) by Hölder's inequality since

ρ pkq ψ N L 1 " 1
and φ is Lipschitz. Up to a subsequence ρ pkq ψ N converges @k P N ˚in the sense of Radon measures. But with (II.6.55) this limit coincides with the one of ρ pkq m ψ N so we obtain (II.1.25). Moreover by (II.6.54) and Proposition II.6.2

E 0 qLL ě ż D qLL E qLL rρs dµpρq `op1q
thus µ only gives mass to minimizers of E qLL .

Chapter III

Semi-classical limit of the 2D Hartree equation in a large magnetic field

Abstract:

We study the dynamic of two dimensional fermionic particles submitted to a magnetic field, assumed to be transverse to the domain and homogeneous. A large magnetic field regime where the gap between Landau levels is of the same order as the other energy contributions is considered. We start from the Hartree equation for the first reduced density matrix, describing the mean field behaviour of a large fermionic system, and derive a gyrokinetic transport equation for the first reduced density. We define a semi-classical density for which the dynamic is computed and compared to the limit gyrokinetic equation. It is shown that the first reduced density almost satisfies the gyrokinetic equation and converges in the sense of measures to a weak solution of this equation.

III.1 Context and results

III.1.1 Model

We consider a large system of interacting fermionic particles in two dimensions. They are placed in a homogeneous magnetic field perpendicular to the domain. In this context the kinetic energy of the particles is quantized into discrete energy levels called Landau levels, separated by a finite energy gap. Our goal is to study the semi-classical limit of the dynamics under high magnetic field. This setup is physically motivated by the quantum hall effect, see [START_REF] Jain | Composite fermions[END_REF] for some physical context. We will start from the Hartree equation appropriate for a large system of fermions, and obtain a gyrokinetic transport equation for the density.

Notation III.1.1: Model

We work on R 2 . The one body kinetic energy operator, also called magnetic Laplacian, is

L b -pih∇ `bAq 2 With Dom pL b q - ␣ ψ P L 2 `R2 ˘|L b ψ P L 2 `R2 ˘(
We work in symmetric gauge, namely the vector potential is

A " 1 2 X K (III.1.1)
where X is the position operator in R 2 . b is the magnetic field amplitude, we associate to it the magnetic length

l b - c h b
Let V be the external potential and w the interaction potential assumed to be radial:

wpx ´yq -r wp|x ´y|q
We denote L p the p th Schatten space. Let γ P L 8 pR `, L 1 pL 2 pR 2 qqq, and ρ γ be the associated reduced density

ρ γ pt, xq -γptqpx, xq (III.1.2)
Here we identified γ with its integral kernel and we will use this convention for the rest of the text. Let W : R 2 Ñ R be a potential, we introduce a notation for the gyrokinetic differential operator:

GYRO W pµqpt, zq -B t µpt, zq `∇K W pzq ¨∇µpt, zq GYRO ρ pµqpt, zq -GYRO V `w‹ρptq pµqpt, zq " B t µpt, zq `∇K pV `w ‹ ρptqq pzq ¨∇µpt, zq

Finally denote

H b ptq -L b `V `1 2 w ‹ ρ γ b ptq (III.1.3)
Our goal is to obtain from the Hartree equation

ihB t γ " rL b `V `w ‹ ρ γ , γs (III.1.4)
the following gyrokinetic transport equation for a density ρ : R `ˆR 2 Ñ R `, B t ρ `∇K pV `w ‹ ρqq ¨∇ρ " 0 (III.1.5)

The Hamiltonian L b `V `w ‹ ρ γ is time dependent hence the total energy is not preserved through the evolution (III.1.4). However, see Proposition III.3.2, the quantity Tr rH b ptqγptqs is constant due to the 1{2 factor in front of the interaction term.

III.1.2 Scaling

In classical mechanics, the movement of a fermion in a transverse magnetic field is decomposed in a cyclotron orbit and the motion of the orbit center. As exposed in Subsection III.1.4 the latter takes place on a time scale of order b. As recalled in Subsection III.2.2, the order of magnitude of the kinetic energy is hb. Our plan is to look at a scaling where all the terms in the Hamiltonian are of order 1 and the time scale is of order b.

Notation III.1.2: scaling

We take a high magnetic field limit define the time rescaled density matrix

@t P R `, γ b ptq -γpbtq (III.1.8)
If γ satisfies (III.1.4), the equation for the time-rescaled density matrix is Hence with (III. 1.6) this confirms that all the terms in the Hamiltonian L b `V `w ‹ ρ γ are of order 1. If one starts from the N -body Schrödinger dynamics, the fermionic characteristic of the system is imposed by the anti-symmetry of the wave-functions. Starting from the mean field dynamics, the fermionic characteristic is imposed directly via the Pauli principle γ b ď 2πl 2 b . As a remark, we give an equivalent formulation of this scaling. If one takes exactly h " 1{b, then (III.1.9) is equivalent to

B t γ b " b ih rL b `V `w ‹ ρ γ b , γ b s " 1 il 2 b rL b `V `w ‹ ρ γ b , γ b s (III.
iB t γ " "`i ∇ `b2 A ˘`b 2 pV `w ‹ ρ γ b q, γ b ‰
In other words with the new scaling

r b -b 2 r γ b - b 2 2π γ we have Tr r r γ b s " r b 2π , r γ b ď 1 iB t r γ b " " ´i∇ `r bA ¯2 `r bV `w ‹ ρ Ă γ b , r γ b ȷ
where all the terms in the Hamiltonian ´i∇ `r bA ¯2 `r bV `w ‹ ρ Ă γ b are of order r b.

III.1.3 Results

The classical counterpart of this work, starting from the Vlasov equation, has been well studied [START_REF] Golse | The Vlasov-Poisson System with Strong Magnetic Field[END_REF] [54] [START_REF] Sonnendrücker | Long time behaviour of the two-dimensional Vlasov equation with a strong external magnetic field[END_REF] [35] [START_REF] Han-Kwan | The three-dimensional finite Larmor radius approximation[END_REF] [2] [14] [START_REF] Griffin-Pickering | Recent Developments on Quasineutral Limits for Vlasov-Type Equations[END_REF]. Some results start from Newton's dynamics [START_REF] Han-Kwan | From Newton's second law to Euler's equations of perfect fluids[END_REF]. In the quantum literature it is known that the Hartree equation can be obtained by a mean field limit from the N -body Schrödinger dynamics [START_REF] Benedikter | Mean-Field Evolution of Fermionic Systems[END_REF] [39] [START_REF] Petrat | A New Method and a New Scaling for Deriving Fermionic Mean-Field Dynamics[END_REF]. It is also known that the Vlasov equation can be derived by a semi-classical limit from the Hartree equation [38] [9]. The mean field and semi-classical limits can be coupled to obtain directly the Vlasov equation from the N -body Schrödinger dynamics [START_REF] Liew | Combined mean-field and semiclassical limits of large fermionic systems[END_REF] [33] [START_REF] Liew | A mixed-norm estimate of the two-particle reduced density matrix of many-body Schrödinger dynamics for deriving the Vlasov equation[END_REF]. More recent results have been dealing with singular potentials [START_REF] Saffirio | Mean-field evolution of fermions with singular interaction[END_REF] [4] [START_REF] Laflèche | From Schrödinger to Hartree-Fock and Vlasov equations with Singular potentials[END_REF]. We also refer to a semi-classical work [START_REF] Porat | Derivation of Euler's equations of perfect fluids from von Neumann's equation with magnetic field[END_REF] obtaining Euler's vorticity equation from the N -body Schrödinger dynamics in a regime where the gap between Landau levels is small compared to the interactions. Note that for large magnetic fields, the limit of the fundamental energy of the N -body Hamiltonian and the associated densities have been well studied [START_REF] Lieb | Ground states of large quantum dots in magnetic fields[END_REF], [START_REF] Lieb | Quantum Dots[END_REF], [START_REF] Lieb | Asymptotics of Heavy Atoms in High Magnetic Fields I. Lowest Landau Band Region[END_REF], [START_REF] Lieb | Asymptotics of Heavy Atoms in High Magnetic Fields II. Semi-classical Regions[END_REF], [START_REF] Lieb | Heavy atoms in the strong magnetic field of a neutron star[END_REF] [44], [21] [46]. Now, with Notation III.1.1 and Notation III.1.2, we can state our main results. The first reduced density approximately satisfies a gyrokinetic dynamics:

Theorem III.1.3: Gyrokinetic limit of the Hartree solution Let γ b P L 8 pR `, L 1 pL 2 pR 2 qqq be a solution of (III.1.9) and assume

Tr rγ b p0qs " 1, 0 ď γ b p0q ď 2πl 2 b Tr rγ b p0qH b p0qs ă 8 If V, w P W 4,8 pR 2 q, then @φ P C 8 c pR `ˆR 2 q, ˇˇˇˇˇˇż R `ˆR 2 ρ γ b pt, zqGYRO ργ b pφqpt, zqdtdz ´ż R 2 φp0, zqρ γ b p0, zqdz ˇˇˇˇˇˇď Cpφ, V,
ρ γ b á bÑ8 ρ P MpR `ˆR 2 q ρ γ b p0q á bÑ8 ρ 0 P MpR 2 q
to a weak solution of (III.1.5), meaning that

@φ P C 8 c pR `ˆR 2 q, ż R `ˆR 2 ρ `Bt φ `∇K pV `w ‹ ρq ¨∇z φ ˘´ż R 2 φp0qρ 0 " 0 III.1.

Classical orbits

In this part we study a classical particle of charge ´1 in a transverse magnetic field of amplitude b in a force field F . Our goal is to extract some useful heuristics from the dynamic of this particle. Newton's fundamental equation of dynamics gives Z 2 ptq " F pt, Zptqq `bZ 1 ptq K (III.1.10) After one integration, assuming " 0

Z 1 ptq " Z 1 p0q `t ż 0 F pτ, Zpτ qqdτ `bZptq K ´bZp0q K
Inserting this in (III.1.10) we obtain

Z 2 ptq `b2 Zptq " F pt, Zptqq `b t ż 0 F pτ, Zpτ qq K dτ `bZ 1 p0q K `b2 Zp0q
For a constant and homogeneous force field, 1.11) is the trajectory satisfying

Zptq " |Z 1 c p0q| b ˆcospbtq sinpbtq loooooooooomoooooooooon -Zcptq `F K b t lo omo on -Z d ptq (III.
Z d p0q " p0, 0q Z c p0q " p1, 0q
The evolution equation for the drift of the orbit center is

Z 1 d ptq " F K b
The characteristic time of this drift is of order b. Indeed if one define

Z d,b ptq -Z d pbtq
the drift is of order 1:

Z 1 d,b ptq " F K
This analysis gives us important information about the motion of the orbit center of particles. First, this motion has to be observed over a time scale of order b, hence the time rescaling of the density matrix (III.1.8). Secondly, the shifting motion takes place in the direction perpendicular to the force field. For instance, if one takes

F -V `w ‹ ρ
this explains the origin of the ∇ K pV `w ‹ ρq in the Gyrokinetic transport equation (III.1.5).

For the cyclotron motion Z c , the characteristic time is 1{b. We see from (III. 1.11) that the radius of the cyclotron orbit

r c - |Z 1 c p0q| b
is small in the large magnetic field limit. This means that this part of the dynamic, even though it consists on fast oscillations, should disappear in the limit equation.

III.1.5 Organisation of the paper

Section III.2 and Section III.3 are preliminaries respectively about the magnetic Laplacian and the conserved properties of the dynamics. In Section III.4 we introduce the semi-classical densities and prove that they approximate the physical density. Then we study the dynamics of the semi-classical densities in Section III.5. Section III.6 contains the conclusion of the proofs of the main theorems. Our method is similar to [START_REF] Liew | Combined mean-field and semiclassical limits of large fermionic systems[END_REF] [33] [START_REF] Liew | A mixed-norm estimate of the two-particle reduced density matrix of many-body Schrödinger dynamics for deriving the Vlasov equation[END_REF] the main difference being about the phase space. Our phase space is N ˆR2 because the gap between Landau level is of the same order as the other energy contributions, and thus doesn't disappears in the limit. For comparison, when one does semi-classical approximations with a small gap between Landau levels, the phase space is the usual position momentum phase space R 2 ˆR2 .

III.2 Quantization

The usual quantization of the magnetic Laplacian gives us a basis of eigenstates indexed by two quantum numbers n and m (III.2.10), n being the index describing the Landau level. With the aim of obtaining a projector on a point in phase space, coherent states are used for a fixed n. In dimension two, the complex parameter in the definition of coherent states can be identified with a position. We thus construct (Definition III.2.11) a one-particle state in the Landau level n located at a point in space. Then, some properties of the associated projector are given.

III.2.1 Spectral analysis of the magnetic Laplacian

In symmetric gauge, the magnetic Laplacian is

L b " ˆih∇ `b 2 ˆ´x 2 x 1

˙˙2

After conjugation by e

i x 1 x 2 2l 2
b , we obtain the magnetic Laplacian in Landau gauge:

e i x 1 x 2 2l 2 b ˆih∇ `b 2 ˆ´x 2 x 1 ˙˙2 e ´i x 1 x 2 2l 2 b " ˆih∇ `b 2 ˆ´x 2 x 1 ˙`h 2l 2 b ˆx2 x 1 ˙˙2 " ˆih∇ `b ˆ0 x 1 ˙˙2 " pihB x 1 q 2 `pihB x 2 `bx 1 q 2
Using the unitary Fourier convention

Fupνq -p upνq - 1 ? 2π ż R upxqe ´iνx dx
one conjugation by the unitary Fourier transform in the variable x 2 , we obtain

F x 2 `pihB x 1 q 2 `pihB x 2 `bx 1 q 2 ˘F´1 x 2 " pihB x 1 q 2 `p´hν 2 `bx 1 q 2
Defining the translation by r P R operator in the x 1 variable T r upx 1 q -upx 1 ´rq after one last conjugation, T ´l2 b ν 2 `pihB x 1 q 2 `p´hν 2 `bx 1 q 2 ˘Tl 2 b ν 2 " pihB x 1 q 2 `pbx 1 q 2 we see that L b is unitary equivalent to the 1D harmonic oscillator. And the 1D harmonic oscillator is well known to have a compact resolvent because of the following embedding.

Proposition III.2.1

H 1 pRq X L 2 pR, |x| dxq is compactly embedded in L 2 pRq.
In this subsection, we set up the usual formalism for the description of the magnetic Laplacian in terms of annihilation and creation operators. More details about these operators and the properties of Landau levels can be found in [START_REF] Rougerie | Holomorphic quantum hall states in higher landau levels[END_REF].

Notation III.2.2

We denote by p 1 , p 2 the coordinates of the magnetic momentum: In classical mechanics, a charged particle in a 2D plane subjected to a transverse magnetic field follows a cyclotron orbit. In the previous section, we quantized the kinetic energy of the cyclotron orbit. To finish the quantization, one can try to quantize the wave functions as a function of the position of the cyclotron orbit, called the guiding center, in the 2D plane. This approach is justified by the fact that in what follows, we will need to reintroduce the potentials V and w, which depend on the position in space.

P h,b -ˆp1 p 2 ˙- ´ˆihB x 1 `

Notation III.2.8

We introduce the following position operators

r -ˆr1 r 2 ˙-P K h,b b " 1 b ˆ´p 2 p 1 Ṙ -X ´r
We can therefore define creation and annihilation operators by:

p b " R 1 ´iR 2 ? 2l b p b : " R 1 `iR 2 ? 2l b
r represents the position of particles in the center of orbit frame. The classical physics meaning of this definition is that, since particles trajectories' are orbits, their momentum is perpendicular to their relative position with respect to the center to the orbit. Electrons are describing clock orbits thus the momentum rotated of π/2 gives us r. Moreover, r is related to the quantization of the cyclotron pulsation of the orbit because

a " p 1 `ip 2 ? 2hb " r 2 ´ir 1 ? 2l b " ´ir ? 2l b a : " ir ? 2l b
From the definition of r, the position R of the orbit center is indeed

X " R `r
and related to the second harmonic oscillator

p b " R ? 2l b p b : " R ? 2l b
We have the following commutation relations:

Proposition III.2.9 r, R and X commutes with one another. Moreover

rr 1 , r 2 s " il 2 b rR 1 , R 2 s " ´il 2 b " p b, p b : ı " Id " a, p b ı " " a, p b : ı " " a : , p b ı " " a : , p b : ı " 0
Proof :

With direct computations " r R 1 , r R 2 ı " 1 b 2 r´p 2 , p 1 s " ihb b 2 " il 2 b " x, r R 1 ı " 1 b rx, ´p2 s " 0 " " y, r R 2 ı " R 1 , r R 1 ı " " x, r R 1 ı " 0 " " R 2 , r R 2 ı " R 1 , r R 2 ı " " x ´r R 1 , r R 2 ı " 1 b rx, p 1 s ´" r R 1 , r R 2 ı " i h b ´il 2 b " 0 " " R 2 , r R 1 ı rR 1 , R 2 s " " x ´r R 1 , y ´r R 2 ı " 1 b prx, ´p1 s `rp 2 , ysq `" r R 1 , r R 2 ı " ´il 2 b " p b, p b : ı " i l 2 b rR 1 , R 2 s " Id
We therefore have two independent harmonic oscillators. By successively applying the creation operators a : and p b : we obtain the desired basis. One can look at [50] [30] [49] [START_REF] Rougerie | Holomorphic quantum hall states in higher landau levels[END_REF] for some references of the following expressions. (III.2.7)

Proof :

In complex coordinates, using the computation in (III.2.6),

a " ´´2ihB x `ib x 2 ?2hb " ´?2il b B x ´i x 2 ? 2l b a : " ´?2il b B x `i x 2 ? 2l b p b " R ? 2l b " x ? 2l b ´r ? 2l b " x ? 2l b `ia : " ? 2l b B x `x 2 ? 2l b p b : " ´?2l b B x `x 2 ? 2l b
Let's determine the common ground state by solving

p bf e ´|x| 2 4l 2
b " 0

We have

? 2l b B x f `ˆ´x 2 ? 2l b `x 2 ? 2l b ˙f " ? 2l b B x f " 0 and since f is holomorphic it is constant. Then φ 0,0 -C 0,0 e ´|x| 2 4l 2 b
Choose C 0,0 such that ∥φ 0,0 ∥ L 2 " 1: Let N 1 ď N 2 , define the truncated projector

∥φ 0,0 ∥ 2 L 2 " C 2 0,0 ż R 2
Π N 1 :N 2 - N 2 ÿ n"N 1 Π n Π ďN -Π 0:N
with similar definitions for Π ąN and Π ďN,z , Π ąN,z .

We also have the following resolutions of identity [START_REF] Champel | Quantum transport properties of two-dimensional electron gases under high magnetic fields[END_REF],

Lemma III.2.12

1 2πl 2 b ż R 2 Π n,z dz " Π n (III.2.8) 1 2πl 2 b ÿ nPN ż R 2 Π n,z dz " Id (III.2.9) ż R 2 Π z dz " Id
Note that ψ n,z is localised around z since

Rψ n,z " zψ n,z

We have the following expressions for the coherent states (see [START_REF] Champel | Quantum transport properties of two-dimensional electron gases under high magnetic fields[END_REF]):

Proposition III. We also obtain the kernel of the projector From (III.2.11) we immediately see that

∇ K z Π z px, yq " y ´x il 2 b
or as operator identity

∇ K z Π z " 1 il 2 b rΠ z , Xs (III.2.13)
This formula will play a key role in the computation of the spacial derivative of the density in Section III.5. The following lemma is an approximation of (III.2.13) for the truncated projector.

Lemma III.2.14

∇ K z Π ďN,z " 1 il 2 b rΠ ďN,z , Xs ´?N `1 ? 2l b ˆ1 1 ´i i ˙ˆ|ψ N,z ⟩ ⟨ψ N `1,z | |ψ N `1,z ⟩ ⟨ψ N,z | ˙(III.2.14) ∇ K z Π n,z " 1 il 2 b rΠ n,z , Xs ´?n `1 ? 2l b ˆ1 1 ´i i ˙ˆ|ψ n,z ⟩ ⟨ψ n`1,z | |ψ n`1,z ⟩ ⟨ψ n,z | ?n ? 2l b ˆ1 1 ´i i ˙ˆ|ψ n´1,z ⟩ ⟨ψ n,z | |ψ n,z ⟩ ⟨ψ n´1,z | ˙(III.2.15)
and

∇ K z b ∇ K z Π ďN,z px, yq " ´1 l 4 b px ´yq b2 Π ďN,z px, yq `Id 2 l 2 b Π N,z px, yq ´?N `1 ? 2l b ∇K z b ˆψN,z pxqψ N `1,z pyq ψ N `1,z pxqψ N,z pyq ˙ˆ1 ´i 1 i ˙`ˆ∇ K z b ˆψN,z pxqψ N `1,z pyq ψ N `1,z pxqψ N,z pyq ˙ˆ1 ´i 1 i ˙˙T Ņ `1 2l 2 b ˆ1 1 ´i i ˙¨c N `2 N `1 ψ N,z pxqψ N `2,z pyq ψ N `1,z pxqψ N `1,z pyq ψ N `1,z pxqψ N `1,z pyq c N `2 N `1 ψ N `2,z pxqψ N,z pyq ‹ ‹ ' ˆ1 ´i 1 i (III.2.16) Proof :
Step 1: Proof of (III.

∇ K z b ˆψN,z pxqψ N `1,z pyq ψ N `1,z pxqψ N,z pyq " i ? N `1? 2l b `∇K z Π N,z px, yq ˘b ˆz-y x-z ˙`i ? N `1? 2l b Π N,z px, yq∇ K z b ˆz-y x-z "i x ´y l 2 b Π ďN,z px, yq b i ? N `1? 2l b ˆz-y x-z ?N `1 ? 2l b ˆ1 1 ´i i ˙ˆψ N,z pxqψ N `1,z pyq ψ N `1,z pxqψ N,z pyq ˙b i ? N `1? 2l b ˆz-y x-z i ? N `1? 2l b Π N,z px, yq ˆ´i ´i 1 ´1" i x ´y l 2 b b ˆψN,z pxqψ N `1,z pyq ψ N `1,z pxqψ N,z pyq ?N `1 ? 2l b ˆ1 1 ´i i ˙¨c N `2 N `1ψ N,z pxqψ N `2,z pyq ψ N `1,z pxqψ N `1,z pyq ψ N `1,z pxqψ N `1,z pyq c N `2 N `1ψ N `2,z pxqψ N,z pyq ‹ ‹ ' `1 ? N `1? 2l b Π N,z px, yq ˆ1 1 i ´i˙( III.2.17) so ∇ K z b ∇ K z Π ďN,z px, yq "∇ K z Π ďN,z px, yq b i x ´y l 2 b ´∇K z b ? N `1 ? 2l b ˆ1 1 ´i i ˙ˆψ N,z pxqψ N `1,z pyq ψ N `1,z pxqψ N,z pyq "´1 l 4 b px ´yq b2 Π ďN,z px, yq ´?N `1 ? 2l b ˆ1 1 ´i i ˙ˆψ N,z pxqψ N `1,z pyq ψ N `1,z pxqψ N,z pyq ˙b i x ´y l 2 b ´?N `1 ? 2l b ∇ K z b ˆψN,z pxqψ N `1,z pyq ψ N `1,z pxqψ N,z pyq ˙ˆ1 ´i 1 i ˙(III.2.18)
From (III.2.17) we can isolate

ˆψN,z pxqψ N `1,z pyq ψ N `1,z pxqψ N,z pyq ˙b i x ´y l 2 b " ˆ∇K z b ˆψN,z pxqψ N `1,z pyq ψ N `1,z pxqψ N,z pyq ˙˙T ´1 ? N `1? 2l b Π N,z px, yq ˆ1 i 1 ´i? N `1 ? 2l b ¨cN `2 N `1 ψ N,z pxqψ N `2,z pyq ψ N `1,z pxqψ N `1,z pyq ψ N `1,z pxqψ N `1,z pyq c N `2 N `1 ψ N `2,z pxqψ N,z pyq ‹ ‹ ' ˆ1 ´i 1 i ṫhus inserting this in (III.2.18), ∇ K z b ∇ K z Π ďN,z px, yq " ´1 l 4 b px ´yq b2 Π ďN,z px, yq ´?N `1 ? 2l b ∇ K z b ˆψN,z pxqψ N `1,z pyq ψ N `1,z pxqψ N,z pyq ˙ˆ1 ´i 1 i ?N `1 ? 2l b ˆ1 1 ´i i ˙ˆ∇ K z b ˆψN,z pxqψ N `1,z pyq ψ N `1,z pxqψ N,z pyq ˙˙T `1 2l 2 b ˆ1 1 ´i i ˙ΠN,z px, yq ˆ1 i 1 ´iṄ `1 2l 2 b ˆ1 1 ´i i ˙¨c N `2 N `1ψ N,z pxqψ N `2,z pyq ψ N `1,z pxqψ N `1,z pyq ψ N `1,z pxqψ N `1,z pyq c N `2 N `1ψ N `2,z pxqψ N,z pyq ‹ ‹ ' ˆ1 ´i 1 i "´1 l 4 b px ´yq b2 Π ďN,z px, yq `Id 2 l 2 b Π N,z px, yq ´?N `1 ? 2l b ∇K z b ˆψN,z pxqψ N `1,z pyq ψ N `1,z pxqψ N,z pyq ˙ˆ1 ´i 1 i ˙`ˆ∇ K z b ˆψN,z pxqψ N `1,z pyq ψ N `1,z pxqψ N,z pyq ˙ˆ1 ´i 1 i ˙˙T Ņ `1 2l 2 b ˆ1 1 ´i i ˙¨c N `2 N `1ψ N,z pxqψ N `2,z pyq ψ N `1,z pxqψ N `1,z pyq ψ N `1,z pxqψ N `1,z pyq c N `2 N `1ψ N `2,z pxqψ N,z pyq ‹ ‹ ' ˆ1 ´i 1 i

III.3 Properties of the time evolution

The goal of this section is to expose some basic properties of the dynamics: conservation fermionic states (Proposition III. First compute

B t H b ptqpzq " 1 2 d dt pw ‹ ρ γ b ptq qpzq " 1 2 d dt Tr rγ b ptqwp' ´zqs " 1 2il 2 b Tr "" L b `V `w ‹ ρ γ b ptq , γ b ptq ‰ wp' ´zq ‰ " 1 2il 2 b Tr "" wp' ´zq, L b `V `w ‹ ρ γ b ptq ‰ γ b ptq ‰ " 1 2il 2 b Tr rrwp' ´zq, L b s γ b ptqs so Tr rγ b ptqB t H b ptqs " ż R 2 ρ γ b ptq pzqB t H b ptqpzqdz " 1 2il 2 b ż R 2 Tr "" ρ γ b ptq pzqwp' ´zq, L b ‰ γ b ptq ‰ dz " 1 2il 2 b Tr "" w ‹ ρ γ b ptq , L b ‰ γ b ptq ‰ (III.3.1)
we can compute

Tr rB t γ b ptqH b ptqs " 1 il 2 b Tr "" L b `V `w ‹ ρ γ b ptq , γ b ptq ‰ H b ptq ‰ " 1 il 2 b Tr "" H b ptq, H b ptq `1 2 w ‹ ρ γ b ptq ȷ γ b ptq ȷ " 1 2il 2 b Tr "" H b ptq, w ‹ ρ γ b ptq ‰ γ b ptq ‰ " 1 2il 2 b Tr "" L b , w ‹ ρ γ b ptq ‰ γ b ptq ‰ (III.3.2)
Then with (III. In Proposition III.5.4 we manage to control the dynamics of a semi-classical density using only the kinetic energy. But we also present some estimates in Proposition III.5.1 and Proposition III.5.3 that only work with higher moments of the kinetic energy. Conservation of higher moments of the kinetic energy is a physical assumption. The next proposition is an attempt at controlling the 2 nd moment, however conservation could not be obtained yet due to the presence of 1{l b factor in the derivative. First we compute As operators

d dt Tr " γ b L 2 b ‰ " 1 il 2 b Tr " rL b `W, γ b s L 2 b ‰ " 1 il 2 b Tr "" L 2 b , W ‰ γ b ‰ (III.
∆W L b ´Lb ∆W ď ϵ |∆W | 2 `1 ϵ L 2 b ď ϵ ∥∆W ∥ 2 L 8 `1 ϵ L 2 b (III.
"" L 2 b , W ‰ γ b ‰ˇˇď h2 ∥∆W ∥ L 8 ˆhb `1 hb Tr " γ b L 2 b ‰ ˙`2h ∥∇W ∥ L 8 b Tr rγ b L 2 b s Tr rγ b L b s Recalling that L b ě hb " op1q we get ˇˇTr "" L 2 b , W ‰ γ b ‰ˇˇď ˆh 2 hb ∥∆W ∥ L 8 `2h ? hb ∥∇W ∥ L 8 ˙Tr " γ b L 2 b ‰ " `l2 b ∥∆W ∥ L 8 `2l b ∥∇W ∥ L 8 ˘Tr " γ b L 2 b ‰ We conclude with (III.3.3). then @φ P C 8 c pR 2 q, ˇˇˇˇˇż R 2 φ `ργ b ´ρsc γ b ˘ˇˇˇˇˇď C a |supppφq| ∥φ∥ W 1,8 b Tr " γ b L k b ‰ ¨$ ' ' ' ' & ' ' ' ' % l 2k´3 b if k ă 2 l b d ln ˆ1 l 2 b ˙if k " 2 l b if k ą 2
Notice that this estimate requires k ą 3 2 for the error on the right side to be be small.

Proof of Proposition III.4.2:

Let φ P C 8 c pR 2 q, ż R 2 φ `ργ b ´ρsc γ b ˘"Tr rφγ b s ´1 2πl 2 b ż R 2 φpzqTr rΠ z γ b s dz " 1 2πl 2 b ż R 2 Tr rpφ ´φpzqq Π z γ b s dz " 1 2πl 2 b ż R 2 ÿ nPN Tr " pφ ´φpzqq Π 2 n,z γ b ‰ dz (III.4.4) so with Young's inequality ˇˇˇˇˇż R 2 φ `ργ b ´ρsc γ b ˘ˇˇˇˇˇď 1 4πl 2 b ż R 2 ÿ nPN ˆ1 ϵ n Tr " |φ ´φpzq| 2 Π n,z ‰ `ϵn Tr " Π n,z γ 2 b ‰ ˙dz " ÿ nPN 1 4πϵ n l 2 b ż R 2 Tr " |φ ´φpzq| 2 Π n,z ‰ dz `ÿ nPN ϵ n 2 Tr " Π n γ 2 b ‰ ď ÿ nPN 1 4πϵ n l 2 b ż R 2 Tr " |φ ´φpzq| 2 Π n,z ‰ dz `ÿ nPN πϵ n l 2 b Tr rΠ n γ b s (III.4.5)
and with the change of variable x -

x ´z ? 2l b Tr " |φ ´φpzq| 2 Π n,z ‰ " ż R 2 |φpxq ´φpzq| 2 Π n,z px, xqdx " 1 2πn!l 2 b ż R 2 |φpxq ´φpzq| 2 ˇˇˇx ´z ? 2l b ˇˇˇ2 n e ´|x´z| 2 2l 2 b dx " 1 πn! ż R 2 ˇˇφ ´z `?2l b x ¯´φpzq ˇˇ2 |x| 2n e ´|x| 2 dx ď 1 πn! 2l 2 b ∥∇φ∥ 2 8 ż R 2 |x| 2pn`1q e ´|x| 2 dx " 2pn `1ql
ÿ nPN pn `1q k Tr rγ b Π n s " ÿ nPN ¨n `1 2hb ˆn `1 2 ˙‹ ‹ ' k Tr " γ b Π n L k b ‰ ď ÿ nPN Tr " γ b Π n L k b ‰ " Tr " γ b L k b ‰ (
R 2 φ `ργ b ´ρsc γ b ˘ˇˇˇˇď C ˜|supppφq| ∥φ∥ 2 W 1,8 ÿ nPN 1 ϵ n min ˆn `1, 1 l 2 b ˙`l 2 b ÿ nPN ϵ n Tr rΠ n γ b s ḑC a |supppφq| ∥φ∥ W 1,8 ˜ÿ nPN 1 ϵpn `1q k min ˆn `1, 1 l 2 b ˙`ϵl 2 b Tr " γ b L k b ‰ ḑC a |supppφq| ∥φ∥ W 1,8 ¨ÿ pn`1ql 2 b ď1 1 ϵpn `1q k´1 `1 ϵl 2 b ÿ pn`1ql 2 b ą1 1 pn `1q k `ϵl 2 b Tr " γ b L k b ‰ ' (III.4.10)
Introducing the notation p λ pxq -x ´λ1 λă0 `lnpxq1 λ"0 `1λą0 (III. 4.11) Recalling the notation in (III. 4.11),

N ÿ n"0 1 pn `1q k´1 " $ ' & ' % O `N 2´k ˘if k ă 2 O plnpN qq if k " 2 O p1q if k ą 2 " O pp k´2 pN qq With ϵ - 1 l b d p k´2 pN q Tr " γ b L k b ‰ we have 1 2πl 2 b N ÿ n"0 ż R 2 |Tr rpφ ´φpzqq Π n,z γ b s| dz ď C a |supppφq| ∥φ∥ W 1,8 b Tr " γ b L k b ‰ l b a p k´2 pN q (III.
Π n s ď ÿ nąN n k N k Tr rγ b Π n s ď 1 N k ÿ nąN Tr " γ b Π n L k b ‰ " 1 N k Tr " γ b Π ąN L k b ‰ so ˇˇˇˇÿ nąN Tr rφΠ n γ b s ˇˇˇˇď ∥φ∥ L 8 N ´k 2 b Tr " γ b Π ąN L k b ‰ (III.4.19)
We conclude by assembling (III.4.17), (III.4.18) and (III.4.17).

III.5 Gyrokinetic dynamics for semi-classical densities

The goal of this section if to prove that the semi-classical densities almost satisfies (III.1.5).

B t γ b ptq " 1 il 2 b rL b `W, γ b ptqs (III.5.2) then @φ P C 8 c pR 2 q, ˇˇˇˇˇż R 2 φpzq `Bt ρ sc γ b pt, zq `∇K W pzq ¨∇ρ sc γ b pt, zq ˘dz ˇˇˇˇď C a |supppφq| ∥φ∥ W 1,8 ∥W ∥ W 3,8 b Tr " γ b ptqL k b ‰ ¨$ ' ' ' ' & ' ' ' ' % l 2k´7 b if k ă 4 l b d ln ˆ1 l 2 b ˙if k " 4 l b if k ą 4 (III.5.3)
This estimate requires k ą 7{2 for the right-hand side to be small. We will later in the text improve this constraint, but we start by Proposition III.5.1 in order to introduce our method of proof. We begin the proof with a technical Lemma.

Lemma III.5.2

Let α P R `, then

I n pαq - 1 πn! ż R 2 |x| α`2n e ´|x| 2 dx " nÑ8 n α 2
and DC ą 0 such that @n P N, I n pαq ď Cpn `1q α 2

Proof :

With polar coordinates and the change of variable

x -x 2 1 π ż R 2 |x| α e ´|x| 2 dx " 2 ż R `xα`1 e ´x2 dx " ż R `x α 2 e ´udu " Γ ´α 2 `1w
here Γ is the Euler Gamma function,

Γpzq - ż R `tz´1 e ´tdt
We have the following equivalent for the Euler Gamma function (as a direct consequence of the Stirling formula)

Γpn `xq Γpnq " xÑ8 n x (III.5.4) so I n pαq " 1 n! Γ ´n `α 2 `1¯" Γ ´n `α 2 `1Γ pn `1q
and by (III.5.4),

I n pαq " nÑ8 pn `1q α 2 " nÑ8 n α 2
Proof of Proposition III.5.1:

Step 1: a direct computation

We start from (III.4.2) and (III.5.2):

B t ρ sc γ b pzq " 1 2πl 2 b Tr rB t γ b Π z s " 1 2iπl 4 b Tr rrL b `W, γ b s Π z s " 1 2iπl 4 b Tr rγ b rΠ z , L b `W ss " 1 2iπl 4 b Tr rγ b rΠ z , W ss (III.5.5)
On the other hand, by (III.2.13) and (III.4.2), Step 2: Taylor expansion of the potential

∇ K W pzq ¨∇ρ sc γ b pzq " 1 2πl 2 b ∇ K W pzq ¨Tr rγ b ∇ z Π z s " ´1 2πl 2 b ∇W pzq ¨Tr " γ b ∇ K z Π z ‰ " ´1 2iπl
We expand W to the second order. Define We introduce another projector:

V z pxq -W pxq ´dW pzqpx ´zq ´1 2
, V z s `1 2 d 2 W pzqprΠ z , Xs , X ´zq `1 2 d 2 W pzqpX ´z, rΠ z , Xsq " rΠ z , V z s `il 2 b 2 d 2 W pzqp∇ K z Π z , X ´zq `il 2 b 2 d 2 W pzqpX ´z, ∇ K z Π z q (III.5.10) Defining the error terms E 1,b pzq - 1 2iπl 4 b Tr rγ b rΠ z , V z ss (III.5.11) E 2,b pzq - 1 4πl 2 b Tr " γ b d 2 W pzqp∇ K z Π z , X ´zq ‰ (III.5.12) r E 2,b pzq - 1 4πl 2 b Tr " γ b d 2 W pzqpX ´z, ∇ K z Π z q ‰ (III.5.
Tr rγ b rΠ n,z , V z ss " Tr rγ b Π n,z rΠ n,z , V z ss `Tr rΠ n,z γ b rΠ n,z , V z ss Since Π n,z , V z , γ b are self-adjoint, with Young's inequality |Tr rγ b rΠ n,z , V z ss| ď2 b Tr " rΠ n,z , V z s rΠ n,z , V z s ˚‰ Tr " γ b Π n,z pγ b Π n,z q ˚‰ "2 ∥rΠ n,z , V z s∥ L 2 b Tr rγ 2 b Π n,z s ď ϵ n ∥rΠ n,z , V z s∥ 2 L 2 `1 ϵ n Tr " γ 2 b Π n,z ‰ ďϵ n ∥rΠ n,z , V z s∥ 2 L 2 `2πl 2 b ϵ n Tr rγ b Π n,z s (III.5.16)
We estimate the first term with the changes of variables x -

x ´z ? 2l b , y - y ´z ? 2l b ∥rΠ n,z , V z s∥ 2 L 2 " ij R 2 ˆR2 |rΠ n,z , V z s px, yq| 2 dxdy " ij R 2 ˆR2 pV z pxq ´Vz pyqq 2 |Π n,z px, yq| 2 dxdy " 1 p2πn!l 2 b q 2 ż R 2 ˆR2
pV z pxq ´Vz pyqq 2 ˇˇˇp x ´zq py ´zq

∥rΠ n,z , V z s∥ 2 L 2 ďC ∥W ∥ 2 W α,8 n! 2 ż R 2 ˆR2 l 2α b `|x| 2α `|y| 2α ˘|xy| 2n e ´|x| 2
˜∥W ∥ 2 W 3,8 ∥φ∥ L 1 ÿ nPN ϵ n min 2ďαď3 ``pn `1ql 2 b ˘α˘`∥ φ∥ L 8 ÿ nPN 2πl 4 b ϵ n Tr rγ b Π n s ḑC ∥φ∥ L 8 ˜|supppφq| ∥W ∥ 2 W 3,8 ÿ nPN ϵ n l 4 b min 2ďαď3 ``pn `1ql 2 b ˘α˘`ÿ nPN 1 ϵ n Tr rγ b Π n s ḑC a |supppφq| ∥φ∥ L 8 ∥W ∥ W 3,8 ˜ϵ ÿ nPN min 2ďαď3 `ppn `1ql 2 b q α l4 b pn `1q k `1 ϵ ÿ nPN pn `1q k Tr rγ b Π n s ḑC a |supppφq| ∥φ∥ L 8 ∥W ∥ W 3,8 ˜ϵ ÿ nPN min 2ďαď3 `ppn `1ql 2 b q α l4 b pn `1q k `1 ϵ Tr " γ b L k b ‰
We split the sum in two part depending on weather pn `1ql 2 b ď 1 or not, for the first part take α " 3, and for the second part α " 2:

ˇˇˇˇˇż R 2 φpzqE 1,b pzqdz ˇˇˇˇˇď C a |supppφq| ∥φ∥ L 8 ∥W ∥ W 3,8 ¨ϵl 2 b ÿ pn`1ql 2 b ď1 1 pn `1q k´3 `ϵ ÿ pn`1ql 2 b ą1 1 pn `1q k´2 `1 ϵ Tr " γ b L k b ‰ ' (III.5.19)
With the asymptotics (III.4.12) and (III.4.13), ˇˇˇˇˇż Hence choosing ϵ -

R 2 φpzqE 1,b pzqdz ˇˇˇˇˇď C a |supppφq| ∥φ∥ L 8 ∥W ∥ W 3,8 ˆϵl 2 b p k´4 `l´2 b ˘`ϵl 2pk´3q
1 l b d Tr " γ b L k b ‰ p k´4 `l´2 b ˘we conclude that ˇˇˇˇˇż R 2 φpzqE 1,b pzqdz ˇˇˇˇˇď C a |supppφq| ∥φ∥ L 8 ∥W ∥ W 3,8 ˆϵl 2 b p k´4 `l´2 b ˘`1 ϵ Tr " γ b L k b ‰ "C a |supppφq| ∥φ∥ L 8 ∥W ∥ W 3,8 b Tr " γ b L k b ‰ l b b p k´4 `l´2 b ˘(III.5.20)
Step 4: Estimate of E 2,b pzq

We start from (III.5.12):

E 2,b pzq - 1 4πl 2 b ij R 2 ˆR2 γ b px, yq∇ K z Π z py, xq ¨d2 W pzqpx ´zqdxdy
Let d denote the tensor contraction defined for n, m ě k by

u 1 b ¨¨¨b u n d k v 1 b ¨¨¨b v m -⟨u n |v 1 ⟩ . . . ⟨u n´k`1 |v k ⟩ u 1 b ¨¨¨b u n´k b v k`1 b ¨¨¨b v m
Identifying d n W pzq with the associated rank n tensor, we notice that

∇ K z ¨d2 W pzqpx ´zq " p∇ d ∇ b ∇W pzqq px ´zq `∇ b ∇W pzq d 2 ∇ K z b px ´zq "d 2 W pzq d 2 ˆ0 1 ´1 0 ˙" 0 because d 2 W pzq is symmetric and ˆ0 1 ´1 0 ˙anti-symmetric. An integration by parts yields ż R 2 φpzqE 2,b pzqdz " ´1 4πl 2 b ż R 2 ∇ K φpzq ¨d2 W pzq ¨ij R 2 ˆR2 γ b px, yqΠ z py, xqpx ´zqdxdy 'dz " ´1 4πl 2 b ż R 2 ∇ K φpzq ¨d2 W pzqTr rγ b Π z pX
C ˜ d 2 W ∇ K φ L 1 ÿ nPN ϵ n pn `1q ` d 2 W ∇ K φ L 8 ÿ nPN l 2 b ϵ n Tr rγ b Π n s ḑC a |supppφq| d 2 W ∇ K φ L 8 ˜ϵ ÿ nPN 1 pn `1q k´1 `l2 b ϵ ÿ nPN pn `1q k Tr rγ b Π n s ḑC a |supppφq| ∥∇φ∥ L 8 d 2 W L 8 ˆϵ `l2 b ϵ Tr " γ b L k b ‰ ˙(III.5.23) Choosing ϵ - l b b Tr " γ b L k b ‰ we conclude that ˇˇˇˇˇż R 2 φpzqE 2,b pzqdz ˇˇˇˇˇď C a |supppφq| ∥∇φ∥ L 8 d 2 W L 8 b Tr " γ b L k b ‰ l b (III.5.24)
Step 5: Conclusion

We can control (III.5.13) with an estimate similar to (III.5.12) by exchanging x and y, obtaining

ˇˇˇˇˇż R 2 φpzq r E 2,b pzqdz ˇˇˇˇˇď C a |supppφq| ∥∇φ∥ L 8 d 2 W L 8 b Tr " γ b L k b ‰ l b (III.5.25)
Finally, using the notation (III.4.11),

1 ď p k´4 `l´2 b ˘" $ ' & ' % l 2pk´4q b if k ă 4 ln `l´2 b ˘if k " 4 1 if k ą 4
so with (III.5.20) (III.5.24) (III.5.25) and (III.5.14) we obtain (III.5.3).

This proof can be adapted for the estimate to work with k ą 2 if we expand W to the second order in (III.5.8) for low Landau levels only, precisely when pn `1ql 2 b ď 1. But this idea works with k " 1 when we consider the dynamics of the truncated semi-classical density (III.4.3) instead of (III.4.2). This is the goal of Proposition III.4.2 and Proposition III.4.3.

III.5.2 Dynamics of the truncated semi-classical density

Let t P R `, k ě 0, γ b ptq P L 1 pL 2 pR 2 qq , W P W 3,8 pR 2 q and assume Tr rγ b ptqs "

1, 0 ď γ b ptq ď 2πl 2 b B t γ b ptq " 1 il 2 b rL b `W, γ b ptqs then @φ P C 8 c pR 2 q, ˇˇˇˇˇż R 2 φpzq `Bt ρ ďN γ b pt, zq `∇K W pzq ¨∇ρ ďN γ b pt, zq ˘dz ˇˇˇˇď C ∥φ∥ L 8 ∥∇W ∥ L 8 1 l b N k´1 2 Tr " γ b L k b pΠ N `ΠN`1 q ‰ `Ca |supppφq| ∥φ∥ W 1,8 ∥W ∥ W 3,8 p N 1´k 2 b Tr " γ b L k b pΠ N `ΠN`1 q ‰ `bTr " γ b ptqL k b ‰ ¨$ ' & ' % l b N 2´k 2 if k ă 4 l b a lnpN q if k " 4 l b if k ą 4
‹ ' (III.5.26)

Proof :

Step 1: a direct computation Using (III.2.14) in (III.5.6),

∇ K W pzq ¨∇z ρ ďN γ b pzq " ´1 2πl 2 b ∇W pzq ¨Tr " γ b ∇ K z Π ďN,z ‰ " ´1 2iπl
rΠ ďN,z , W ´X ¨∇W pzqs " rΠ ďN,z , V z s `1 2 d 2 W pzqprΠ ďN,z , Xs , X ´zq `1 2 d 2 W pzqpX ´z, rΠ ďN,z , Xsq " rΠ ďN,z , V z s `il 2 b 2 d 2 W pzqp∇ K z Π ďN,z , X ´zq `il 2 b 2 d 2 W pzqpX ´z, ∇ K z Π ďN,z q `i? N `1l b 2 ? 2 d 2 W pzq ˆˆ1 1 ´i i ˙ˆ|ψ N,z ⟩ ⟨ψ N `1,z | |ψ N `1,z ⟩ ⟨ψ N,z | ˙, X ´zi ? N `1l b 2 ? 2 d 2 W pzq ˆX ´z, ˆ1 1 ´i i ˙ˆ|ψ N,z ⟩ ⟨ψ N `1,z | |ψ N `1,z ⟩ ⟨ψ N,z | ˙˙(III.5.28)
Defining the error terms

E 0,b pzq - ? N `1 2 ? 2πl 3 b ∇W pzq ¨ˆ1 1 ´i i ˙ˆTr rγ b |ψ N,z ⟩ ⟨ψ N `1,z |s Tr rγ b |ψ N `1,z ⟩ ⟨ψ N,z |s ˙(III.5.29) E 1,b pzq - 1 2iπl 4 b Tr rγ b rΠ ďN,z , V z ss (III.5.30) E 2,b pzq - 1 4πl 2 b Tr " γ b d 2 W pzq `∇K z Π ďN,z , X ´z˘‰ (III.5.31) r E 2,b pzq - 1 4πl 2 b Tr " γ b d 2 W pzq `X ´z, ∇ K z Π ďN,z ˘‰ (III.5.32) E 3,b pzq - ? N `1 4 ? 2πl 3 b Tr " γ b d 2 W pzq ˆˆ1 1 ´i i ˙ˆ|ψ N,z ⟩ ⟨ψ N `1,z | |ψ N `1,z ⟩ ⟨ψ N,z | ˙, X ´z˙ȷ (III.5.33) r E 3,b pzq - ? N `1 2 ? 4πl 3 b Tr " γ b d 2 W pzq ˆX ´z, ˆ1 1 ´i i ˙ˆ|ψ N,z ⟩ ⟨ψ N `1,z | |ψ N `1,z ⟩ ⟨ψ N,z | ˙˙ȷ (III.5.34)
and inserting (III.5.28) in (III.5.27) we have

B t ρ ďN γ b `∇K W ¨∇ρ ďN γ b " E 0,b `E1,b `E2,b `r E 2,b `E3,b `r E 3,b (III.5.35)
Step 2: Estimate of E 0,b

As operators, with Young's inequality

2 ||ψ N `1,z ⟩ ⟨ψ N,z || ď Π N,z `ΠN`1,z 2 ||ψ N,z ⟩ ⟨ψ N `1,z || ď Π N,z `ΠN`1,z so |E 0,b pzq| ď C ∥∇W ∥ L 8 ? N l 3 b pTr rγ b Π N `1,z s `Tr rγ b Π N,z sq and ˇˇˇˇˇż R 2 φpzqE 0,b pzqdz ˇˇˇˇˇď C ∥φ∥ L 8 ∥∇W ∥ L 8 ? N l b pTr rγ b Π N s `Tr rγ b Π N `1sq (III.5.36) Note that Tr " γ b L k b Π N ‰ " ˆ2hb ˆN `1 2 ˙˙k Tr rγ b Π N s " N Ñ8 2 k N k Tr rγ b Π N s Tr " γ b L 2 b Π N `1‰ " N Ñ8 2 k N k Tr rγ b Π N `1s
Inserting this in (III.5.36) we have ˇˇˇˇˇż

R 2 φpzqE 0,b pzqdz ˇˇˇˇˇď C ∥φ∥ L 8 ∥∇W ∥ L 8 1 l b N k´1 2 Tr " γ b L k b pΠ N `ΠN`1 q ‰ (III.5.37)
Step 3: Estimate of E 1,b

Following the same proof as for (III.5.11) we obtain instead of (III.5.19):

ˇˇˇˇˇż R 2 φpzqE 1,b pzqdz ˇˇˇˇˇď C a |supppφq| ∥φ∥ L 8 ∥W ∥ W 3,8 ˜ϵl 2 b N ÿ n"0 1 pn `1q k´3 `1 ϵ Tr " γ b L k b ‰ With the asymptotic (III.4.13), taking ϵ - 1 l b d Tr " γ b L k b ‰ p k´4 pN q we have ˇˇˇˇˇż R 2 φpzqE 1,b pzqdz ˇˇˇˇˇď C a |supppφq| ∥φ∥ L 8 ∥W ∥ W 3,8 ˆϵl 2 b p k´4 pN q `1 ϵ Tr " γ b L k b ‰ "C a |supppφq| ∥φ∥ L 8 ∥W ∥ W 3,8 b Tr " γ b L k b ‰ l b a p k´4 pN q (III.5.38)
Step 4: Estimate of E 2,b

Following the same proof as for (III.5.12) we obtain instead of (III.5.23)

ˇˇˇˇˇż R 2 φpzqE 2,b pzqdz ˇˇˇˇˇď C a |supppφq| ∥∇φ∥ L 8 d 2 W L 8 ˜ϵ N ÿ n"0 1 pn `1q k´1 `l2 b ϵ Tr " γ b L k b ‰ With the asymptotic (III.4.13), taking ϵ -l b d Tr " γ b L k b ‰ p k´2 pN q we have ˇˇˇˇˇż R 2 φpzqE 2,b pzqdz ˇˇˇˇˇď C a |supppφq| ∥∇φ∥ L 8 d 2 W L 8 ˆϵp k´2 pN q `l2 b ϵ Tr " γ b L k b ‰ "C a |supppφq| ∥∇φ∥ L 8 d 2 W L 8 b Tr " γ b L k b
‰ l b a p k´2 pN q (III.5.39)

Step 5: Estimate of E b which is a possible choice of N . Hence we will be able to control the dynamics of the truncated semi-classical density for k ą 1. One could conclude the proof of the main results with the

if k " 2 l b `l2 b N 5´k 2 if 2 ă k ă 5 l b `l2 b a lnpN q if k " 5 l b if k ą 5 ‹ ‹ ‹ ‹ ‹ ‹ '
where cpφ, W q -p1 `|supppφq|q ∥φ∥ W 1,8 ∥W ∥ W 4,8

Proof :

Step The first term is of the same form as (III.5.46), for the second we proceed to an integration by parts and obtain two terms of the form (III.5.33). The last two terms are also of the form (III.5.33). So using (III. Finally, since GYRO ργ b pφq P C 8 c pR 2 q , supp `GYRO ργ b pφq ˘Ă supppφq, and @t P r0, T s, GYRO ργ b pφqpt, 'q W 1,8 " B t φpt, 'q `∇K pV `w ‹ ρ γ b ptq q ¨∇φptq W 1,8 The truncated semi-classical densities are bounded in M pr0, T s ˆR2 q so up to a subsequence we have weak star convergence in the sense of measures:

ď ∥φ∥ W 1,8 `1 ` V `w ‹ ρ γ b ptq W 1,8 ď ∥φ∥ W 1,8 p1 `∥V ∥ W 1,8 `∥w∥ W 1,
ρ sc,ďN γ b á bÑ8
ρ T P M `r0, T s ˆR2 By uniqueness of the limit T 2 ě T 1 ùñ ρ T 2 |r0,T 1 s " ρ T 1 hence we constructed a limit ρ P M pr0, T s ˆR2 q such that @φ P C 0 c pR `ˆR 2 q, By [16, 1. theorem 6], the embedding MpΩ n q Ă W ´1,1 pΩ n q is compact. So after a diagonal extraction we can have @n P N, ρ sc,ďN γ b Ñ bÑ8 ρ n P W ´1,1 pΩ n q but W 1,8 pΩ n q Ă C 0 pΩ n q, so ρ n " ρ |Ωn . Hence, @n P N, ρ ´ρsc,ďN ´ρ˘G YRO ρ pφq (III.6.10)

Step 3: `∇z φ ¨ρsc,ďN

γ b ˘‹ ∇ K w P W 1,8 pR `ˆR 2 q
The goal of this part is to prove that `∇z φ ¨ρsc,ďN

γ b ˘‹ ∇ K is bounded in W 1,

u

  τ j,k piq " ÿ σPS N ϵ ´pσqu τ j,k σpiq " ÿ σPS N ϵ ´pτ j,k σqu σpiq " ´N â ȋ"i u i donc si u 1:N est liée, le produit antisymétrique est nul et si u 1:N est orthonormale, on peut calculer la norme N â ȋ"i

  où k 1:d est un vecteur d'onde dans le réseau réciproque de la maille r0, Ls d : k 1:d P 1 L Z d (I.1.5) En sommant les N plus petites valeurs propres π 2 |k 1:d | 2 de l'énergie cinétique, on obtient O ´N 1`d 2 comme ordre de grandeur. Pour ramener le terme d'énergie cinétique au même ordre que les contributions du potentiel extérieur et des interactions et dans (I.1.3) on prend un régime où la constante de Planck réduite est petite. Explicitement h -N ´1 d (I.1.6) ainsi par (I.1.4),

2 ˆ´y x '" ¨0 0 1 '

 x1 Le potentiel vecteur A engendre le champ magnétique ∇ ^A " ∇ ^¨´y x 0 En présence d'un champ magnétique, on remplace le Laplacien par le Laplacien magnétique, (I.1.1) devient alors

sur L 2

 2 pR 2 q b ´N . On suppose que le potentiel extérieur est confinant V pxq Ñ |x|Ñ8 8 et que le potentiel d'interaction est le potentiel Coulombien : wpx ´yq " 1 |x ´y| L'hypothèse du potentiel confinant empêche les particules de s'échapper à l'infini. Concernant l'effet Hall quantique, les particules sont confinées dans des puits quantiques, par exemple grâce à des semi-conducteurs. On définit la longueur magnétique l b -c h b

Notation I. 3 . 1 : 1

 311 Définition du régimeOn considère une limite fort champ magnétique Soit γ P L 8 pR `, L 1 pL 2 pR 2 qq, tel que Tr rγp0qs " 1 and 0 ď γp0q ď 2πl 2 b (I.3.6)

3 . 7 ) 2 ρ 2 b

 3722 La contrainte (I.3.6) se propage en temps (voir Proposition III.3.1). De plus ż R γ b ptq " Tr rγ b ptqs Donc le principe de Pauli γ b ď 2πl 2 b garantit que le système occupe un volume d'ordre 1 dans la limite. En effet, un état fermionique typique satisfaisant (I.3.6) est une projection sur un produit antisymétrique de N fonctions orthogonales avec N -O ˆ1 2πl Un tel état à N particules occupe un volume d'ordre

Theorem I. 3 . 2 : 2 b

 322 Gyro-cinétique de la solution de l'équation de Hartree Soit γ b P L 8 pR `, L 1 pL 2 pR 2 qqq une solution de (I.3.7) telle que Tr rγ b p0qs " 1, 0 ď γ b p0q ď 2πl Tr " γ b p0q ˆLh,b `V `1 2 w ‹ ρ γ b p0q ˙ȷ ă 8

ˇˇˇĎu

  Théorème de Wick peut être déduit le principe de Pauli pour la k ième matrice densité réduite. Theorem I.4.3: Principe de Pauli Soit γ N P L 1 pH b ˘N q tel que Tr rγ N s " 1 et γ N ě 0, 0 ď γ pkq N ď pN ´kq!k! N ! De plus la borne supérieure est atteinte pour produit symétriques/antisymétriques.

Theorem II. 1 . 5 :

 15 Mean field limit with magnetic periodic conditions With the definitions introduced in Notation II.1.1, Notation II.1.2, Notation II.1.3, Notation II.1.4,

  Lemma II.2.6 L h,b defines the Sobolev space pDom pL h,b q , x'y L q with xχ|ψy L -xL h,b χ|ψy which is equivalent to pH 2 mp pΩq, x'y H 1 q. The quadratic form defined by x'y L is continuous, Hermitian and coercive on Dom pL h,b q.

  section 1] as a reference. Proposition II.2.10: Zeros of LLL wave-functions If ψ P LLL, then ψ has exactly d zeros inside Ω. Proof : Let ψf e ´ϕ l 2

  29, Chapter V Theorem 8]. Proposition II.2.11: Degeneracy of Landau levels Landau levels have a finite degeneracy and @n P N, Dim(nLL) " d Proof :

"

  d1 p"k pmod dq (II.3.5)

3 . 4 :

 34 Convergence of the integral kernelThe kernel (II.3.4) satisfies Π n pz, zq "

αq| m e ´cpa`αqq 2 ď 8 ż 1 p2 `2αuq m e ´cpa`αuq 2 ď ż R

 81ż p2 `2uq m e ´cpa`uq 2 du ď Cpc, mq the term for q " 1 is α |a `b `α| m e ´cpa`αq 2 ď Cfor the negative q, we see thatα ÿ qď´1 |a `b `αq| m e ´cpa`αqq 2 " α ÿ qě1|´a ´b `αq| m e ´cp´a`αqq 2 ď Cpc, mq because ´a, ´b P r´1, 1s.

  With the variational principle and the Lieb-Thirring inequality (II.4.20), Tr rpL h,b `1q γs "Tr rpL h,b `1 ´Vq γs `Tr rVγs ě ∥γ∥ L 8 Tr " 1 pLh,b`1´Vqď0 pL h,b `1 ´Vq ı `Tr rVγs ě ´CLT ∥γ∥ L 8 h2

ργ ďc ρ γ 2 (

 2 and we get Tr rpL h,b `1q γs ě h2 4C LT ∥γ∥ L 8 ż II.4.22) Since L h,b ě hb, Tr rL h,b γs ě hbTr rγs so because hb Ñ 8, Tr rpL h,b `1q γs ď ˆ1 `1 hb ˙Tr rL h,b γs ď CTr rL h,b γs

  Opl b qq Then, with the resolution of identity (II.3.3) we have ż pNˆΩq k m γ k dη bk " Tr rγ k s Since @X P N ˆΩ, Π X and m k are positive γ m k is also positive. (II.3.3) also implies

NˆΩΩΩΩ ∆g λ p' ´Rqg λ p' ´RqdR " ´ż Ω p∇g λ q 2 " ´λ4 ż Ω p∇gpλxqq 2 dx " ´λ2 ∥∇g∥ 2 L 2 (ψ

 222 1: Kinetic energy With (II.4.4) in mind, we start by computing the kinetic term. Inserting the resolution of identity (II.3.3), we have Tr " b g λ p' ´RqΠ n g λ p' ´Rqγ p1q ψ N ı dηpn, Rq Now, we use the diagonalization of the magnetic Laplacian L h,b Π n " E n Π n by commuting L h,b with g λ p' ´Rq to obtain Tr " rL h,b , g λ p' ´Rqs Π n g λ p' ´RqdR rL h,b , g λ p' ´Rqs g λ p' ´RqdR fi fl (II.5.12) We compute rP h,b , g λ p' ´Rqs " ih∇g λ p' ´Rq and rL h,b , g λ p' ´Rqs " rP h,b , g λ p' ´Rqs ¨Ph,b `Ph,b ¨rP h,b , g λ p' ´Rqs " 2ih∇g λ p' ´Rq ¨Ph,b ´h 2 ∆g λ p' ´Rq " P h,b ¨2ih∇g λ p' ´Rq `h 2 ∆g λ p' ´Rq (II.5.13) inserting this in (II.5.12), we find Tr " ∆g λ p' ´Rqg λ p' ´RqdR fi fl But because g has a fixed L 2 norm and is periodic ∇ ż Ω g λ p' ´Rq 2 dR " 0 " 2 ż Ω ∇g λ p' ´Rqg λ p' ´RqdR Moreover ż N pn, Rqdηpn, Rq ´phλq 2 ∥∇g∥ 2 L 2

Notation II. 6 . 3 :

 63 Hartree Fock theory Let s, t, u, v P L 2 pΩq, if one define the exchange operator on L 1 pL 2 pΩ 2 qq as Ex |s b t⟩ ⟨u b v| -|s b t⟩ ⟨v b u| (II.6.1) Let γ P L 1 pL 2 pΩqq, define γ 2 -N N ´1 p1 ´Exq γ b2 (II.6.2) Define the Hartree-Fock energy E HF rγs -Tr rpL h,b `V q γs `Tr rwγ 2 s (II.6.3) With Wick's theorem definitions (II.6.2) and (II.6.3) are actual statements for Slater states.

  HF rγs `ˆ1 `Tr rL h,b γs hb ˙O pl b q " E HF rγs `Tr rL h,b γs hb O pl b q

3 . 6 :

 36 Tr rL h,b γ m s "2πl 2 b ż NˆΩ mpXqTr rL h,b Π X s dηpXq rrL h,b , g λ p' ´Rqs Π n g λ p' ´Rqs dηpn, Rq " ż NˆΩ E n mpXqdηpXq `Ophbl b q `2πl

  rL h,b γ m s `ˆ1 `1 4ϵh 2 ˙Tr rγ m s Ẇith (II.6.16), Tr rL h,b γ m s " O phbq and using Lemma II.5.2, Tr rγ m s " ∥m∥ L 1 `Opl b q so the choice of ϵ is the same as in (II.6.11) thus Tr " wExγ b2 m ‰ " Opl b q

E 2 b

 2 sc,hb r r ms " E sc,hb rms `O phbl b q `O phb p1 ´∥m∥ L 1 qq (II.6.19) Proof : First, by Corollary II.3.6, DE : N ˆΩ Ñ R such that 2πl Tr rΠ X s " 1 `lb EpXq |Epn, Rq| ď Cpnq So Tr rγ m s " ż NˆΩ mpXq p1 `lb EpXqq dηpXq If Tr rγ m s " 1 then m has the desired properties.

  , n 1 q " Tr rγ m s ă 1 and Tr rγ r m s is Lipschitz in τ , so by the intermediate value theorem we can conclude Dτ ě 0 such that if we define r mr mpτ, n 1 q lb EpXqq dηpXq " 1

"so

  Tr rγ m s ą 1 and Tr rγ Dτ ě 0 such that if one defines r mr mpτ q mq dη " ∥m∥ L 1 ´1 `lb

ď 1 pL 2

 12 pq `rqq k (II.6.28) b) M p1q pq, 'q P D qLL and M p1q pn, 'q " 1 năq 1 L 2 pq `rq `1n"q M p1q pq, 'q' (II.6.29) c) M pkq are the reduced densities of a symmetric measure M on pN ˆΩq N and M pkq L 1 " 1 d) in the sense of Radon measures have convergence of the potential terms:

6 )" 1 and 0 ď γp0q ď 2πl 2 b(

 62 Let γ P L 8 pR `, L 1 pL 2 pR 2 qq, such that Tr rγp0qs III.1.7)

1 . 9 ) 2 ρ0 2 b

 1922 Due to the constraint (III.1.7) known to propagate in time (see Proposition III.3.1), from (III.1.2) we see that ż R γ b ptq " Tr rγ b ptqs Moreover the Pauli principle γ b ď 2πl 2 b guarantees that the system occupies a volume of order 1 in the limit l b Ñ bÑ8 Indeed it is known, [22] or [46, subsection I.4], that the degeneracy per area inside a Landau level is of order l ´2 b . A typical fermionic state satisfying (III.1.7) is a projection onto a N -body Slater determinant of N orthonormal one body wave-functions with N -O ˆ1 2πl Ṡuch a N -particles state occupies a volume of order N l ´2 b " Op1q

wq 1 b ln ` l´1 b

 l´1 Moreover the reduced density converges to a weak solution of the gyrokinetic transport equation. Theorem III.1.4: Convergence of densities Under the same assumptions as Theorem III.1.3, up to a subsequence, ρ γ b converges in the sense of measures:

Proposition III. 2 . 10

 210 In symmetric gauge (III.1.1), the family defined by φ n,m -Hilbert basis of L 2 pR 2 q. The full expression, isφ n,m pxq " `p´2il 2 b B x `ixq n x m ?πn!m! `?2l b ˘n`m`1 e ´|x| 2 4l 2 b

  is an orthonormal Hilbert basis of L 2 pR 2 q. Since we are interested the creation and annihilation on the polynomial part of the wave-functions, define for an operator u Thereforeφ n,m -C 0,0 P n,m e 2l b B x `x ? 2l b ˙m Id " p´2il 2 b B x `ixq n x m ? n!m! `?2l b ˘n`m III.2.4 Coherent states Definition III.2.11 Define the coherent state ψ n,z -e z p b : ´zp b ? 2l b φ n,0 and the associated projector Π n,z -|ψ n,z ⟩ ⟨ψ n,z | the Landau level projector Π n -ÿ mPN |φ n,m ⟩ ⟨φ n,m |

Π 2 `2 b and 2

 22 n,z px, yq "ψ n,z pxqψ n,z pyq " |x| 2 `|y| 2 ´2pzx`zyq´2px -zqpy -zq 4l |z| 2 `|x| 2 `|y| 2 ´2pzx `zyq ´2 px -zq py -zq " |x| 2 `|y| 2 ´2 pzx `zy `xy ´xz ´zyq " |x ´y| 2 ´2iIm rxy `2zpx -yqs so Π z px, yq " 1 2πl 2 b e ´|x´y| 2 ´2i p x K ¨y`2z K ¨px´yq q 4l 2 b

First, the trace is preserved by the evolution because d dt Tr rγ b ptqs " 1 il 2 b 0 Upt 2 , t 1 q -e 1 il 2 b ş t 2 t 1 γp0q ď 2πl 2 b ùñ 2πl 2 b 2 bIII. 3 . 2 2 b"

 2022122322 3.1), conservation of the energy (Proposition III.3.2) and of the kinetic energy (Proposition III.3.3). III.3.1 Conservation of fermionic states Proposition III.3.1 Assume γ b P L 8 pR `, L 1 pL 2 pR 2 qqq solves B t γ b ptq " 1 il 2 b " L b `V `w ‹ ρ γ b ptq , γ b ‰ and satisfies Tr rγ b p0qs " 1, 0 ď γ b p0q ď 2πl 2 b then @t P R `, Tr rγ b ptqs " 1, 0 ď γ b ptq ď 2πl 2 b Proof : Tr "" L b `V `W ‹ ρ γ b ptq , γ b ptq ‰‰ " Moreover γ b ptq " Upt, 0qγ b p0qUp0, tq with pLb`V `w‹ρ γ b pτ qq dτ so γp0q ě 0 ùñ γ b ptq " Upt, 0qγ b p0qUpt, 0q ˚ě 0 ´γb ptq " Upt, 0q `2πl 2 b ´γb p0q ˘Upt, 0q ˚ě 0 ùñ γ b ptq ď 2πl Energy conservation Recalling the notation (III.1.3), we state that the quantity Tr rγ b H b s is preserved through the dynamics. Proposition III.3.2: Energy conservation Assume γ b P L 8 pR `, L 1 pL 2 pR 2 qqq solves B t γ b ptq " 1 il L b `V `w ‹ ρ γ b ptq , γ b ‰ then d dt Tr rγ b ptqH b ptqs " 0Proof :

3 . 1 )

 31 and (III.3.2) we conclude that d dt Tr rγ b ptqH b ptqs " Tr rB t γ b ptqH b ptqs `Tr rγ b ptqB t H b ptqs " 0 Next we use the energy conservation to control the kinetic energy. In Section III.4 and Section III.5 we will estimate error terms using this control. Proposition III.3.3: Kinetic energy conservation Let γ b P L 1 pL 2 pR 2 qq , W P L 8 pR 2 q and assume Tr rγ b s " 1 then Tr rγ b L b s ď |Tr rγ b pL b `W qs| `∥W ∥ L 8 Proof : The kinetic energy is bounded by Tr rγ b L b s " Tr rγ b pL b `W qs ´Tr rγ b W s ď |Tr rγ b pL b `W qs| `∥W ∥ L 8

Proposition III. 3 . 4 : 2 b

 342 2 nd moment of the Kinetic energy bound Let t P R `, γ b ptq P L 1 pL 2 pR 2 qq , W P W 2,8 pR 2 q and assume Tr rγ b ptqs " 1, 0 ď γ b ptq B t γ b ptq " 1 il rL b `W, γ b ptqs then

3 . 3 ) 2

 332 With a direct computation rP b , W s " rih∇ `bA, W s " ih∇W so rL b , W s " " P 2 b , W ‰ " P b ¨rP b , W s `rP b , W s ¨Pb " P b ¨pih∇W q `ih∇W ¨Pb "ih rP b , ∇W s `2ih∇W ¨Pb " ´h 2 ∆W `2ih∇W ¨Pb b rL b , W s γ b s `Tr rrL b , W s L b γ b s " ´h 2 pTr rL b ∆W γ b s `Tr r∆W L b γ b sq `2ih pTr rL b ∇W ¨Pb γ b s `Tr r∇W ¨Pb L b γ b sq But rP b , ∇W s -P b ¨∇W ´∇W ¨Pb " pTr r∆W L b γ b s ´Tr rL b ∆W γ b sq `2ih pTr rL b ∇W ¨Pb γ b s `Tr rP b ¨∇W L b γ b sq (III.3.4)

3 . 5 ) 8 b

 358 and with Cauchy-schwarz inequality|Tr rL b ∇W ¨Pb γ b s `Tr rP b ¨∇W L b γ b s| " b |∇W | 2 L b ‰ Tr rγ b L b s ď ∥∇W ∥ L Tr rγ b L 2 b sTr rγ b L b s (III.3.6) Inserting (III.3.5)with ϵhb ∥∆W ∥ L 8 and (III.3.6) in (III.3.4) we obtain ˇˇTr

  4.18) Moreover ˇˇˇˇÿ nąN Tr rφΠ n γ b s ˇˇˇˇ" |Tr rφΠ ąN γ b s| ď b Tr " γ b |φ| 2 ‰ Tr rγ b Π ąN s ď ∥φ∥ L 8 a Tr rγ b Π ąN s and Tr rγ b Π ąN s " ÿ nąN Tr rγ b

III. 5 . 1 1 :

 511 Dynamics of the semi-classical density Proposition III.5.Gyrokinetic equation for the semi-classical density Let t P R `, k ą 3, γ b ptq P L 1 pL 2 pR 2 qq , W P W 3,8 pR 2 q and assume Tr rγ b ptqs " 1, 0 ď γ b ptq ď 2πl 2 b (III.5.1)

4 b 4 b

 44 Tr rγ b rΠ z , X ¨∇W pzqss (III.5.6)Putting together (III.5.5) and (III.5.6),B t ρ sc γ b pzq `∇K W pzq ¨∇ρ sc γ b pzq " 1 2iπlTr rγ b rΠ z , W ´X ¨∇W pzqss (III.5.7)

Step 3 :FromnPN

 3 [START_REF] Lieb | The stability of matter in quantum mechanics[END_REF] and inserting (III.5.10) (III.5.11) (III.5.12) (III.5.13) in (III.5.7) we obtain B t ρ sc γ b pzq `∇K W pzq ¨∇ρ sc γ b pzq " E 1,b pzq `E2,b pzq `r E 2,b pzq (III.5.14) Estimate of E 1,b pzq Tr rγ b rΠ n,z , V z ss (III.5.15)

2 ż R 2

 22 ˆR2 ´Vz pz `?2l b xq ´Vz pz `?2l b yq ¯2 |xy| 2n e ´|x| 2 ´|y| 2 dxdy (III.5.17) Let 2 ď α ď 3, using the expansion (III.5.8), ˇˇV z pz `?2l b xq ˇˇ" ˇˇW pz `?2l b xq ´?2l b dW z pxq ´l2 b d 2 W z px, xq ˇˇď C ∥W ∥ W α,8 |l b x| α and similarly with y instead of x. With Lemma III.5.2,

´|y| 2 dxdy ďC ∥W ∥ 2 W 3 , 8 `pn `1ql 2 b˘α 4 b∥ 2 W 3 2 φpzqE 1

 238242321 Inserting this along with (III.5.16) in (III.5.15) we get |E 1,b pzq| ď C l ˜∥W Tr rγ b Π n,z s ¸(III.5.18) Integrating against φ P C 8 c pR 2 q, choosing ϵ n -ϵ a |supppφq| ∥W ∥ W 3,8 pn `1q k and using (III.4.9) (III.2.8) (III.5.1), we obtain ˇˇˇˇˇż R

4 b 4 b

 44 Tr rγ b rΠ ďN,z , X ¨∇W pzqss `rγ b |ψ N,z ⟩ ⟨ψ N `1,z |s Tr rγ b |ψ N `1,z ⟩ ⟨ψ N,z |s ṡo with the same computation as for (III.5.7), B t ρ ďN γ b pzq `∇K W pzq ¨∇z ρ ďN γ b pzq " 1 2iπl Tr rγ b rΠ ďN,z , W ´X ¨∇W pzqss rγ b |ψ N,z ⟩ ⟨ψ N `1,z |s Tr rγ b |ψ N `1,z ⟩ ⟨ψ N,z |s ˙(III.5.27) Using (III.5.10) for Π ďN,z instead of Π z and then inserting (III.2.14),

3 bN l 3 bϵ 2 b `2πl 2 bϵ 2 φE 3 ,b pzqdz ˇˇˇˇˇď C d 2 W L 8 ? N l 3 bˆ∥φ∥ L 1 ϵN l 2 b `∥φ∥ L 8 l 4 bϵ 2 φE 3 ,b pzqdz ˇˇˇˇˇď C a |supppφq| ∥φ∥ L 8 d 2 W L 8 N 1´k 2 b 2 φpzq 1 l b N k´1 2 2 φpzq 1 l b N k´1 2 2 ! 1 , l b N 2´k 2 ! 1 When 1 ă k ă 4 ,

 3322238328423222122122114 We only need to estimate terms of the form ? N l Tr rγ b |ψ N,z ⟩ ⟨ψ N `1,z | pX i ´zi qs or ? Tr rγ b |ψ N `1,z ⟩ ⟨ψ N,z | pX i ´zi qs with i P t1, 2u. With Young's inequality and the same inequality as in (III.4.6), |Tr rγ b |ψ N `1,z ⟩ ⟨ψ N,z | pX i ´zi qs| ďϵTr " pX i ´zi q 2 Π N,z Tr rγ b Π N `1,z s (III.5.40) and similarly Tr rγ b |ψ N,z ⟩ ⟨ψ N `1,z | pX i ´zi qs ď 2ϵpN `2ql Tr rγ b Π N,z sSince N `1 " N `2 " N , integrating against φ P C 8 c pR 2 q, we obtainˇˇˇˇˇż R pTr rγ b Π N `1s `Tr rγ b Π N sq ďC ∥φ∥ L 8 d 2 W L 8 ˜|supppφq| ϵ N k N k Tr rγ b pΠ N `ΠN`1 qs ḑC ∥φ∥ L 8 d 2 W L 8 ˜|supppφq| ϵ N Tr " γ b L k b pΠ N `ΠN`1 q ‰ (III.5.41)Step 6: Conclusion Exchanging x and y in (III.5.32) and (III.5.34) we obtain with the same arguments as for (III.5.39) and (III.5.41): b pzqdz ˇˇˇˇˇď C a |supppφq| ∥φ∥ L 8 d 2 W L 8 N 1´k 2 b Tr " γ b L k b pΠ N `ΠN`1 q ‰ (III.5.43) Putting together (III.5.37) (III.5.38) (III.5.39) (III.5.41) (III.5.42) (III.5.43) with (III.5.35) we get ˇˇˇˇˇż R `Bt ρ sc,ďN γ b pzq `∇K W pzq ¨∇ρ sc,ďN γ b pzq ˘dz ˇˇˇˇďC ∥φ∥ L 8 ∥∇W ∥ L 8 Tr " γ b L k b pΠ N `ΠN`1 q ‰ `Ca |supppφq| ∥φ∥ W 1,8 ∥W ∥ W 3,8 p .4.11) that if x ě e, α ă β ùñ p α pxq ě p β pxq (III.5.44) so p k´2 pN q ď p k´4 pN q and we conclude that ˇˇˇˇˇż R `Bt ρ sc,ďN γ b pzq `∇K W pzq ¨∇ρ sc,ďN γ b pzq ˘dz ˇˇˇˇď C ∥φ∥ L 8 ∥∇W ∥ L 8 Tr " γ b L k b pΠ N `ΠN`1 q ‰ `Ca |supppφq| ∥φ∥ W 1,8 ∥W ∥ W 3,8 p a subsequence, whose choice will be the object to Lemma III.6.1, we can assume Tr " γ b L k b Π N ‰ " O ˆ1 N Ṫhis time we have a small error in (III.5.26) if l b N k`1 2 " 1, N 1´k this is equivalent to

4 b 2 b" γ b d 2 W 2 ,b pzq - 1 4πl 2 b" γ b d 2 Wγ b d 2 γ b d 2 WE 4 ,b pzq - 1 12πl 2 b" γ b d 3 W 4 ,b pzq - 1 12πl 2 b" γ b d 3 WE 4 ,b pzq - 1 12πl 2 b" γ b d 3 Wγ b d 3 γ b d 3 Wγ b d 3 WStep 2 : 2 φpzqE 1 Step 3 :E 4 ,b pzq " 1 12πl 2 b´1 0 "ˆ∇b3 W pzq d 2 2 ∇ 4 b 2 φpzqE 2 ,b pzqdz ˇˇˇˇˇď C a |supppφq| ∥∇φ∥ L 8 d 3 W L 8 ˜ϵl 2 |supppφq| ∥∇φ∥ L 8 d 3 W L 8 l 2 b ˆϵp k´3 pN 2 φpzqE 2 ,b pzqdz ˇˇˇˇˇď C a |supppφq| ∥∇φ∥ L 8 d 3 W L 8 b" γ b L k b ‰ a p k´3 pN ql 2 b(III.5. 50 ) 4 : 2 φpzqE 4 ,b pzqdz " ´1 12πl 2 b ż R 2 ∇CN l 4 b and ˇˇˇˇˇż R 2 φpzqE 4 Step 5 :ϵ 2 φE 5 ,b pzqdz ˇˇˇˇˇď C a |supppφq| ∥φ∥ L 8 d 3 Wγ b d 3 W 2 2

 422222224234234233332213420224223232282504242245253322 rγ b |ψ N,z ⟩ ⟨ψ N `1,z |s Tr rγ b |ψ N `1,z ⟩ ⟨ψ N,z |s Tr rγ b rΠ ďN,z , V z ss E 2,b pzq -1 4πlTr pzq `∇K z Π ďN,z , X ´z˘‰ r ETr pzq `X ´z,∇ K z Π ďN,z ˘‰ E 3,b pzq -W pzq ˆˆ1 1 ´i i ˙ˆ|ψ N,z ⟩ ⟨ψ N `1,z | |ψ N `1,z ⟩ ⟨ψ N,z | ˙, X ´z˙ȷ r E 3,b pzqpzq ˆX ´z, ˆ1 1 ´i i ˙ˆ|ψ N,z ⟩ ⟨ψ N `1,z | |ψ N `1,z ⟩ ⟨ψ N,z | ˙˙ȷ Tr pzq `∇K z Π ďN,z , X ´z, X ´z˘‰ r E Tr pzq `X ´z, ∇ K z Π ďN,z , X ´z˘‰ r r Tr pzq `X ´z, X ´z, ∇ K z Π ďN,z ˘‰ E 5,b pzq -W pzq ˆˆ1 1 ´i i ˙ˆ|ψ N,z ⟩ ⟨ψ N `1,z | |ψ N `1,z ⟩ ⟨ψ N,z | ˙, X ´z, pzq ˆX ´z, ˆ1 1 ´i i ˙ˆ|ψ N,z ⟩ ⟨ψ N `1,z | |ψ N `1,z ⟩ ⟨ψ N,z | ˙, X ´z˙ȷ r r E 5,b pzqpzq ˆX ´z, X ´z, ˆ1 1 ´i i ˙ˆ|ψ N,z ⟩ ⟨ψ N `1,z | |ψ N `1,z ⟩ ⟨ψ N,z |˙˙ȷ and inserting (III.5.45) in (III.5.27) we haveB t ρ ďN γ b `∇K W ¨∇ρ ďN γ b "E 0,b `E1,b `E2,b `r E 2,b `E3,b `r E 3,b `E4,b `r E 4,b `r r E 4,b `E5,b `r E 5,b `r r E 5,b (III.5.47) Estimate of E 1,bUsing that the fourth derivative of W is bounded, we getˇˇV z px `?2l b xq ˇˇď C ∥W ∥ W 4,8 |l b x| 4so following the same proof as for (III.5.11) we obtain instead of (III.5.19) ˇˇˇˇˇż R ,b pzqdz ˇˇˇˇˇď C a |supppφq| ∥φ∥ L 8 ∥W ∥ W 4,8 ˜ϵl 4 Estimate of E 4,bBy symmetry of the derivatives of W ,Tr " γ b ∇ K z Π ďN,z d ∇ b3 W pzq d 2 pX ´zq b2 ‰Next we notice that∇ K z d ∇ b3 W pzq d 2 pX ´zq b2 " `∇K z d ∇ b3 W pzq ˘d2 pX ´zq b2 `∇b3 W pzq d 3 ∇ K z b pX ´zq b2 "∇ b3 W pzq d 3 ˆ0 1 ´1 0 ˙b pX ´zq `∇b3 W pzq d 3 pX ´zq b ˆ0 1 ˆ0 1 ´1 0 ˙˙d pX ´zq ``∇ b3 W pzq d pX ´zq ˘d ˆ0 1 ´1 0 ˙" 0 since ∇ b3 W pzq, ∇ b3 Wpzq d pX ´zq are symmetric and ˆ0 1 ´1 0 ˙anti-symmetric. Hence with an integration by parts, K φpzq d d 3 W pzq d 2 Tr " γ b Π ďN,z pX ´zq b2 ‰ (III.5.49) From this point on, we follow the same proof as for (III.5.12) with one difference in (III.5.22) where we use Tr " Π n,z |X ´z| 4 ‰ ď Cpn `1q 2 l and obtain instead of (III.5.23), ˇˇˇˇˇż R Tr Step Estimate of r E 4,b Similarly as for E 4,b we proceed to an integration by parts and instead of (III.5.49) we have ż R K φpzq d d 3 W pzq d 2 Tr rγ b pX ´zq b Π ďN,z pX ´zqs But |Tr rγ b pX ´zq b Π ďN,z pX ´zqs| ď Cl 2 b Tr " Π n,Z |X ´z| 2 ‰ ď ,b pzqdz ˇˇˇˇˇď C |supppφq| ∥∇φ∥ L 8 d 3 W L 8 l 2 b N (III.5.51) Estimate of E 5,bWe follow the same proof as for (III.5.33), the only difference being in (III.5.40) that we replace withˇˇTr " γ b |ψ N `1,z ⟩ ⟨ψ N,z | pX i ´zi q 2 ‰ˇˇď ϵTr " pX i ´zi q 4 Π N,z Tr rγ b Π N `1,z s (III.5.52)So we gain a factor ? N l b and instead of (III.5.41) we obtainˇˇˇˇˇż R L 8 l b N pzq ˆX ´z, ˆ1 1 ´i i ˙ˆ|ψ N,z ⟩ ⟨ψ N `1,z | |ψ N `1,z ⟩ ⟨ψ N,z | ˙, X´z˙ȷWe only need to estimate terms of the form Tr rγ b pX i ´zi q |ψ N `1,z ⟩ ⟨ψ N,z | pX j ´zj qs with i, j P t1, 2u. But with Young's inequality, 2 |Tr rγ b pX i ´zi q |ψ N `1,z ⟩ ⟨ψ N,z | pX j ´zj qs| ďTr rγ b pX i ´zi qΠ N `1,z pX i ´zi qs `Tr rγ b pX j ´zj qΠ N,z pX j ´zj qs We are going to control the second term with j " 1, all the others terms being estimated the same way. With (III.2.15), Tr rγ b pX 1 ´z1 qΠ N,z pX 1 ´z1 qs ď Tr " γ b Π N,z pX 1 ´z1 q 2 ‰ `Tr rγ b rX 1 , Π N,z s pX 1 ´z1 qs ď Tr " γ b Π N,z pX 1 ´z1 q 2 ‰ `il 2 b Tr rγ b B z 2 Π N,z pX 1 ´z1 qs ´i ? N `1l b ? Tr rγ b p|ψ N,z ⟩ ⟨ψ N `1,z | `|ψ N `1,z ⟩ ⟨ψ N,z |q pX 1 ´z1 qs `i ? N l b ? Tr rγ b p|ψ N ´1,z ⟩ ⟨ψ N,z | `|ψ N,z ⟩ ⟨ψ N ´1,z |q pX 1 ´z1 qs

2 2 b 2 b 2 b 2 φpzq 1 l b N k´1 2 2 φpzqa p k´5 pN q 1 `´N 1´k 2 ´N 1´k 2 1 lnpN q ¯1k" 5 `lb 1 ką5 3 : 2 φp0,R `ˆR 2 φGYRO 2 b 2 w ‹ ρ γ b ptq L 8 " 2 w ‹ ρ γ b ptq L 8 ď 2 φpt 2 φGYRO0R `ˆR 2 φGYRO ργ b pρ sca

 222221221215132222828222 5.53) and (III.5.41) we obtain ˇˇˇˇˇż R φpzq r E 5,b pzqdz ˇˇˇˇďC a |supppφq| ∥φ∥ L 8 d 3 W L 8 l b N 3´k Tr " γ b L k b Π N ‰ `Ca |supppφq| ∥φ∥ W 1,8 ∥W ∥ W 4,8 l 2 b N 1´k 2 b Tr " γ b L k b Π N ‰ `Ca |supppφq| ∥φ∥ L 8 d 3 W L 8 l b N 3´k Tr " γ b L k b pΠ N ´1 `ΠN `ΠN`1 q ‰ ďC a |supppφq| ∥φ∥ W 1,8 ∥W ∥ W 4,8 l b N 3´k Tr " γ b L k b pΠ N ´1 `ΠN `ΠN`1 q ‰ (III.5.54)Step 7: ConclusionE 0,b , E 2,b , r E 2,b , E 3,b , rE3,b are the same than in Proposition III.5.3. We control r r E 4,b , r r E 5,b with respectively the same arguments as for E 4,b , E 5,b by exchanging x and y. Therefore, putting together (III.5.37), (III.5.39), (III.5.41), (III.5.48), (III.5.50), (III.5.51), (III.5.53), (III.5.54) with (III.5.47) we obtain ˇˇˇˇˇż R `Bt ρ sc,ďN γ b pzq `∇K W pzq ¨∇ρ sc,ďN γ b pzq ˘dz ˇˇˇˇďC ∥φ∥ L 8 ∥∇W ∥ L 8 Tr " γ b L k b pΠ N `ΠN`1 q ‰ `Ca |supppφq| ∥φ∥ W 1,8 ∥W ∥ W 4,8 p k b pΠ N ´1 `ΠN `ΠN`1 q ‰ ˙`C |supppφq| ∥∇φ∥ L 8 d 3 W L 8 l 2 b N With (III.5.44), we get ˇˇˇˇˇż R `Bt ρ sc,ďN γ b pzq `∇K W pzq ¨∇ρ sc,ďN γ b pzq ˘dz ˇˇˇˇˇď cpφ, W q ˆ1 l b N k´1 2 Tr " γ b L k b Π N ´1:N `1‰ `bTr " γ b L k b ‰ ´lb a p k´2 pN q `l2 b kă2 `alnpN q1 k"2 `1ką2 ¯`l 2Proof of Theorem III.1.With an integration by part,ż R zqρ γ b p0, zqdz ´ż R `ˆR 2 ρ γ b pt, zqGYRO ργ b pφqpt, zqdtdz " zqGYRO ργ b pφqpt, zqdtdz `ż R 2 φp0, zq `ργ b ´ρsc,ďN γ b ˘p0, zqdz ´ż R `ˆR 2 `ργ b ´ρsc,ďN γ b ˘pt, zqGYRO ργ b pφqpt, zqdtdz " ż ργ b pρ sc,ďN γ b qdtdz `ż R 2 φp0, zq `ργ b ´ρsc,ďN γ b ˘p0, zqdz ´ż R `ˆR 2 `ργ b ´ρsc,ďN γ b ˘GYRO ργ b pφqdtdz (III.6.1)Let T ě 0 be such that supppφq Ă r0, T s ˆR2 . By Proposition III.3.1, @t P r0, T s , Tr rγ b ptqs " 1 and 0 ď γ b ptq ď 2πl Let t P r0, T s, then by Proposition III.3.3 applied to W " V `1 2 w ‹ ρ γ b ptq and Proposition III.3.2, Tr rγ b ptqL b s ď |Tr rγ b ptqH b ptqs| ` V `1 |Tr rγ b p0qH b p0qs| ` V `1 |Tr rγ b p0qH b p0qs| `∥V ∥ L 8 `1 2 ∥w∥ L 8 ď CpV, wq (III.6.2) and ∥V `w ‹ ρ γ b ptq∥ W 8,4 ď ∥V ∥ W 8,4 `∥w∥ W 8,4 so from Proposition III.5.4 for k " 1, W " V `w ‹ ρ γ b ptq we get ˇˇˇˇˇż R , zqGYRO γ b `ρďN γ b ˘pt, zqdz ˇˇˇˇˇď Cpφ, V, wq ˆ1 l b ? N Tr rγ b ptqL b Π N ´1:N `1s `´? N `lb N ¯aTr rγ b ptqL b Π N ´1:N `1s `l2 b N `aTr rγ b ptqL b s ´lb ? ργ b pρ sc,ďN γ b qdtdz ˇˇˇˇˇˇď Cpφ, V, wq ¨1 l b ? N T ż Tr rγ b ptqL b Π N ´1:N `1s dt and ˇˇˇˇˇˇż Tr rγ b p0qΠ ąN L b s `aTr rγ b p0qL b sl b ?

5 ) 3 ă α, α ă β ă 1 (

 531 Inserting (III.6.3), (III.6.4), (III.6.5) in (III.6.1), and then takingN ´-l ´α b , N `-l ´β b , with 2 γ b p0, zqdz ´ż R `ˆR 2 ρ γ b pt, zqGYRO ργ b pφqpt,

7 )

 7 Let pΩ n q nPN be an increasing sequence of open bounded subsets of R ˚ˆR 2 , such thatď nPN Ω n " R ˚ˆR 2

`R `ˆR 2 ` 2 ` 2 `

 222 also bounded in M pR 2 q, so after another extraction one has ρ M `R2 ˘(III.6.9) in the sense of measures.Step 2: Error decompositionLet φ P C 8 GYRO ργ b pφq ´GYRO ρ pφq ż `ˆR 2 ρ sc,ďN γ b ∇ z φ ¨pρ γ b ´ρq ‹ ∇ K w `ż R `ˆR `ˆR 2 pρ γ b ´ρq `∇z φ ¨ρsc,ďN γ b ˘‹ ∇ K w `ż R `ˆR z φ ¨ρsc,ďN γ b ˘‹ ∇ K w `ż R `ˆR 2 `ρsc,ďN γ b ´ρ˘`∇ z φ ¨ρsc,ďN γ b ˘‹ ∇ K w `ż R `ˆR 2`ρsc,ďNγ b

  Soit u 1:N P H, si u 1:N est liée, leur produit antisymétrique est nul et si u 1:N est une famille orthogonale,

	Soit
		P ˘" 1 N !	ÿ σPS N
	ȋ"1	u i "	σPS N ÿ	i"1 ˘N â ϵ ˘pσqu σpiq " N !P	u i

1.2: Projection sur les espaces bosonique/fermionique ϵ ˘pσqU σ est une projection orthogonale de H bN vers H b ˘N . Définissons le produit tensoriel symétrique/antisymétrique N â

  L'effet Hall est un phénomène qui se produit dans certains matériaux conducteurs lorsqu'ils sont soumis à un champ magnétique transverse au matériau. Si l'on impose un courant électrique I à travers le matériau, l'effet Hall se manifeste par l'apparition d'une tension électrique transversale au courant dans le matériau, appelée tension de Hall, notée V H . Elle est due à la déviation des porteurs de charge, électrons ou trous, par la force magnétique. La conductance de Hall est alors Pour des systèmes classiques cette conductance est simplement proportionelle à l'amplitude du champ magnétique. Le phénomène peut alors s'expliquer par le fait que des particules chargées soumises à un champ magnétique transverse décrivent des orbites, appelés orbites cyclotron. En mécanique quantique l'énergie cinétique d'une particule dans un champ magnétique transverse et homogène est quantifiée en niveau d'énergie discrets, séparés par un gap constant, nommés niveaux de Landau. Pour des systèmes aux échelles de la mécanique quantique, la conductance de Hall se retrouve quantifiée en multiples du quantum de conductance:

	σ Hall -	I V H

  2 q est le potentiel vecteur générant le champ magnétique transverse et homogène. Par exemple: Pour une fonction ψ P L 2 pΩq les conditions magnétiques périodiques s'écrivent

	A Sym -	1 2	p´y, xq (gauge symétrique)
	A

lan -p´y, 0q (gauge de Landau) Le Laplacien magnétique est L h,b -pih∇ `bAq 2 (I.2.1) En identifiant R 2 avec C on utilise une notation complexe pour le vecteur px, yq P R 2 , c'est-à-dire px, yq " x `iy P C Soit z 0 P C, ∇ ^pA ´Ap' ´z0 qq " 0 on peut donc choisir φ z 0 P C 8 pR 2 , Rq tel que A ´Ap' ´z0 q -l 2 b ∇φ z 0 @t P r0, Ls, # ψpL `itq " e iφ L pL`itq ψpitq ψpt `iLq " e iφ iL pt`iLq ψptq (I.2.2) Et le domaine du Laplacien magnétique est Dom pL h,b q -␣ ψ P H 2 pΩq tel que (I.2.2) soit vérifié ( L'Hamiltonien à N -corps est

  Le théorème pour la convergence des densités est :

	de Landau sont générées par des translations magnétiques. Dans la Section II.3, on définit
	la projection orthogonale sur un niveau de Landau. Cette projection sera ensuite localisée
	dans l'espace donnant ainsi un projecteur sur un point de l'espace des phases. Le projecteur
	sur un niveau de Landau est alors étudié et comparé à celui sur R 2 . Ensuite, une inégalité de
	Lieb-Thirring est démontrée dans la Section II.4 dans le but contrôler la norme L 2 des densités
	réduites et ainsi traiter des potentiels L 2 . Enfin, les Section II.5 et Section II.6 contiennent la
	preuve à proprement parler des théorèmes de convergence de l'énergie fondamentale (Theorem
	I.2.6) et des densités réduites (Theorem I.2.7). Dans la Section II.5 on justifie que l'énergie
	semi-classique (I.2.6) est une bonne approximation de l'énergie quantique à N corps. Puis, la
	Section II.6 porte sur l'approximation de champ moyen et consiste principalement à contrôler
	les corrélations. Cette partie se partage en la preuve d'une borne supérieure et d'une borne
	inférieure pour l'énergie quantique à N corps.
	Theorem I.2.7: Convergence des densités
	Si V, w P L 2 pΩq, et pψ N q N PN est une suite de minimiseurs de (I.2.3). En notant ρ	pkq N la k ième
	densité réduite de |ψ N ⟩ ⟨ψ N |, il existe une mesure de probabilité µ sur D qLL telle que
	• µ ne donne de la masse qu'aux minimiseurs de (I.2.7)
	• @k P N ˚, au sens des mesures,
	ρ pkq N	á N Ñ8	ż	ˆq L 2 pq `rq	`ρ˙b k	dµpρq
			D qLL
	La densité de particules converge vers une somme convexe de densités de la forme
			q L 2 pq `rq	`ρ
	De par la forme du principe de Pauli, il est visible que la constante dans ce terme correspond
	aux particules dans les niveaux remplis. Ensuite, dans le niveau de Landau partiellement rempli,
	la densité est donnée par un minimiseur de (I.2.7).
	I.2.3 Stratégie
	La première étape pour démontrer les résultats ci-dessus est la quantification du Laplacien
	Magnétique (I.2.1). On obtient en Section II.2 une base où les fonctions propres d'un niveau

  A k q b Γ N ´ks " Tr rΓ k A k s Tr rΓ N ´ks " Tr rTr rΓ N ´ks Γ k A k s "Tr rTr k`1:N rΓ k b Γ N ´ks A k s " Tr rTr k`1:N rγ N s A k s " Tr

						HbpN´kq ˘puis revenir au
	cas général par linéarité. Ainsi,
	Tr	" γ N A k b Id bpN ´kq H	ı	"Tr rpΓ k " γ N A k pkq	ı
	La trace et la positivité des matrices densités réduites est une conséquence directe de ces
	propriétés pour γ N . Par (I.4.1) appliqué à A k -Id H bk ,
				Tr	" γ N pkq	ı	" Tr rTr k`1:N rγ N ss " Tr rγ N s " 1
	Puis, si ψ P H bk ,
			xψ|γ N ψy " xψ b 1 bpN ´kq γ N ψ b 1 bpN ´kq y ě 0 pkq
	Dans le cas où la matrice densité est un projecteur sur un produit symétrique/antisymétrique,
	il existe une formule exacte permettant d'exprimer les densités réduites en fonctions des vecteurs
	du produit.
	Theorem I.4.2: Théorème de Wick
	Si

  Dpαq " e ´1 2 rαa : ,´αas e αa : e ´αa " e

	où |n⟩ -	a : n ? n!	|0⟩ est normé. L'opérateur nombre d'excitations est
					N -a : a
	On a les propriétés suivantes pour les états cohérents :
	Proposition I.4.4
			⟨ψ α |ψ α ⟩ " 1
			ra, Dpαqs " α
			aψ α " αψ α								(I.4.2)
												Soit a : et a des
	opérateurs de création et d'annihilation :
					" a, a : ‰	" Id H
	où H est l'espace de Hilbert généré par application successives de a : : `a: n |0⟩ ˘nPN sur le vide
	|0⟩ -1 C . Soit α P C, l'opérateur état cohérent est
					Dpαq -e αa : ´αa
	Par la formule de Baker-Campbell-Hausdorff,
										´|α| 2 2 e αa : e ´αa
	On définit alors les états cohérents
		ψ α -Dpαq |0⟩ " e	´|α| 2 2 e αa :	|0⟩ " e	´|α| 2 2	ÿ nPN	α n n!	a : n	|0⟩ " e ´|α| 2 2	ÿ nPN	α n ? n	|n⟩

  Cpb, h, Dq `Dk 4 with Parseval's theorem we obtain ∥∇ψ∥ 2 ď Cpb, h, Dq ∥ψ∥ 2 `D ∥∆ψ∥ 2

	Choose D "	ˆ1 2h ∥A∥ 8	¨h 2 2	˙2 so that
	2h ∥A∥ 8 ∥∇ψ∥ ď Cpb, hq ∥ψ∥ `2h ∥A∥ 8	? D ∥∆ψ∥ " Cpb, hq ∥ψ∥	`h 2 2	∥∆ψ∥	(II.2.12)
	and we can conclude that
				∥L h,b ψ∥ ě	h2 2	∥∆ψ∥ ´Cpb, hq ∥ψ∥	(II.2.13)
	Finally ∥ψ∥ can also be controlled:
		∥L h,b ψ∥ 2 " phbq 2 ˆN	`1 2	˙ψ 2	ě	ˆhb 2	˙2 ∥ψ∥ 2	(II.2.14)

since N is positive (II.2.19). Combining (II.2.12), (II.2.13), (II.2.14) gives the desired embedding.

  2 mp pΩq, so a and a : are adjoint of one another and ∥P h,b ψ∥ L 2 is equivalent to the H 1 norm. A and its gradient are bounded so `H2 mp pΩq, x'y H 1 ˘is continuously embedded in pH 2 mp pΩq, ∥'∥ L q. Moreover ∥P h,b ψ∥ L 2 ěh ∥∇ψ∥ L 2 ´∥bAψ∥ L 2 ě h ∥∇ψ∥ L 2 ´b ∥A∥ L 8 ∥ψ∥ L

	xψ|N ψy " xaψ|aψy ě 0	(II.2.19)
	Let ψ P H 2 mp pΩq, now we prove that the norm
		b
	∥ψ∥ L -	xψ|ψy L "

2 And ∥ψ∥ L 2 can be controlled with (II.2.18) and (II.2.19) ∥ψ∥ 2 L " xψ| p2hbN `hbq ψy ě hb ∥ψ∥ 2 L 2 (II.2.20) so ∥h∇ψ∥ L 2 ď ∥ψ∥ L `b ? hb ∥A∥ L 8 ∥ψ∥ L Therefore we have the desired continuous embedding: ∥ψ∥ H 1 ď Cpb, hq ∥ψ∥ L (II.2.21)

  positivity of L h,b follows from that of N . Using the Lax-Milgram theorem, see results of [40, Section 2.5], and Lemma II.2.6 the operator L of domain DompLqtu P Dom pL h,b q , such that @v P Dom pL h,b q |xu|vy L | ď Cpuq ∥v∥ L 2 u DompLq, u P Dom pL h,b q , xu|vy L -xLu|vy is closed and self adjoint. But this operator is equal to pL h,b , Dom pL h,b qq because it coincides with L h,b on Dom pL h,b q and the required inequality in the definition of DompLq is satisfied taking Cpuq -∥L h,bA u∥ L 2 , thus Dom pL h,b q " DompLq. The continuity of Dom pL h,b q ãÑ L 2 pΩq has been proved in (II.2.20). Then, we have the canonical embeddings

	defined by
	@v P

  l e ´2πkl´πdk 2 ´πd´2πpl`kdq " c l e ´2πpk`1ql´πdpk`1q 2 at this point we are done for k ě 0, but c l " c l`kd e 2πkl`πdk 2 so c l´kd " c l e 2πkpl´kdq`πdk 2 " c l e 2πkl´πdk 2 " c l e ´2πp´kql´πdp´kq 2

		4 Ll b ?	e 2iπl x L	kPZ ÿ	e 2iπkd x L ´1 2l 2 b	py`kL`l L d q 2	(II.2.32)
	"	π ´1 4 ? Ll b	e ´πl 2 d	´y2 2l 2 b	`2iπl z L θ	´d z L	`il, id ¯(II.2.33)
	and we obtain (II.2.34) for k ď 0. Inserting (II.2.34) in (II.2.30) gives
	f pzq "

Proof :

With the same notation as in the proof of Proposition II.2.11, we prove by induction that c l`kd " c l e ´2πkl´πdk 2 (II.2.34) This is satisfied for k " 0. Using (II.2.31) and assuming the relation (II.2.34) for k P N, c l`pk`1qd " c l`kd e πd´2πpl`rk`1sdq " c d´1 ÿ l"0

  L ψ 0l pzq and obtain the magnetic periodic boundary conditions in Landau gauge (II.2.4). The lowest Landau level is generated by successive magnetic translations:In order to obtain a full basis of L 2 , we only need to apply successively a : to generate the Landau levels and τ ´i L d to generate the wave-functions inside a Landau level. The successive applications of a : bring out Hermite polynomials.For n P N, we define the n th Hermite polynomial byH n -p´1q n e x 2ˆd dx ˙n e

	Corollary II.2.14: Generation of nLL with magnetic translations
	If l P rr0, d ´1ss,
		ψ 0l " ´τ´i L d	¯l ψ 00 " τ ´il L d	ψ 00	(II.2.35)
	Proof :
	φ ,Lan "	´y0 x l 2
		τ ´i L d	" e	i Lx dl 2 b T ´i L d	" e 2iπ x L T ´i L d
	and
		τ ´il L d	" ´τ´i L d	¯l " e 2iπl x L ´T´i L
	Notation II.2.15: Hermite polynomials

b defined in (II.2.4) is linear in z 0 and independent of y, thus with (II.2.1), d ¯l With this, (II.2.32) can be written as (II.2.35). ´x2 (II.2.36)

  3.5).

	Notation II.2.17: Fourier transform
	We use the convention
	Fgpνq -ĝpνq -	2π ? 1	ż
							´x2 2	(II.2.42)
	In this convention the Poisson summation formula is
	ÿ	gpkq "	? 2π	ÿ	ĝ p2πkq	(II.2.43)
	kPZ					kPZ
	h Fh n " p´iq n h n	(II.2.44)
	with the following normalization
		∥h n ∥ 2 L 2 "	? π2 n n!	(II.2.45)
	With this we are ready for the next computation:
	Proposition II.2.18: Poisson summation of eigenfunctions

R

gpxqe ´iνx dx for which F is unitary on L 2 pRq. And denote the Hermite function h n pxq -H n pxqe n are the eigenfunctions of the one dimensional harmonic oscillator and of the Fourier transform:

  Moreover, if P n , Q n are complex polynomials of degree n, the function

	ÿ
	Ξpzq -
	k,qPZ
	, b P r´1, 1s, α	ÿ qPZ ˚|a `b `αq| m e ´cpa`αqq 2	ď Cpc, mq	(II.3.8)
	@α P r0, 1s, a P R, b P r´1, 1s, α	ÿ	|a `b `αk| m e ´cpa`αkq 2	ď Cpc, mq	(II.3.9)
			kPZ

  4.5: Diamagnetic inequality for Green functionsFor all x P Ω and for almost every y P Ω,|G bA,λ px, yq| ď G λ px, yq If φ n Ñ φ in L 2 pΩq, by Young's inequality ∥ρ n ˚φn ´φ∥ L 2 ď ∥ρ n ˚pφ n ´φq∥ L 2 `∥ρ n ˚φ ´φ∥ L 2 ď ∥ρ n ∥ L 1 ∥φ n ´φ∥ L 2 `∥ρ n ˚φ ´φ∥ L 2 Ñ

			ij
	|G bA,λ px, 'q ˚ρn | ˚ρn pyq ď	ρ n px ´zqρ n pz ´tqG λ pt ´yqdzdt
			Ω 2
			ij
	Proof :	"	ρ n p2x ´y ´zqρ n pz ´tqG λ pt ´xqdzdt
	Let φ P C 8 pΩq, by definition	Ω 2
	ż	"ρ n ˚ρn ˚Gλ px, 'qp2x ´yq	(II.4.19)
				(II.4.18)
	in a weak sense on C 8 p pΩq	˚.
	Similarly as (II.4.17),
		p´h 2 ∆ `λqρ n ˚Gλ p', yq " ρ n py ´'q
	Thus testing inequality (II.4.18) on ρ n ˚Gλ p', yq P C 8 p pΩ, R `q we get
	ż

Ω

G bA,λ px, 'q pL h,b `λq φ " G bA,λ pL h,b `λq φ " φ so, in the distributional sense pL h,b `λq G bA,λ px, 'q " δ x (II.4.16)

Let ρ P C 8 pR 2 , R `q radial with support included in the ball Bp0, L{2q such that ∥ρ∥ L 1 " 1.

Let n P N ˚, define the localizer ρ n P C 8 pTq defined by

ρ n pxq -# n 2 ρpnxq if x P B `0, L 2n 0 else Since ρ n is periodic, the regularisation u x -G bA,

λ px, 'q ˚ρn P C 8 p pΩq Thus, equation (II.4.16) becomes pL h,b `λq u x " δ x ˚ρn " ρ n px ´'q (II.4.17) We estimate Re rspu x q pL h,b `λq u x s " Re rspu x qρ n px ´'qs ď ρ n px ´'q Then we apply Kato's inequality (II.4.7) to u x , use spu x qu x " |u x | and obtain `´h 2 ∆ `λ˘| u x | ď Re rspu x qL h,b u x s `λ |u x | ď ρ n px ´'q Ω |u x pzq| ρ n py ´zqdz ď ż Ω ρ n px ´zqρ n ˚Gλ p', yqpzqdz With the changes of variables tt `x ´y, zz `x ´y and Property II.4.4, nÑ8 0 Fix x P Ω, with Property II.4.4 up to a subsequence, |ρ n ˚GbA,λ px, 'q| Ñ |G bA,λ px, 'q| in L 2 pΩq so |G bA,λ px, 'q ˚ρn | ˚ρn Ñ nÑ8 |G bA,λ px, 'q| in L 2 pΩq and up to another subsequence almost everywhere. Similarly, almost everywhere ρ n ˚ρn ˚Gλ px, 'q Ñ nÑ8 G λ px, 'q So passing to the limit in (II.4.19), for almost every y P Ω, |G bA,λ px, yq| ď G λ px, 2x ´yq " G λ py ´xq " G λ px, yq

  1: Birman-Schwinger principleWe denote N λ the number of eigenvalues of L h,b `1 less than or equal to λ.

					From [13,
	Section 4.3],
		"		ı	ż
	´Tr	1 pLh,b`1´Vqď0 pL h,b `1 ´Vq	"	R `Nλ dλ
	Define the Birman-Schwinger operator
		K λ -	? VG bA,λ	? V
	and let B Hence with Proposition II.4.5,
	ˇˇˇˇˇż
	ˇˇG bA,λ px, yq ˇˇ" 3 2				3 2

λ be the number of eigenvalues of K λ greater or equal to 1. We use the diamagnetic inequality to restrict to the non magnetic case. Since G bA,λ is positive, we can define its square root. Using the arguments of

[START_REF] Lieb | The stability of matter in quantum mechanics[END_REF] Theorem 4.4]

we can deduce from Proposition II.4.5 the diamagnetic inequality for a G bA,λ : ˇˇaG bA,λ px, yq ˇˇď a G λ px, yq Ω G bA,λ px, zq a G bA,λ pz, yqdz ˇˇˇˇˇď ż Ω G λ px, zq a G λ pz, yqdz " G

  Rq " phλq 2 ∥∇g∥ 2 L 2 ∥m∥ L 1 `phλq 2 O pl b q " O `phλq 2 ˘(II.6.14)

	"4iπl 2 b	h ż NˆΩ	mpn, Rq	¨ż Ω	∇g λ px ´Rq ˆCpnq	b l b	`b r Epn, Rq ˙gλ px ´Rqdx 'dηpn, Rq
	"O phbλl b q						(II.6.15)
	Inserting (II.6.14) and (II.6.15) in (II.6.13), we obtain
							ż
			Tr rL h,b γ m s "	E n mpXqdηpXq `O phbλl b q	`O `phλq 2 ˘(II.6.16)
							NˆΩ
						ż
				2
				b
						ż
			´2πl 2 b			mpn, RqTr	" h2 ∆g λ p' ´RqΠ n g λ p' ´Rq	‰	dηpn, Rq	(II.6.13)
					NˆΩ
	With (II.5.14),
		ż
	´2πl 2 b	mpn, RqTr	"	h2 ∆g λ p' ´RqΠ n g λ p' ´Rq	‰	dηpn, Rq
	NˆΩ
	" ´h 2 ∆g And by (II.3.7), D r ż NˆΩ mpn, Rq ¨ż Ω E : N ˆΩ Ñ R such that
							P h,b Π n px, xq "	b l b	Cpnq `b r Epn, xq
							|Epn, xq| ď r Cpnq
	so
		ż
	2πl 2 b

NˆΩ

mpn, RqTr r2ih∇g λ p' ´RqP h,b Π n g λ p' ´Rqs dηpn, Rq

Using (II.3.6), DE : N ˆΩ Ñ R such that 2πl 2 b Π n px,

xq " 1 `lb Epn, xq |Epn, xq| ď Cpnq λ px ´Rq p1 `lb Epn, xqq g λ px ´Rqdx 'dηpn, Rq " phλq 2 ∥∇g∥ 2 L 2 ∥m∥ L 1 ´h 2 l b ż NˆΩ mpn, Rq ¨ż Ω λ 3 ∆gpλxqEpn, x `Rqλgpλxqdx 'dηpn, NˆΩ mpn, RqTr r2ih∇g λ p' ´RqP h,b Π n g λ p' ´Rqs dηpn, Rq

  Finally we prove the estimate on semi-classical energies (II.6.19):E n ∥m ´r m∥ L 8 `pn 1 `1q ∥V ∥ L 1 ∥m ´r m∥ L 8

		n 1		ż		ż
	|E sc,hb r r ms ´Esc,hb rms| ď	ÿ	E n	| r mpn, 'q ´mpn, 'q|	`n1 ÿ	|V | | r mpn, 'q ´mpn, 'q|
		n"0	Ω		n"0	Ω
					ż
		`n1 ÿ	|wpx ´yq| | r mpn, xq r mpr n, yq ´mpn, xqmpr n, yq| dxdy
			n,r n"0	Ω 2
				n 1
	ďL 2	ÿ
			n"0
					n 1
					ÿ
		`L2 ∥w∥ L 1	∥ r mpn, 'q r mpr n, 'q ´mpn, 'qmpr n, 'q∥ L 8
					n,r n"0
					n 1
					ÿ
					n"0

3: Comparison of the Semi-classical energies With inequalities (II.6.21) and (II.6.23) we know that ∥m ´r m∥ L 8 " Opl b q `O p1 ´∥m∥ L 1 q (II.6.24) Moreover ∥ r mpn, 'q r mpr n, 'q ´mpn, 'qmpr n, 'q∥ L 8 ď ∥ r mpn, 'q∥ L 8 ∥ r mpr n, 'q ´mpr n, 'q∥ L 8 `∥mpr n, 'q∥ L 8 ∥ r mpn, 'q ´mpn, 'q∥ L 8 ď ∥ r m∥ L 8 ∥ r m ´m∥ L 8 `∥m∥ L 8 ∥ r m ´m∥ L 8 so with (II.6.20) and (II.6.22) |E sc,hb r r ms ´Esc,hb rms| ď ˜L2

  6.8 associated with m ρ,N , it satisfies E sc,hb r r m ρ,N s " E sc,hb rm ρ,N s `O phbl b q `hbO ˆ1

								N ´dpq `rq	˙(II.6.26)
	and Tr	"	γ m ρ,N	‰	" 1, 0 ď γ m ρ,N ď	1 N	. Moreover by (II.6.16),
						Tr	"	L h,b γ m ρ,N	‰	" O phbq
	Thus, we can apply Propositions Proposition II.6.6, Proposition II.6.7 and (II.6.26):

  mpq, 'q then µ P P pD qLL q and

	ρ	pkq M "	ż	ˆq L 2 pq `rq	`ρ˙b k	dµpρq	(II.6.46)
			D qLL

  bA 1 ihB x 2 `bA 2 , p 2 s " ih pB x 1 A 2 ´Bx 2 A 1 q " ihbTake ψ P LLL, we saw in (III.2.5) that aψ " 0 so 0 "pp 1 `ip 2 qψ " ´pihB x 1 `A1 ´hB x 2 `ibA 2 qψ " ´ˆ2ihB x

	ȧnd define the annihilation and creation operators respectively as a -p 1 `ip 2 ? 2hb a : -p 1 ´ip 2 ? 2hb and the number of excitation operator N -a : a. The following is standard: Property III.2.3: Commutation relations We have the commutation relations: rp 1 , p 2 s " ihb " a, a : ‰ " Id (canonical commutation relation) and L b " 2hb ˆN `Id 2 ṡppN q " N Proof : Start with rp 1 so " a, a : ‰ " rp 1 , ´ip 2 s `rip 2 , p 1 s 2hb " ´2i rp 1 , p 2 s 2hb " Id and we compute N " p 2 1 `p2 2 `irp 1 , p 2 s 2hb " L b 2hb ´Id 2 Then, " x 2 ˙ψ `i b 2 x 1 " ´2ih ˆBx `x 4l 2 b ˙ψ (III.2.6) Hence Df P O, such that N , a ´b 2 ψ " f e ´|x| 2 4l 2 b

:

‰ " a : aa : ´a: a : a " a : " a, a : ‰ " a :

Proof :

  2.14) Π n,z px, yq " pB z `Bz q Π n,z px, yq pxqψ n,z pyq ´ψn,z pxqψ n´1,z pyq ānd B z 2 Π n,z px, yq "i pB z ´Bz q Π n,z px, yq

	we get	" "	y -x 2l 2 b y -x 2l 2 b	ψ n,z pxqψ n,z pyq ψ n,z pxqψ n,z pyq ψ N `1,z pyq " `x -z 2l 2 b ´i ? n `1 ψ n,z pxqψ n,z pyq ? 2l b ψ n`1,z pxqψ n,z pyq `z -x 2l 2 b i ? N `1 z -y ? ψ N,z pyq 2l b `i ? ψ n´1,z pxqψ n´1,z pyq n ? 2l b ψ n,z pxqψ n´1,z pyq
	and B z Π n,z px, yq " " " inserting (III.2.14), x -z 2l 2 b Π n,z px, yq ˆψN,z pxqψ N `1,z pyq ψ N `1,z pxqψ N,z pyq ˙" `z -y 2l 2 b Π n´1,z px, yq i ? N `1? 2l b x -y 2l 2 b ψ n,z pxqψ n,z pyq `y -z 2l 2 b ψ n,z pxqψ n,z pyq Π N,z px, yq b ˆz-y Ṫhus x-z x -y 2l 2 b ψ n,z pxqψ n,z pyq `i ? n `1 ? 2l b ψ n,z pxqψ n`1,z pyq n ? 2l b ψ n´1,z pxqψ n,z pyq ´i ? 2l 2 ψ n´1,z pxqψ n´1,z pyq `z -y
	So
	B z 1 "i ´ψn´1,z "i x 2 ´y2 l 2 b Π n,z px, yq ? n ? 2l b y 1 ´x1 l 2 b Π n,z px, yq	`i ? n ? 2l b `1 `?n `1 ? 2l b	ī ?n ´ψn,z pxqψ n`1,z pyq ´ψn`1,z pxqψ n,z pyq ´ψn,z pxqψ n`1,z pyq `ψn`1,z pxqψ n,z pyq
					? 2l b	´ψn´1,z pxqψ n,z pyq `ψn,z pxqψ n´1,z pyq We
	deduce that
			∇ K z Π n,z px, yq "i	x	´y l 2 b	Π n,z px, yq	´?n 2l b `1 ?	´i i ˆ1 1	ψ n`1,z pxqψ n,z pyq ˙ˆψ n,z pxqψ n`1,z pyq ?n
							2l b ?	´i i ˆ1 1	ψ n,z pxqψ n´1,z pyq ˙ˆψ n´1,z pxqψ n,z pyq Ȧfter
	Using	summation over n, we conclude that
			ψ n,z pxq " ´i ? n ψ n,z pyq " i ? n Π ďN,z px, yq ´?N z -x ? 2l b `1 ψ n´1,z pxq ˆ1 1 ? 2l b ´i i z -y ? 2l b ψ n´1,z pyq 2: Proof of (III.2.16) z Π ďN,z px, yq " i ∇ K x ´y l 2 b	Ṡtep ˙ˆψ N,z pxqψ N `1,z pyq ψ N `1,z pxqψ N,z pyq
	With (III.2.12) compute Noticing that
	B z Π n,z px, yq "	y -z 2l 2 b	Π n,z px, yq ψ N `1,z pxq " `z -x 2l 2 b Π n´1,z px, yq ´i ? N `1 z -x ? 2l b	ψ N,z pxq

  Tr rΠ n,z s " 4 ∥φ∥ 2 Next we identify the k nd moment of the kinetic energy in the sum:

	We can also write
		Tr	"	|φ ´φpzq| 2 Π n,z	‰	ď 4 ∥φ∥ 2 L 8 L 8	(III.4.7)
	Since
						Π n,z px, xq " Π n,x pz, zq
	with (III.4.6) and (III.4.7), by seeing that the integrand is symmetric in x and z we get
	ż						ż
	Tr	" |φ ´φpzq| 2 Π n,z	‰	dz "	|φpxq ´φpzq| 2 Π n,z px, xqdxdz
	R 2					psupppφqˆR 2 qYpR 2 ˆsupppφqq
						ż
						ď2	|φpxq ´φpzq| 2 Π n,z px, xqdxdz
						R 2 ˆsupppφq
						ż
						"2	Tr	" |φ ´φpzq| 2 Π n,z	‰	dz
						supppφq
						ďC |supppφq| ∥φ∥ 2 W 1,8 min `pn `1ql 2 b , 1 ˘(III.4.8)
							2 b ∥∇φ∥ 2 8	(III.4.6)

  Where the notation d n W stand for the n th differential of W , meaning that d n W pzq is a n-linear form. Notice that

		d 2 W pzqpx ´z, x ´zq	(III.5.8)
	so that
	W pyq ´W pxq ´py ´xq ¨∇W pzq "V z pyq ´Vz pxq
	`1 2	d 2 W pzqpy ´z, y ´zq	´1 2	d 2 W pzqpx ´z, x ´zq

d 2 W pzqpy ´z, y ´zq "d 2 W pzqpy ´x, y ´zq `d2 W pzqpx ´z, y ´zq "d 2 W pzqpy ´x, y ´zq `d2 W pzqpx ´z, y ´xq `d2 W pzqpx ´z, x ´zq so rΠ z , W ´X ¨∇W pzqs px, yq " Π z px, yq pW pyq ´W pxq ´py ´xq ¨∇W pzqq "Π z px, yq ˆVz pyq ´Vz pxq `1 2 d 2 W pzqpy ´x, y ´zq `1 2 d 2 W pzqpx ´z, y ´xq " rΠ z , V z s px, yq `1 2 d 2 W pzqpΠ z px, yqpy ´xq, y ´zq `1 2 d 2 W pzqpx ´z, Π z px, yqpy ´xqq (III.5.9) Thus with (III.2.13), we have the operator identity rΠ z , W ´X ¨∇W pzqs " rΠ z

  1: a direct computation Redefine V z pxq -W pxq ´dW pzqpx ´zq ´1 2 d 2 W pzqpx ´z, x ´zq ´1 6 d 3 W pzqpx ´z, x ´z, x ´zq Notice that d 3 W pzqpy ´z, y ´z, y ´zq "d 3 W pzqpy ´x, y ´z, y ´zq `d3 W pzqpx ´z, y ´z, y ´zq "d 3 W pzqpy ´x, y ´z, y ´zq `d3 W pzqpx ´z, y ´x, y ´zq `d3 W pzqpx ´z, x ´z, y ´zq "d 3 W pzqpy ´x, y ´z, y ´zq `d3 W pzqpx ´z, y ´x, y ´zq `d3 W pzqpx ´z, x ´z, y ´xq `d3 W pzqpx ´z, x ´z, x ´zq so (III.5.9) becomes rΠ ďN,z , W ´X ¨∇W pzqs px, yq " Π ďN,z px, yq ˆVz pyq ´Vz pxq

	`d2 W pzq 2	px ´z, y ´xq	`d3 W pzq 6	py ´x, y ´z, y ´zq	`d3 W pzq 6	px ´z, y ´x, y ´zq
	6 `d3 W pzq	px ´z, x ´z, y ´xq Ṫhen
	inserting (III.2.14),
	rΠ ďN,z , W ´X ¨∇W pzqs
	" rΠ ďN,z , V z s	`d2 W pzq 2	prΠ ďN,z , Xs , X ´zq	`d2 W pzq 2	pX ´z, rΠ ďN,z , Xsq
	`d3 W pzq 6	prΠ ďN,z , Xs , X ´z, X ´zq	`d3 W pzq 6	pX ´z, rΠ ďN,z , Xs , X ´zq
	`d3 W pzq 6	pX ´z, X ´z, rΠ ďN,z , Xsq
	" rΠ ďN,z , V z s	`il 2 b 2	d 2 W pzqp∇ K z Π ďN,z , X ´zq	`il 2 b 2	d 2 W pzqpX ´z, ∇ K z Π ďN,z q
	`il 2 b 6	d 3 W pzqp∇ K z Π ďN,z , X ´z, X ´zq	`il 2 b 6	d 3 W pzqpX ´z, ∇ K z Π ďN,z , X ´zq
	`il 2
	`i? N `1l b 2 ? 2 ? N `1l b 2 ? 2 ? N `1l b 6 ? 2 ? N `1l b 6 ? 2 ? N `1l b 6 ? 2	d 2 W pzq d 2 W pzq ˆX ´z, ˆˆ1 1 ´i i ˆ1 1 ˙ˆ|ψ N,z ⟩ ⟨ψ N `1,z | |ψ N `1,z ⟩ ⟨ψ N,z | ˙, X ˙ˆ|ψ N,z ⟩ ⟨ψ N `1,z | ´zi ˙i ´i i |ψ N `1,z ⟩ ⟨ψ N,z | d 3 W pzq ˆˆ1 1 ´i i ˙ˆ|ψ N,z ⟩ ⟨ψ N `1,z | ˙, X ´z, X ´zi |ψ N `1,z ⟩ ⟨ψ N,z | d 3 W pzq ˆX ´z, ˆ1 1 ´i i ˙ˆ|ψ N,z ⟩ ⟨ψ N `1,z | ˙, X ´zi |ψ N `1,z ⟩ ⟨ψ N,z | ´i i |ψ N `1,z ⟩ ⟨ψ N,z | d 3 W pzq ˆX ´z, X ´z, ˆ1 1 ˙ˆ|ψ N,z ⟩ ⟨ψ N `1,z | ˙˙(III.5.45)
	Defining the errors terms
							`d2 W pzq 2	py ´x, y ´zq

b 6 d 3 W pzqpX ´z, X ´z, ∇ K z Π ďN,z q

  8 pR `ˆR 2 q uniformly in t and b. Let φ P C 8 ptq ˘‹ ∇ K w W 1;8 ď ∇φptqρ sc,ďN γ b ptq L 1 ∥w∥ W 1;8 ď ∥φ∥ W 1,8 ∥w∥ W 1;8 (III.6.11) and with B t φ instead of φ `∇B t φptq ¨ρsc,ďN γ b ptq ˘‹ ∇ K w L 8 ď ∇B t φptqρ sc,ďN γ b ptq L 1 ∥w∥ L 8 ď ∥φ∥ W 1,8 ∥w∥ L 8 (III.6.12) Let x P R 2 , with an integration by parts, ∇ K`V `w ‹ ρ γ b ptq ˘pzq ¨∇z ρ γ b pt, zq∇ z φpt, zq ¨∇K wpx ´zqdz `ż R 2 GYRO ργ b pρ sc,ďN γ b qpt, zq∇ z φpt, zq ¨∇K wpx ´zqdz " ż R 2 ρ γ b pt, zq∇ K `V `w ‹ ρ γ b ptq ˘pzq ¨∇z `∇z φpt, zq ¨∇K wpx ´zq ˘dz `ż R 2 GYRO ργ b pρ sc,ďN γ b qpt, zq∇ z φpt, zq ¨∇K wpx ´zqdz " ż R 2 ρ γ b pt, zq∇ K `V `w ‹ ρ γ b ptq ˘pzq ¨∇b2 z φpt, zq∇ K wpx ´zqdz `ż R 2 ρ γ b pt, zq∇ K `V `w ‹ ρ γ b ptq ˘pzq ¨∇z b ∇ K z wpx ´zq∇ z φpt, zqdz `ż R 2 GYRO ργ b pρ sc,ďN γ b qpt, zq∇ z φpt, zq ¨∇K wpx ´zqdz (III.6.13) For the two first terms in (III.6.13) we use Hölder inequality, and for the last term Proposition III.5.4 with ∇ z φptq ¨∇K wpx ´'q as test function. Hence with the choices (III.6.6), `∇φptq ¨Bt ρ sc,ďN γ b ptq ˘‹ ∇ K w L 8 ď ∇V `∇w ‹ ρ γ b ptq L 8 d 2 φptq L 8 ∥∇w∥ L 8 ` ∇V `∇w ‹ ρ γ b ptq L 8 d 2 w L 8 ∥∇φptq∥ L 8 `C p1 `|supppφq|q ∇φptq ¨∇K wpx ´'q W 1,8 V `w ‹ ρ γ b ptq W 4,8 ∇φptq ¨∇K wpx ´'q W 1,8 ď ∥φptq∥ W 2,8 ∥w∥ W 2,8 ∥φptq∥ W k,8 ď ∥φ∥ W k,8 Step 4: Conclusion With the choices (III.6.6), by (III.6.3), Using (III.6.15) and (III.6.8) for n large enough so supppφq Ă Ω n , Hence the five terms in (III.6.10) converge to 0 respectively because of (III.6.16), (III.6.17), (III.6.18), (III.6.19), (III.6.20) hence the conclusion

	`∇φptq ¨Bt ρ sc,ďN γ b	ptq ˘‹ ∇ K wpxq
	ż	ˇˇˇˇˇˇż					ˇˇˇˇˇˇď
	" " With (III.6.9), since φp0q P C 0 R 2 B t ρ sc,ďN γ b pt, zq∇ z φpt, zq ¨∇K wpx ´zqdz φGYRO ργ b pρ sc,ďN qdtdz Cpφ, V, wq γ b R `ˆR 2 ´ż R 2 c pR 2 q,	1 ln `l´1 b b	˘(III.6.16)
						ż
						φp0qpρ 0	´ρsc,ďN γ b	p0qq Ñ bÑ8	0	(III.6.17)
						R 2
	Using (III.6.11), similarly as for (III.6.4) we have
	ˇˇˇˇˇˇż							ˇˇˇˇˇˇď
	R `ˆR 2	`ργ b	´ρsc,ďN γ b	˘`∇ z φ ¨ρsc,ďN γ b	˘‹ ∇ K w	Cpφ, V, wq	1 ln `l´1 b
		ˇˇˇˇˇˇż						ˇˇˇˇˇď
			`ρsc,ďN γ b	´ρ˘`∇	z φ ¨ρsc,ďN γ b	˘‹ ∇ K w
		R `ˆR 2
		ρ ´ρsc,ďN γ b	W ´1,1 pΩnq `∇z φ ¨ρsc,ďN γ b	˘‹ ∇ K w W 1,8 Ñ bÑ8	0	(III.6.19)
	And since GYRO ρ pφq P C 0 c pR ˆR2 q, (III.6.7) gives
					ż
						`ρsc,ďN γ b	´ρ˘G YRO ρ pφq Ñ bÑ8	0	(III.6.20)
				R `ˆR 2
								1
								b
								ln `l´1
					ż
					R 2	c φp0qρ 0 `R˚ˆR2 ˘, then @t P R `, ´ż R `ˆR 2 ρGYRO ρ pφq " 0
	`∇φptq ¨ρsc,ďN γ b hence
			`∇φptq ¨Bt ρ sc,ďN γ b	ptq ˘‹ ∇ K w L 8 ď Cpφ, V, wq	(III.6.14)

b ȃnd @t P R `, With (III.6.11), (III.6.12), (III.6.14) we conclude that

`∇z φ ¨ρsc,ďN γ b ˘‹ ∇ K w W 1,8 ď Cpφ, V

, wq (III.6.15) b ˘(III.6.18)

commutateurs, entre γ b et l'énergie cinétique ou des moments de l'opérateur position, pourraient aussi être utile. Ces méthodes devraient aussi permettre d'améliorer la vitesse de convergence logarithmique dans le Theorem I.3.2.Ensuite, un but principal est de prendre comme point de départ l'équation de Schrödinger à N -corps. Une première avancée dans cette direction serait de considérer l'équation de Hartree Fock à la place de celle de Hartree, c'est-à-dire chercher à contrôler le terme d'échange dans la limite de fort champ magnétique. Ce terme ne devrait pas poser de problème car il est petit dans un régime semi-classique.En étudiant le problème 3d, on pourrait obtenir pour la dynamique un terme supplémentaire de transport dans la direction du champ magnétique.

2

Remerciements

Proof :

Let pu n q nPN be bounded in H 1 pRq X L 2 pR, |x| dxq, in others words:

Since pu n q nPN and pp1 `|'|qu n q nPN are bounded in L 2 pΩq, after extraction, we can assume The fist term goes to 0 because of (III.2.2), (III.2.3) and dominated convergence. The second term goes to 0 when R Ñ 8 uniformly in n because p u P L 2 pRq and px u n q nPN is a tight sequence because from (III.2.1),

Therefore we known that L b has a pure point spectrum and that L 2 pΩq is a Hilbertian direct sum of its eigenspaces.

III.2.2 Landau quantization and similarly rN , as " ´a

Let n ě 0 and assume u n is a normalised eigenvector of eigenvalue n, then N a : u n " a : aa : u n " a : pa : a `1qu n " a : pN `1qu n " pn `1qa : u n and compute the norm

Therefore

is also a normalised eigenvector of eigenvalue n ´1. Iterating this process we must have n P N otherwise we construct eigenvectors of negative eigenvalues which is impossible since N is positive. Thus sppN q Ă N.

Since the spectrum is non empty it contains one eigenvector of integer eigenvalue. And successive applications of a and a : to this vector generates the whole spectrum N. Moreover, if u 0 is an eigenvector of egeinvalue 0, au 0 satisfies N au 0 " ´au 0 but au 0 cannot be an eigenvector so au 0 " 0 (III.2.5)

Notation III.2.4: Landau levels

We define the n th Landau level as the eigenspace associated to n P N:

The ground level, denoted LLL for Lowest Landau Level has energy E 0 " hb.

Next, we state that the Landau levels are isomorphic.

Proposition III.2.5: Mapping between Landau levels

The Landau levels are isomorphic, and the operator a : ? n `1 is a unitary mapping from nLL to (n+1)LL of inverse a ? n `1 .

Proof :

Because of (III.2.4),

˙aψ so L b aψ P L 2 pΩq and aψ P H 2 pΩq. Thus aψ P nLL and a : ? n `1 : nLL Ñ (n+1)LL is surjective. Finally the mapping is unitary:

`1 xχ|pN `1qψy " xχ|ψy Therefore we can extend a basis of LLL using a : to higher Landau levels.

III.2.3 Landau levels

It is know [START_REF] Rougerie | Holomorphic quantum hall states in higher landau levels[END_REF] that the Lowest Landau level consists of holomorphic functions pondered by a Gaussian factor.

Notation III.2.6

For the rest of the text, we will identify

with the complex notation xx 1 `ix 2 and use

Proposition III.

III.4 Semi-classical approximations

In this section, we introduce the semi-classical density (III.4.2) and the truncated semi-classical density (III.4.3) that only takes into account Landau level under a certain threshold. One main difficulty in our method is that the coherent states (Definition III.2.11) are not well localized on high Landau levels, hence the introduction of the truncated semi-classical density. The Kinetic energy is used to control the number of particles in high Landau levels. The truncated semi-classical density turns out to be a good approximation of the physical density (Proposition III.4.3). It is also possible to show that the semi-classical (untruncated) density is a good approximation of the physical density (Proposition III.4.2) but this requires higher moments of the kinetic energy.

Definition III.4.1

Let γ P L 1 pL 2 pR 2 qq. We define the semi-classical phase space density associated to γ, so-called Husimi function by The parameter N will represent the number of Landau levels we take into account for the semi-classical approximations in Section III.4 and Section III.5. we have the asymptotics The second term in (III.4.10) requires k ą 1 for the convergence of the series. If we only assume that the kinetic energy is bounded (k " 1q, in order to deal with the divergent series in (III.4.10) we need to consider the truncated semi-classical density (III.4.3) instead of (III.4.2). This is the goal of the next section.

III.4.1 Semi-classical density

III.4.2 Truncated semi-classical density

This time, the right-hand side term is small for k " 1 if we choose N such that ? N l b ! 1. This constraint has a physical meaning. Indeed from the expression of the coherent state (III.

III.6 Proofs of the main results

This last section contains the conclusion of the proofs of Theorem III.1.3 and Theorem III.1.4. What is left to do is passing to the limit in the estimates of Section III.4 and Section III.5.

Lemma III.6.1: Choice of a convenient N Assume for contradiction that

which yields the desired contradiction.

We may now conclude the proof of our main result.