Arnaud Triay

DERIVATION OF THE DIPOLAR GROSS-PITAEVSKII ENERGY

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Introduction

Bose-Einstein Condensation (BEC) is a phenomenon occurring at very low temperature for a highly dilute gas of bosons. In the proper experimental conditions, most of the particles get to occupy the same quantum state. In 1925, Bose [START_REF] Bose | Plancks Gesetz und Lichtquantenhypothese[END_REF] and Einstein [START_REF] Einstein | Quantentheorie des einatomigen idealen Gases, Akademie der Wissenshaften[END_REF] proved condensation for ideal particles, that is, under the important assumption that the particles do not interact with each other. Seventy years later, Cornell and Wieman [START_REF] Anderson | Observation of Bose-Einstein condensation in a dilute atomic vapor[END_REF] obtained the first experimental realization of a full BEC with Rubidium atoms. This major achievement has triggered a new interest in the theoretical study of condensation. In 1999 the first mathematical proof of BEC was provided by Lieb, Seiringer and Yngvason [START_REF] Lieb | Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional[END_REF][START_REF] Lieb | The mathematics of the Bose gas and its condensation[END_REF] for confined particles interacting via positive, radial and short range pair interaction, in the dilute regime.

The experimental study of BEC remains a very active field of research today. A challenging task is to realize condensates of particles with diverse interactions. In 2005, a first dipolar condensate has been observed by Griesmaier et al using Chromium particles [START_REF] Griesmaier | Bose-Einstein Condensation of Chromium[END_REF]. The dipolar interaction differs in many points from the ones encountered before. It is long range, anisotropic and has an attractive and a repulsive part. Its study is both experimentally and mathematically intricate but it opens the way to new physical effects. For instance, it is believed that dipolar condensates exhibit a roton-maxon excitation spectrum [START_REF] Santos | Roton-maxon spectrum and stability of trapped dipolar Bose-Einstein condensates[END_REF] which is not observed for simpler interactions. See for example [START_REF] Lahaye | The physics of dipolar bosonic quantum gases[END_REF] for a review of the physical properties of the dipolar Bose gas.

In this article, we give the first derivation of the Gross-Pitaevskii (GP) theory for dipolar Bose gases, starting from the many-particle linear Schrödinger problem. Our method is based on mean field limits and de Finetti theorems introduced in [START_REF] Lewin | Derivation of Hartree's theory for generic mean-field Bose gases[END_REF][START_REF] Lewin | The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases[END_REF] as well as new techniques developed to deal with negative pair interaction that can been found in [START_REF] Thành Nam | Ground states of large bosonic systems: The Gross-Pitaevskii limit revisited[END_REF][START_REF]A note on 2d focusing many-boson systems[END_REF].

More precisely, we will show that the particles in the condensate have a common state which can be computed by minimizing the dipolar Gross-Pitaevskii functional:

E a,b GP (u) = R 3 |(∇ + iA(x))u(x)| 2 dx + R 3 V (x)|u(x)| 2 dx + a 2 R 3 |u(x)| 4 dx + b 2 R 3 ×R 3 (K ⋆ |u| 2 (x))|u(x)| 2 dx. (1)
The first term in [START_REF] Anderson | Observation of Bose-Einstein condensation in a dilute atomic vapor[END_REF] represents the kinetic energy where A : R 3 → R 3 is a vector potential (modeling a magnetic field or the Coriolis force due to the rotation of the atoms). The second term is the one-body potential energy where V : R 3 → R is a trapping potential, that is, V (x) → ∞ when |x| → ∞. The third term is the short range interaction in the gas where a ∈ R is proportional to the scattering length or an approximation of it. The last term is the dipolar energy and b ∈ R is proportional to the norm of the dipoles. The dipolar interaction potential

K dip (x) = 1 -3 cos 2 (θ x ) |x| 3 = Ω dip (x/|x|) |x| 3 , (2)
represents the interaction between two aligned dipoles located at distance |x|. The parameter θ x is the angle between x and the direction n of all the dipoles, namely cos(θ x ) = n • x/|x|. In this paper we consider more general long-range potentials of the form

K(x) = Ω(x/|x|) |x| 3 (3)
where Ω is a even function satisfying the cancellation property on S 2 , the unit sphere of R 3 ,

S 2 Ω(ω)dσ(ω) = 0, (4)
where dσ denotes the Lebesgue measure on S 2 . Note that the convolution is an improper integral which is not absolutely convergent. It will be discussed later in Lemma 10. If all the particles share the same quantum state u, the latter should minimize the functional in [START_REF] Anderson | Observation of Bose-Einstein condensation in a dilute atomic vapor[END_REF] under the normalization constraint

K ⋆ |u| 2 (x) =
R 3 |u(x)| 2 dx = 1.
We therefore introduce the ground state energy e GP (a, b) := inf

u L 2 =1 E a,b GP (u). (5)
This variational problem has been extensively studied, both theoretically [START_REF] Carles | On the Gross-Pitaevskii equation for trapped dipolar quantum gases[END_REF][START_REF] Carles | Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases[END_REF][START_REF] Antonelli | Existence of solitary waves in dipolar quantum gases[END_REF] and numerically [START_REF] Bao | Gross-Pitaevskii-Poisson equations for dipolar Bose-Einstein condensate with anisotropic confinement[END_REF][START_REF] Bao | Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates[END_REF] in the dipolar case [START_REF] Antonelli | Existence of solitary waves in dipolar quantum gases[END_REF].

Our aim is to justify the validity of the Gross-Pitaevskii minimization [START_REF] Carles | Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases[END_REF], starting with the exact many-body problem based on the Hamiltonian

H N = N j=1 -∇ x j + iA(x j ) 2 + V (x j ) + 1 N -1 1≤i<j≤N N 3β w(N β (x i -x j )). (6)
This operator acts on N s L 2 (R 3 ), the symmetric tensor product of N copies of L 2 (R 3 ). We denote by e N := inf

Ψ∈ N s L 2 (R 3 ) Ψ L 2 =1
Ψ, H N Ψ N the many-body ground state energy per particle. We investigate the limit of a large number of particles, N → ∞. In [START_REF] Carles | On the Gross-Pitaevskii equation for trapped dipolar quantum gases[END_REF], the scaling chosen for the interaction between the particles is very common. On the one hand, the L 1 -preserving scaling of the potential

w N (x) := N 3β w(N β x)
will retain in the limit only its short and long range parts. In our case, we will have

w N ⇀ aδ + bK,
since K and δ behave the same under scaling. On the other hand, the coupling factor (N -1) -1 is typical of mean field limits and ensures that the potential is of the same order as the kinetic energy. The function w is the real interaction between the atoms in the gas. It will be assumed to be repulsive at short distances (hence a > 0) and of dipolar type (close to bK) at large distances.

The parameter β ∈ [0, 1] interpolates between the pure mean-field regime β = 0 and the Gross-Pitaevskii regime β = 1 which is more difficult to handle. The case β < 1/3 corresponds to a high density regime with the interaction length N -β being larger than the mean distance N -1/3 between the particles. This is a natural setting for the law of large numbers to apply. The case β > 1/3 is more subtle and corresponds to a low density regime where the particles meet rarely but interact with intensity proportional to N 3β-1 . The difficulties culminate at β = 1 where now the details of the scattering process play a role. When w has a negative part, it is natural to assume that the true interaction w is (classically) stable of the second kind, which means that 1≤i<j≤N w(x ix j ) ≥ -CN, for all N ≥ 2 and all space configurations x 1 , ..., x N ∈ R 3 of the N particles [START_REF] Ruelle | Statistical mechanics. Rigorous results[END_REF]. This assumption ensures automatically that e N is bounded from below when β ≤ 1/3. The case β > 1/3 is particularly difficult and the proof that e N is bounded from below requires to use the kinetic energy, that is, the quantum feature of the system. The scaled interaction alone is unstable.

For β = 0 the convergence of e N to the Hartree energy was proven in [START_REF] Lewin | Derivation of Hartree's theory for generic mean-field Bose gases[END_REF]. In this paper, we prove that lim

N →∞ e N = e GP (a, b) (7)
as N → ∞ for

0 < β < 1 3 + s 45 + 42s
under the sole assumption that w is stable as in [START_REF] Fannes | Finite size mean-field models[END_REF] and that V (x) ≥ C|x| s at infinity. In that case, we are able to show that e N ≥ -C but it is unclear if additional assumptions on w are needed for higher β's. Under the additional condition that e N ≥ -C our result [START_REF] Chang | Methods in nonlinear analysis[END_REF] applies up to β < 2/3. In addition to the convergence of the energy, we are also able to prove the convergence of the minimizers. Loosely speaking, we prove that the ground state of (5) factorizes in the limit, that is,

Ψ(x 1 , ..., x N ) ≃ u(x 1 ) • • • u(x N ) as N → ∞, (8)
where u is a minimizer of the dipolar Gross-Pitaevskii functional. The meaning of ( 8) is in the sense of density matrices and not in L 2 norm as we will recall.

The paper is organized as follows. In Section 2 we state our main contributions, including some results on the existence and uniqueness of minimizers of the Gross-Pitaevskii functional (1) as well as our main convergence property [START_REF] Chang | Methods in nonlinear analysis[END_REF]. In Section 3 we prove our main result, Theorem 5.

Main results

2.1.

Properties of the dipolar Gross-Pitaevskii functional. As discussed before, the dipolar GP equation has already been widely studied and we slightly extend the results of [START_REF] Bao | Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates[END_REF][START_REF] Carles | Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases[END_REF] to deal with more general K and V as well as a magnetic potential or a Coriolis force A. The main adjustment concerns the uniqueness of minimizers which, in order to remain true, requires some extra assumptions on the intensity of the magnetic field.

Theorem 1 (Existence of GP minimizers).

Let us assume that

V ∈ L 1 loc (R 3 ), A ∈ L 2 loc (R 3
) and that there is some s > 0 such that

V (x) ≥ C -1 (|A(x)| 2 + |x| s ) -C, ∀x ∈ R 3 (9)
for some C > 0. Let

K(x) = Ω(x/|x|) |x| 3
with Ω ∈ L q (S), for some q > 1, a even function satisfying the cancellation property

S 2 Ω(w)dσ(w) = 0.
Let K be the Fourier transform of K in the sense of the principal value (as defined later in Lemma 10)

. i) If    b > 0 and a ≥ b (inf K) -, or b < 0 and a ≥ -b (sup K) + , (10)
then e GP (a, b) > -∞ and E GP has minimizers. ii) If    b > 0 and a < b (inf K) -, or b < 0 and a < -b (sup K) + , (11)
then e GP (a, b) = -∞.

Remark 2.

In the dipolar case where

Ω(x/|x|) = 1-3 cos 2 (θ x ), then K(k) = 4π 3 3 cos 2 (θ k ) -1 , (inf K) -= 4π/3 and (inf K) + = 8π/3.
The understanding of the effective theory is necessary to grasp the forthcoming difficulties in the derivation from the many-body theory. The condition i) on a and b aims to make the interaction stable which in this case is equivalent to be positive, mainly because then a+b K ≥ 0. Indeed, one can easily see that if the interaction term is negative for some configuration (a, b) and some wave function u, then by scaling one gets that e GP (a, b) = -∞.

In fact, most of our results remains true if we change (∇ + iA(x)) 2 by any abstract self-adjoint operator

h such that h ≥ C -1 (-∆ + |x| s ) -C.
We first state some lemma that will be useful in the proof of Theorem 1.

Lemma 3 (Magnetic Laplacian).

Under assumption [START_REF] Christandl | One-and-a-half quantum de Finetti theorems[END_REF], one has the operator inequality on L 2 (R 3 )

C -1 (-∆ + V ) -C ≤ (-i∇ + A) 2 + V ≤ C(-∆ + V + 1),
for some constant C > 0.

Proof of Lemma 3. Using the Cauchy-Schwarz inequality for operators (45) and inequality (9), we have the upper bound

(p + A) 2 = p 2 + pA + Ap + |A| 2 ≤ C(p 2 + V ) + C.
To get the lower bound, we choose some 0 < η < 1 sufficiently close to 1 but fixed and we use again the Cauchy-Schwarz inequality

(p + A) 2 + V ≥ (1 -η)p 2 + (1 -η -1 )|A| 2 + V ≥ C -1 (p 2 + V ) -C.
Proof of Theorem 1. We start by proving i). First, since V ∈ L 1 loc (R 3 ) and A ∈ L 2 loc (R 3 ) the quadratic form associated with the first two terms of the energy E a,b GP is closed on a domain included in H 1 (R 3 ) and provides a self-adjoint realization of H 0 = -(∇ + iA) 2 + V by the method of Friedrichs [START_REF] Leinfelder | Schrödinger operators with singular magnetic vector potentials[END_REF]. Then, to prove i), it suffices to follow the same proof as in [START_REF] Carles | Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases[END_REF] and to use, when necessary, that H 0 = -(∇ + iA) 2 + V has compact resolvent [START_REF] Iwatsuka | Magnetic Schrödinger operators with compact resolvent[END_REF]. More precisely, let us take a minimizing sequence (u n ) for E GP . The condition on a and b ensures that the interaction part of the energy is non-negative

a |u| 4 + b K ⋆ |u| 2 |u| 2 = a + b K | |u| 2 | 2 ≥ 0.
Thus E GP is bounded below on the unit sphere of L 2 (R 3 ) and u n , H 0 u n is bounded. We can then extract of (u n ) a converging subsequence in L 2 ∩ L 4 . Indeed, thanks to Lemma 3, H 0 has compact resolvent. We can write

u n = H -1/2 0 H 1/2
0 u n and since (H

1/2 0 u n ) is bounded in L 2 (R 3 ), because (u n ) is a minimizing sequence of E a,b GP , then (u n ) is precompact in L 2 (R 3
). We can then extract a converging subsequence that we still denote by (u n ) and we write u its limit. Fatou's lemma gives

E(u) ≤ lim inf E GP (u n ) = inf E GP .
For ii), it suffices to employ Lemma 3 above to adapt the proof of [START_REF] Carles | Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases[END_REF]. Without loss of generality, we can assume that e 3 ∈ {a + b K < -δ} = ∅. Let f be a smooth function compactly supported with f L 2 = 1 and let us denote by ρ

(x) = |f | 2 (x) and f λ (x) = f (λx 1 , λx 2 , λ -2 x 3 ). We have f λ L 2 = f L 2 = 1, ρ 2 λ L 1 = ρ 2 L 1 = f 4 L 4 and ρ λ L ∞ ≤ ρ λ L 1 = 1 for all λ > 0.
For any r > 0 we denote by C r the cone with vertex at the origin, with direction e 3 and with angle r. For any η > 0, there exists λ 0 such that for any 0 < λ ≤ λ 0 , we have

R 3 \Cr ρ 2 λ < η. (12)
According to Lemma 10 below, K is an homogeneous function which is continuous except at the origin. Since K is continuous at e 3 , we can find r > 0 such that B(e 3 , r) ⊂ {a + b K < -δ}. Now for any η > 0, there exists some λ 0 > 0 such that for all λ ≤ λ 0 we have [START_REF] Einstein | Quantentheorie des einatomigen idealen Gases, Akademie der Wissenshaften[END_REF], then

a + b K | ρ λ | 2 = R 3 \Cr a + b K | ρ λ | 2 + Cr a + b K | ρ λ | 2 ≤ η a + b K L ∞ -( f 4 L 4 -η)δ. If we take η < δ f 4 L 4 /(2δ + 2 a + b K L ∞ ), we obtain a|f λ | 4 + bK ⋆ |f λ | 2 |f λ | 2 = a + b K ρ 2 λ ≤ - f 4 L 4 δ 2 < 0.
Now, let us denote by ϕ = f λ 0 and by ϕ ℓ = ℓ 3/2 ϕ(ℓx). Since V is L 1 loc and ϕ is compactly supported, with, say, supp ϕ ⊂ B(0, R), we have

R 3 V |ϕ ℓ | 2 ≤ ℓ 3 B(0,Rℓ -1 ) |V | = o(ℓ 3 ).
Applying Lemma 3 we obtain

E GP (ϕ ℓ ) ≤ C |∇ϕ ℓ | 2 + V |ϕ ℓ | 2 + a|ϕ ℓ | 4 + bK ⋆ |ϕ ℓ | 2 |ϕ ℓ | 2 .
And for ℓ large enough, we have

e GP (a, b) ≤ Cℓ 2 |∇ϕ| 2 + ℓ 3 B(0,Rℓ -1 ) |V | + ℓ 3 a|ϕ| 4 + bK ⋆ |ϕ| 2 |ϕ| 2 ≤ Cℓ 2 - f 4 L 4 δ 2 ℓ 3 .
Taking the limit ℓ → ∞, the last inequality gives e GP (a, b) = -∞.

If A = 0, we can assume u > 0, because |∇u| ≥ |∇|u|| almost everywhere. Then since a + b K ≥ 0, the functional is strictly convex with respect to |u| 2 and uniqueness follows as in [START_REF] Bao | Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates[END_REF] where the case K = K dip was considered. On the other hand, if A = 0, the functional is no more convex and uniqueness can fail. Indeed, a strong magnetic field can create vortices which are a sign of rotational symmetry breaking. This situation was already encountered in the non dipolar case K = 0 in [START_REF] Seiringer | Gross-Pitaevskii theory of the rotating Bose gas[END_REF][START_REF]Ground state asymptotics of a dilute, rotating gas[END_REF]. A small magnetic field A does not create any vortex and uniqueness remains true, as we will show in Theorem 4.

In the non-dipolar case K = 0, uniqueness has been shown in several situations [START_REF] Lieb | The mathematics of the Bose gas and its condensation[END_REF][START_REF] Seiringer | Gross-Pitaevskii theory of the rotating Bose gas[END_REF][START_REF]Ground state asymptotics of a dilute, rotating gas[END_REF]. These works can all be adapted to the dipolar case K = 0 but here for shortness we only discuss an extension of [START_REF] Lieb | The mathematics of the Bose gas and its condensation[END_REF]Ch. 7]. In the following result we apply the implicit function theorem for A small enough without getting any information on how small A has to be. The second part deal with a radial magnetic field and gives an explicit range of validity as in [START_REF] Lieb | The mathematics of the Bose gas and its condensation[END_REF]Ch. 7].

Theorem 4 (Uniqueness of the minimizer for the dipolar GP functional with magnetic field). Let A, V and K satisfy the assumptions of Theorem 1 and let (a, b) be admissible parameters as in [START_REF] Combes | Classical bounds and limits for energy distributions of hamilton operators in electromagnetic fields[END_REF].

(1) For t ∈ R, we denote by E a,b,t GP the Gross-Pitaevskii functional where A has been replaced by tA. Then, there exists t 0 > 0 such that for all |t| < t 0 , E a,b,t GP has a unique minimizer, up to a phase factor, in the sector of mass R 3 |u| 2 = 1.

(2) Assume A(x) = a(r, z)e θ , V (x) = V (r, z) where (r, θ, z) are the cylindrical coordinates with e z = n (the commmon direction of all the dipolar moments). If ra ∞ < 1/2 then E GP has a unique minimizer, up to a phase factor, in the sector of mass R 3 |u| 2 = 1. It is non negative and axially symmetric, with axis e z = n.

The proof is given in Appendix C.

2.2.

Derivation of the dipolar Gross-Pitaevskii energy and minimizers. The phenomenon of condensation can be detected through the convergence of the ground state and the ground state energy towards those predicted by the Gross-Pitaevskii theory. We want to give general assumptions on w to be stable and to behave like the dipolar interaction at long distances.

Investigating this problem, one can first notice that taking w = w 0 + K with w 0 ∈ L 1 is not permitted. Indeed, the two particles hamiltonian H 2 is ill defined due to the singularity of K at the origin. The dipolar interaction is only physically relevant at large distances and therefore it is reasonable to assume that w is close to K outside of a ball.

Theorem 5 (Convergence of the energy).

Let A ∈ L 2 loc (R 3 ) and 0 ≤ V ∈ L 1 loc (R 3 ) satisfying V (x) ≥ C -1 (|A(x)| 2 + |x| s ) -C, ∀x ∈ R 3 . Let K = Ω(x/|x|)|x| -3 with Ω ∈ L q (S 2
), for some q ≥ 2, a even function satisfying the cancellation property (4). Let w, R > 0 and b ∈ R, be such that

w -b1 |x|>R K ∈ L 1 (R 3 ) ∩ L 2 (R 3 ).
We also assume w to be classically stable, that is

1≤i<j≤N w(x i -x j ) ≥ -CN, ∀N ≥ 2, ∀x 1 , ..., x N ∈ R 3 , (13)
for some C > 0 independent of N . If

β < 1 3 + s 45 + 42s (14)
then lim

N →∞ e N = e GP (a, b), (15)
for

a = R 3 w(x) -b1 |x|>R K(x) dx.
In addition, a and b satisfy the stability property [START_REF] Combes | Classical bounds and limits for energy distributions of hamilton operators in electromagnetic fields[END_REF].

Remark 6.

The convergence rate given by the proof is the sum of the error in (62) and of another one depending on w (see Lemma 16) which comes from the approximation of the Gross-Pitaevskii energy by the Hartree one. We do not state any quantitative estimate here for shortness.

Notice that the value of the constant R > 0 in ( 5) has no importance because of the cancellation property of K, see Lemma 10 below.

If V (x) → ∞ when |x| → ∞ faster than any polynomial (e.g. V = +∞ outside a bounded domain) then the condition ( 14) reduces to

β < 1 3 + 1 42 .
We now turn to the convergence of states, which is expressed in terms of the k-particle reduced density matrices. For Ψ ∈ N s L 2 (R 3 ), let us define its k-particle density matrix by γ

Ψ := Tr k+1→N |Ψ Ψ| , or, in terms of its kernel,

γ (k) Ψ (x 1 , ..., x k , y 1 , ..., y k ) = (R 3 ) N-k Ψ(x 1 , ..., x k , z k+1 , ..., z N )Ψ(y 1 , ..., y k , z k+1 , ..., z N )dz k+1 ...dz N .
From now on, when we consider a ground state Ψ N of H N , we will denote by γ (k) N its k-particle reduced density matrix for simplicity.

Theorem 7 (Convergence of states).

Under the assumptions of Theorem 5, for any sequence of ground states (Ψ N ), there exists a Borel probability measure µ supported on M GP (a, b), the set of ground states of E a,b GP , such that, up to a subsequence (N ′ ),

γ (k) N ′ -→ N ′ →∞ M GP |u ⊗k u ⊗k | dµ(u), ∀k ≥ 1, ( 16
)
where the convergence is in the trace norm. Besides, if E a,b GP has a unique minimizer, then there is convergence of the whole sequence (γ [START_REF] Harrow | The church of the symmetric subspace[END_REF]. Proof. Since the energy per particle is bounded by Theorem 5, the convergence of states can be proved exactly as in [START_REF] Lewin | The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases[END_REF]Theorem 2.5].

(k) N ) N in
We now make several remarks concerning our two main results, Theorem 5 and 7.

The main novelty of these two results is that the interaction potential has a negative part, is anisotropic and long range. Furthermore the derivation holds for some β > 1/3 where the stability is of quantum nature (due to the kinetic term). Unfortunately, we are not yet able to push the analysis to larger β's, but we conjecture that similar results hold for β < 1, possibly under more stringent assumptions on w.

In fact, the assumption ( 13) is not essential.What we need is that the kinetic energy per particle is bounded, or more precisely,

Tr h ⊗ hγ (2) Ψ N ≤ C (17)
for all N , where h = (-i∇ + A) 2 + V . For example, if H N,ε ≥ -CN , where H N,ε is defined as H N but replacing w by (1ε) -1 w in ( 6), for some ε > 0, we can prove [START_REF] Iwatsuka | Magnetic Schrödinger operators with compact resolvent[END_REF] for any β < 2/3 and therefore have convergence of both the ground states and ground state energies. If, on the other hand, we assume [START_REF] Fannes | Finite size mean-field models[END_REF], the boundedness of the moment (17) holds immediately for β ≤ 1/3 and with a bootstrap argument borrowed from [START_REF]A note on 2d focusing many-boson systems[END_REF] we extend it to β < 1/3 + s/(45 + 42s).

The assumption ( 13) is very natural. If a potential w satisfies w -b1 |x|>R K dip ∈ L 1 ∩ L 2 then by increasing its value in a neighborhood of the origin, we can make it classically stable as in [START_REF] Fannes | Finite size mean-field models[END_REF]. We explain this in Appendix B.

A natural question arises: are all admissible configurations (a, b) (i.e. those satisfying (10) of Theorem 1) reachable by the derivation? In the case of the dipolar potential

K dip (x) = 1 -3 cos 2 (θ x ) |x| 3 ,
the answer is yes. We are able to construct a potential w dip (resp. w dip ) satisfying the assumptions of Theorem 5 in the case b > 0 (resp. b < 0) and such that the corresponding a and b satisfy the equality case a = 4π 3 b (resp. a = -8π 3 b). Then, any configuration (a, b) satisfying the strict inequality [START_REF] Combes | Classical bounds and limits for energy distributions of hamilton operators in electromagnetic fields[END_REF] is reached by adding a well chosen non negative function to w dip (resp. w dip ). The interaction w dip is defined as follows. Let d ∈ R,

w dip (x) := 2W (x) -W (x + dn) -W (x -dn) (18)
where

W (x) = 1 -e -|x| |x| .
The potential w dip represents the interaction of a couple of dipoles interacting with the smeared Coulomb potential W . The potential w dip is defined in Appendix A, in the proof of the following proposition. One could think that the case b < 0 has less physical meaning as it will turn out that b = d 2 , but it is experimentally feasible to tune the parameter b to be negative [START_REF] Giovanazzi | Tuning the dipolar interaction in quantum gases[END_REF].

Proposition 8 (Full range of parameters in the dipolar GP functional).

For any admissible parameters (a, b), i.e. satisfying [START_REF] Combes | Classical bounds and limits for energy distributions of hamilton operators in electromagnetic fields[END_REF], there exists a potential w satisfying the assumptions of Theorem 5 and in particular a = (w -b1 |x|>R K dip ). Hence we have lim

N →∞ e N = e GP (a, b).
The proof is provided in Appendix A. The rest of the paper is dedicated to the proof of our main result.

Proof of Theorem 5: derivation of the dipolar Gross-Pitaevskii energy

This section is dedicated to the proof of Theorem 5.

3.1. Preliminaries. Our method in this paper is to follow the path exposed in [START_REF] Lewin | The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases[END_REF] which consists in a two steps argument. We first approximate the N -body theory by Hartree's theory and then pass from the latter to the Gross-Pitaevskii theory. We use and adapt techniques developed in [START_REF] Thành Nam | Ground states of large bosonic systems: The Gross-Pitaevskii limit revisited[END_REF][START_REF]A note on 2d focusing many-boson systems[END_REF] for the non-negative short range 3D case and the short range 2D case to our 3D dipolar problem. We denote by h = -∇ x + iA(x)

2 + V (x) the one body operator which is the Friedrichs extension of the operator defined similarly on C ∞ 0 (R 3 ). Furthermore, h has a compact resolvent under the assumption that V is confining: lim |x|→∞ V (x) = ∞ , which we make, see [START_REF] Iwatsuka | Magnetic Schrödinger operators with compact resolvent[END_REF] for more details on h.

The Hartree functional is given by the energy of an condensed state

E N H (u) = u ⊗N , H N u ⊗N N = R 3 |(∇ + iA)u| 2 + R 3 V |u| 2 + 1 2 R 3 ×R 3 (w N ⋆ |u| 2 )|u| 2 . ( 19
)
We denote by e H,N := inf

u L 2 =1 E N H (u)
the Hartree ground state energy. If a potential W is classically stable then it is Hartree stable [START_REF] Lewin | The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases[END_REF]:

R 3 ×R 3 W (x -y)ρ(x)ρ(y)dxdy ≥ 0, ∀ρ ≥ 0. ( 20
)
It is easy to see that Hartree stability is needed in order to have lim inf e H,N > -∞. Indeed, if the quantity in ( 20) is negative for some ρ ≥ 0, by taking u

N (x) = ρ N (x) = N 3β/2 ρ(N β x), one can see that e H,N → ∞ as N → ∞. If, besides, W (0) < ∞, Hartree stability implies classical stability.
Notice that both in the Hartree functional and in the GP functional, stability is equivalent to the non negativity of the interaction energy, compare Lemma 10 and condition (10) of Theorem 1.

The two steps of the proof are as follows. The derivation of the Hartree theory follows arguments of [START_REF] Lewin | Derivation of Hartree's theory for generic mean-field Bose gases[END_REF], using de Finetti theorems combined with a stability argument. The way from Hartree theory to Gross-Pitaevskii theory is essentially based on the observation that for w ∈ L 1 , w N → δ 0 w in the sense of measures.

3.1.1.

Estimating the pair-potential by the kinetic energy. This section is dedicated to the proof of Proposition 9 which is a generalization of [START_REF] Thành Nam | Ground states of large bosonic systems: The Gross-Pitaevskii limit revisited[END_REF]Lemma 3.2] to a larger class of interactions including the dipolar potential. The latter has singularities at 0 and infinity and is therefore not a L 1 function as in [START_REF] Thành Nam | Ground states of large bosonic systems: The Gross-Pitaevskii limit revisited[END_REF].

Proposition 9 (Domination of the interaction potential by the kinetic energy). Define w = w 0 + 1 |x|≥R K with w 0 ∈ L 1 (R 3 ) ∩ L 2 (R 3 ), K as in Theorem 5 and R > 0. Denote w N = N 3β w(N β •) for any 0 ≤ β ≤ 1. We have the following estimates.

|w N (x -y)| ≤ CN β h x (21) ±w N (x -y) ≤ C ε (h x ) 3/4+ε ⊗ (h y ) 3/4+ε , ∀ǫ > 0, (22)
± (h x w N (x -y) + w N (x -y)h x ) ≤ CN 3β/2 h x ⊗ h y . (23)
We will first state some lemmas to deal with the dipolar potential K. The first two have been adapted from [START_REF] Duoandikoetxea | Fourier analysis[END_REF] and their proof will not be given here. In this paper we use the following Fourier transform

f (k) = R 3 f (x)e -k•x dx.
Lemma 10 (Fourier transform of the potential K, Corollary 4.5 of [START_REF] Duoandikoetxea | Fourier analysis[END_REF]). Let q > 1 and Ω ∈ L q (S 2 ) be a even function satisfying the cancellation property

S 2 Ω(ω)dσ(ω) = 0. For any R ′ > R ≥ 0 we define K R ′ R (x) = 1 R ′ >|x|>R Ω(x/|x|) |x| 3 . ( 24
)
Then for any

R ′ > R > 0, K R ′ R (k) = S 2 R ′ R cos(rk • ω) r Ω(ω)drdσ(ω) = S 2 R ′ R cos(rk • ω) -cos(r|k|) r Ω(ω)drdσ(ω),
and as R ′ → ∞ and R → 0 it converges for k = 0 to

K(k) = S 2 log |k| |k • ω| Ω(ω)dσ(ω). ( 25
)
Moreover K R ′ R L ∞ (R 3 ) ≤ C q Ω L q (S 2 ) for some constant C q > 0 independent of R ′ > R ≥ 0 and therefore K L ∞ (R 3 ) ≤ C q Ω L q (S 2 )
Note that the function K defined in ( 25) is homogeneous of degree zero and continuous except at the origin.

Lemma 11 (Definition and continuity in L p , Theorem 4.12 of [START_REF] Duoandikoetxea | Fourier analysis[END_REF]). Under the assumptions of the previous lemma, for any 1 < p < ∞, we define for f ∈ L p (R 3 )

K ⋆ f (x) := lim R ′ →∞ R→0 K R ′ R ⋆ f (x)
which exists for almost every x ∈ R 3 . Then, there exists some constant

C p > 0, independent of 0 ≤ R < R ′ ≤ ∞ such that K R ′ R ⋆ f L p (R 3 ) ≤ C p C q Ω L q (S 2 ) f L p (R 3 ) , ∀f ∈ L p (R 3
). Now we prove an inequality allowing to control long range interactions having bounded Fourier transform, by the kinetic energy.

Lemma 12 (Domination of the long range potential by the kinetic energy). Let W ∈ L 2 (R 3 ) be such that W ∈ L ∞ (R 3 ). Then, for all ε > 0, there exists a constant C ε > 0, independent of W , such that the following operator inequality holds

± W (x -y) ≤ C ε W L ∞ (R 3 ) (1 -∆ x ) 3/4+ε ⊗ (1 -∆ y ) 3/4+ε . ( 26
)
Proof. To prove [START_REF] Lieb | The mathematics of the Bose gas and its condensation[END_REF] it is sufficient to show that for any R > 0 we have

(1 -∆ x ) -3 8 -ε 2 (1 -∆ y ) -3 8 -ε 2 W (x -y)(1 -∆ x ) -3 8 -ε 2 (1 -∆ y ) -3 8 -ε 2 ≤ C ε W L ∞ (R 3 ) (27)
with C ε independent of W . We define

G = F -1 1 (1 + p 2 ) 3/8+ε/2 ,
where we denote by F -1 the inverse Fourier transform. Then, inequality ( 27) is equivalent to

f (x, y)G(x -x ′ )G(y -y ′ ) W (x ′ -y ′ ) W L ∞ (R 3 ) × × G(x ′ -x ′′ )G(y ′ -y ′′ )g(x ′′ , y ′′ )dx dy dx ′ dy ′ dx ′′ dy ′′ ≤ C f L 2 (R 3 ) g L 2 (R 3 ) , (28)
for all f, g ∈ L 2 (R 3 × R 3 ) and for some constant C > 0. We will prove inequality [START_REF] Rougerie | De Finetti theorems, mean-field limits and Bose-Einstein condensation[END_REF] with Y ∈ S, the Schwartz space, instead of G and we will conclude by a density argument.

Let f, g ∈ L 2 (R 3 × R 3 ) and let Y ∈ S then we have

f (x, y)Y (x -x ′ )Y (y -y ′ )W (x ′ -y ′ )Y (x ′ -x ′′ )× (29) × Y (y ′ -y ′′ )g(x ′′ , y ′′ )dx dy dx ′ dy ′ dx ′′ dy ′′ = W (w) f (x, y)Y (x -(w + y ′ ))Y (y -y ′ )× × Y ((w + y ′ ) -x ′′ )Y (y ′ -y ′′ )g(x ′′ , y ′′ )dx dy dy ′ dx ′′ dy ′′ dw = W (w)h(w)dw = W (p) h(p)dp,
where we have made the change of variable x ′ = w + y ′ and we have defined

h(w) = f (x, y)Y (x -(w + y ′ ))Y (y -y ′ )Y ((w + y ′ ) -x ′′ )Y (y ′ -y ′′ )g(x ′′ , y ′′ ) dxdydy ′ dx ′′ dy ′′ .
Since W L ∞ < ∞, it suffices to prove that h ∈ L 1 (R 3 ) to get the result. We then compute

h(p) = f (x, y)Y (y -y ′ ) Y (q)e iq•(y ′ -x) Y (p -q)e i(p-q)•(y ′ -x ′′ ) dq × × Y (y ′ -y ′′ )g(x ′′ , y ′′ )dx dy dy ′ dx ′′ dy ′′ = f (x, y) Y (q)e -iq•x Y (p -q)e -i(p-q)•x ′′ Y (k)e ik•y Y (p -k)e i(p-k)•y ′′ dk × × g(x ′′ , y ′′ )dq dx dy dx ′′ dy ′′ = f (q, -k) g(q -p, k -p) Y (q) Y (p -q) Y (k) Y (p -k) g(q -p, k -p)dk dq
We then have

| h(p)|dp ≤ f (q, -k) g(q -p, k -p) Y (q) Y (p -q) Y (k) Y (p -k) dpdqdk ≤ 1 2 f (q, -k) g(q -p, k -p) | Y | 2 (q)| Y | 2 (p -q) + | Y | 2 (k)| Y | 2 (p -k) dpdqdk ≤ 1 2 | f | 2 (q, -k)| Y | 4 (p -q)dpdqdk 1/2 | g| 2 (q -p, k -p)| Y | 4 (q)dpdqdk 1/2 + | f | 2 (q, -k)| Y | 4 (p -k)dpdqdk 1/2 | g| 2 (q -p, k -p)| Y | 4 (k)dpdqdk 1/2 ≤ f L 2 (R 6 ) g L 2 (R 6 ) Y 4 L 4 (R 3 ) ≤ C 4/3 f g Y 4 L 4/3 (R 3 )
. Where C 4/3 is the Lipschitz modulus of the Fourier transform from L 4/3 (R 3 ) to L 4 (R 3 ). We have just proved that the expression (29) defines a continuous 4-linear form on L 4/3 (R 3 ) by density of smooth functions. Recalling that G ∈ L 4/3 (R 3 ) proves (28) and ends the proof.

Thanks to Lemma 12 and Lemma 3 we are now able to prove Proposition 9.

Proof of Proposition 9. The first inequality ( 21) is exactly the same as in [34, Lemma 3.2] and the last inequality ( 23) is easily adapted from [START_REF] Thành Nam | Ground states of large bosonic systems: The Gross-Pitaevskii limit revisited[END_REF]. The major improvement concerns [START_REF] Lewin | Derivation of Hartree's theory for generic mean-field Bose gases[END_REF] where in [START_REF] Thành Nam | Ground states of large bosonic systems: The Gross-Pitaevskii limit revisited[END_REF] the potential was assumed to be in L 1 which the dipolar potential is not.

Proof of [START_REF] Lewin | The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases[END_REF]. Using the Sobolev inequality one has

|W (x -y)| ≤ C W L 3/2 (-∆ x ),
for any W ∈ L 3/2 (R 3 ). Then using w N L 3/2 = N β w L 3/2 and Lemma 3 we obtain [START_REF] Lewin | The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases[END_REF].

Proof of [START_REF] Lewin | Derivation of Hartree's theory for generic mean-field Bose gases[END_REF]. Let us write w(x) = w 0 (x) +

1 |x|>R K(x) with w 0 ∈ L 1 . Then, w L ∞ ≤ w 0 L ∞ + K R L ∞ ≤ w 0 L 1 + C q Ω L q (S 2
) thanks to Lemma 10. We can now apply Lemma 12 which proves [START_REF] Lewin | Derivation of Hartree's theory for generic mean-field Bose gases[END_REF].

Proof of [START_REF]Remarks on the quantum de finetti theorem for bosonic systems[END_REF]. The proof of [START_REF] Thành Nam | Ground states of large bosonic systems: The Gross-Pitaevskii limit revisited[END_REF]Lemma 3.2] gives

± (h x W (x -y) + W (x -y)h x ) ≤ C( W L 3/2 + W L 2 )h x ⊗ h y .
and since w N L 3/2 = N β w L 3/2 and w N L 2 = N 3β/2 w L 2 , inequality (23) follows.

3.2.

From the many-body problem to Hartree theory. Our proof is inspired of the one for the 2D case of [START_REF]A note on 2d focusing many-boson systems[END_REF] with techniques first introduced in [START_REF] Lewin | The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases[END_REF]. The method is the following: we derive a lower bound on the many-body energy per particle e N making two approximations. First we project the ground state on a finite (but varying) number of low energy levels of the one-body operator, and then we use the quantitative de Finetti theorem in finite dimension to approximate the projection of the ground state by a Hartree state. The error terms involve L (the dimension of the low energy subspace the ground state has been projected on), N the number of particles and e N,ε the energy per particle itself (with kinetic energy decreased by a factor (1ε)) which is not yet known to be bounded. But, by a bootstrap argument, first introduced in [START_REF]A note on 2d focusing many-boson systems[END_REF], we get a condition on β ensuring the boundedness of e N . As in [START_REF] Thành Nam | Ground states of large bosonic systems: The Gross-Pitaevskii limit revisited[END_REF][START_REF]A note on 2d focusing many-boson systems[END_REF] the main improvement regarding [START_REF] Lewin | The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases[END_REF] is the use of moment estimates on ground states rather than pure operator estimates.

3.2.1. Moment estimates. Now, we prove some moment estimates which will be useful to control the errors. Namely, we adapt [24, Lemma 5] to the 3D case, the main difference being the assumption 0 < β < 1 which has to be strengthened due to the negative part of the potential w. We define the Hamiltonian

H N,ε := H N -ε N j=1 h j
where h is replaced by (1ε)h, and we denote by N × e N,ε its ground state energy.

Lemma 13 (Moments estimates).

Let 0 < β < 2/3 and Ψ N ∈ h N be a ground state of H N . Then for all ε ∈ (0, 1) we have

Tr hγ

(1)

Ψ N ≤ C 1 + |e N,ε | ε ( 30
)
and

Tr h ⊗ hγ (2)
Ψ N ≤ C (1 + |e N,ε |) 2 ε 2 , ( 31
)
for N sufficiently large, and where the constant C is independent of N .

Proof. We have which shows the first moment estimate since e N is upper bounded by a constant. For the second inequality, let us write

H N,ε = H N -ε N j=1 h j ≥ N e N,ε (32)
1 N 2 Ψ N ,     N j=1 h j   H N + H N   N j=1 h j     Ψ N = 2 e N N Ψ N , N j=1 h j Ψ N ≤ C (1 + |e N,ε |) 2 ε ( 33
)
where we used that |e N | 1 + |e N,ε | and the first moment estimate. Let us find a lower bound on [START_REF] Bose | Plancks Gesetz und Lichtquantenhypothese[END_REF]. One has

1 N 2     N j=1 h j   H N + H N   N j=1 h j     = 2 N 2   N j=1 h j   2 + 1 N 2 (N -1) N i=1 j<k (h i w N (x j -x k ) + w N (x j -x k )h i ) . (34)
We split the second term in [START_REF] Thành Nam | Ground states of large bosonic systems: The Gross-Pitaevskii limit revisited[END_REF] in two terms, depending whether i = j < k = i or i ∈ {j, k}. For any 1 ≤ i 0 ≤ N , we have

1 N -1 1≤j<k≤N j =i 0 ,k =i 0 w N (x j -x k ) = H N,ε -(1 -ε) N j=1 h j - 1 N -1 j =i 0 w N (x i 0 -x j ) ≥ N e N,ε -1 -ε + C N β N -1   N j=1 h j   , (35)
where [START_REF] Lewin | The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases[END_REF] of Lemma 9 has been used. Then multiplying (35) by h i , we obtain

1 N 2 (N -1) N i=1 i =j<k =i h i w N (x j -x k ) + w N (x j -x k )h i ≥ 2 e N,ε N   N j=1 h j   - 2 N 2 1 -ε + C N β N   N j=1 h j   2 . (36)
On the other hand, for j = k, we have by ( 23)

h j w N (x j -x k ) + w N (x j -x k )h j ≥ -CN 3β/2 h j h k
and, after summing for 1 ≤ j < k ≤ N , we obtain

j<k h j w N (x j -x k ) + w N (x j -x k )h j ≥ -CN 3β/2   N j=1 h j   2 . ( 37
)
Combining ( 36) and (37) we arrive at the estimate

1 N 2   N j=1 h j   H N + H N   N j=1 h j   ≥ 2 N 2 ε -C N β N -C N 3β/2 N   N j=1 h j   2 -C 1 + |e N,ε | N   N j=1 h j   . ( 38
)
Taking the trace of (38) against |Ψ N Ψ N | and using the first moment estimate, we conclude that

ε -C N β N -C N 3β/2 N Tr h ⊗ hγ (2) Ψ N ≤ C (1 + |e N,ε |) 2 ε .
Now, if β < 2/3, for N large enough one has ε -CN β-1 -CN 3β/2-1 ≥ ε/2 > 0 and the second moment estimate is proved.

3.2.2. Lower bound via de Finetti. We now state some quantitative version of the de Finetti theorem [24, Lemma 3] originally proven in [START_REF] Christandl | One-and-a-half quantum de Finetti theorems[END_REF] (see also [START_REF] Chiribella | On quantum estimation, quantum cloning and finite quantum de Finetti theorems[END_REF][START_REF] Harrow | The church of the symmetric subspace[END_REF][START_REF]Remarks on the quantum de finetti theorem for bosonic systems[END_REF] for variants of the proof and [START_REF] Fannes | Finite size mean-field models[END_REF] for new results). For a summary of the use of de Finetti theorems in the mathematics of ultra-cold atomic gases, see [START_REF] Rougerie | De Finetti theorems, mean-field limits and Bose-Einstein condensation[END_REF]. Heuristically, the result states that the density matrices of symmetric wave functions are well approximated by convex combinations of product states when N is large.

Theorem 14 (Quantitative quantum de Finetti in finite dimension).

Let Ψ ∈ H N = N s L 2 (R 3
) and P be an orthogonal projection of finite rank. Then, there exists a positive Borel measure dµ Ψ on the unit sphere SP H such that

Tr H P ⊗2 γ (2) Ψ P ⊗2 - SP H |u ⊗2 u ⊗2 | dµ Ψ (u) ≤ 8 dim P H N and SP H dµ Ψ (u) ≥ Tr P γ (1) Ψ 2 .
We then denote by

K N 2 = h ⊗ 1 + 1 ⊗ h + 1 2 w N (x -y) and P := P (L) = 1 (-∞,L] (h)
, for any L > 0, the projection onto the subspace of energy levels lower than L of the one-particle operator. The ground state energy can be written

Ψ N , H N Ψ N N = Tr K 2 γ (2)
Ψ N = SL 2 u ⊗2 , K 2 u ⊗2 dµ Ψ N (u) + Tr K 2 γ
(2)

Ψ N -P ⊗2 γ (2)
Ψ N P ⊗2 + Tr K 2 P ⊗2 γ (2)
Ψ N P ⊗2 - SL 2 |u ⊗2 u ⊗2 | dµ Ψ N (u)) .
Where we have denoted the unit sphere of L 2 (R 3 ) by SL 2 . The first term is the sought after approximation leading to the Hartree energy :

E N H (u) = u ⊗2 , K N 2 u ⊗2
. The two others are error terms which have to be controlled. We summarize our estimates in the following lemma.

Lemma 15 (Lower bound via de Finetti).

For any 0 < β ≤ 1 and any L, ε, δ > 0, there exists a constant C > 0 and N 0 > 0 such that, we have for all N ≥ N 0

SL 2 u ⊗2 , K 2 u ⊗2 dµ Ψ N (u) ≥ e H,N -C (1 + |e N,ε |) εL , ( 39
)
Tr K 2 P ⊗2 γ (2)
Ψ N P ⊗2 - SL 2 |u ⊗2 u ⊗2 | dµ Ψ N (u) ≥ -C δ L 3(1+ 1 s )+2δ N ( 40
)
and

Tr K 2 γ (2)
Ψ N -P ⊗2 γ

(2)

Ψ N P ⊗2 ≥ -C δ (1 + |e N,ε |) 13 8 + 3δ 2 εL 1 8 - δ 2 . ( 41
)
Combining the inequalities (39), ( 40) and ( 41) we find

e N ≥ e H,N -C (1 + |e N,ε |) εL -C δ   L 3(1+ 1
s )+2δ N + (1 + |e N,ε |) 13 8 + 3δ 2 ε 13 8 + 3δ 2 L 1 8 - δ 2   .
Proof. Let us take 0 < β ≤ 1, L, ε, δ > 0 and N 0 > 0 such that Lemma 13 holds.

Main term (39). Since u ⊗2 , K 2 u ⊗2 = E H (u), we have

SL 2 u ⊗2 , K 2 u ⊗2 dµ Ψ N (u) ≥ e H,N SL 2 dµ Ψ N (u). But SL 2 dµ Ψ N (u) ≤ Tr P γ (1) Ψ N 2
and Q := 1 -P ≤ L -1 h and therefore

SL 2 dµ Ψ N (u) ≥ (1 -Tr Qγ (1) Ψ N ) 2 ≥ 1 - 2 L Tr hγ (1)
Ψ N .

So we obtain

SL 2 u ⊗2 , K 2 u ⊗2 dµ Ψ N (u) ≥ e H,N - C L Tr hγ (1)
Ψ N ≥ e H,N -C (1 + |e N,ε |) εL ( 42
)
where we have used [START_REF] Santos | Roton-maxon spectrum and stability of trapped dipolar Bose-Einstein condensates[END_REF].

First error term (41). Using the quantitative quantum de Finetti Theorem 14 and the bound P h ≤ LP we obtain

Tr (h 1 + h 2 ) P ⊗2 γ (2)
Ψ N P ⊗2 - SL 2 |u ⊗2 u ⊗2 | dµ Ψ N (u) ≤ C Ld L N , (43)
where d L = dim Im P . Besides, from [START_REF] Lewin | Derivation of Hartree's theory for generic mean-field Bose gases[END_REF] in Corollary 9 we deduce that for δ > 0,

±P ⊗2 w N (x 1 -x 2 )P ⊗2 ≤ C δ (P h 1 ⊗ P h 2 ) 3/4+δ ≤ C δ L 3/2+2δ P ⊗2
which, combined with (43), gives

Tr K 2 P ⊗2 γ (2)
Ψ N P ⊗2 - SL 2 |u ⊗2 u ⊗2 | dµ Ψ N (u) ≥ -C δ L 3/2+2δ dim P H N . ( 44
)
We now use [10, Theorem 2.1] according to which there is some constant C > 0 such that dim P H ≤ CL 3(1/2+1/s) and which shows (40).

Second error term (40). The operator inequality h ≥ P h gives

Tr (h 1 + h 2 ) γ (2)
Ψ N -P ⊗2 γ

(2)

Ψ N P ⊗2 = Tr (h 1 + h 2 ) -P ⊗2 (h 1 + h 2 ) P ⊗2 γ (2)
Ψ N ≥ 0.

Then, we follow and adapt [START_REF]A note on 2d focusing many-boson systems[END_REF]. Using the Cauchy-Schwarz inequality for operators

± (AB + B * A * ) ≤ η -1 AA * + ηB * B, ∀η > 0, (45)
we obtain

± 2 γ (2) Ψ N -P ⊗2 γ (2)
Ψ N P ⊗2 = ± 1 -P ⊗2 γ (2) Ψ N + γ (2)
Ψ N 1 -P ⊗2 + P ⊗2 γ

(2)

Ψ N 1 -P ⊗2 + (1 -P ⊗2 )γ (2)
Ψ N P ⊗2 ≤ 2η -1 1 -P ⊗2 γ (2)
Ψ N 1 -P ⊗2 + η γ (2)
Ψ N + P ⊗2 γ (2)
Ψ N P ⊗2 , ∀η > 0. (46)
On the other hand, using [START_REF] Lewin | Derivation of Hartree's theory for generic mean-field Bose gases[END_REF] and the fact that

t r = inf ρ>0 rρ -1 t + (1 -r)ρ r 1-r , ∀t ≥ 0, ∀r ∈ (0, 1), (47)
we get

± w N (x -y) ≤ C δ (h x h y ) 3/4+δ ≤ C δ ρ -1 h x h y + ρ 3+4δ 1-4δ . (48)
Now, combining (46) and (48) we have

2 Tr w N γ (2)
Ψ N -P ⊗2 γ

(2)

Ψ N P ⊗2 ≥ -2C δ η -1 Tr (h x h y ) 3/4+δ 1 -P ⊗2 γ (2)
Ψ N 1 -P ⊗2 -C δ η Tr ρ -1 h x h y + ρ 3+4δ 1-4δ γ (2)
Ψ N + P ⊗2 γ (2)
Ψ N P ⊗2 .

After optimizing over η > 0 we find

Tr w N γ (2)
Ψ N -P ⊗2 γ (2)
Ψ N P ⊗2 ≥ -C δ Tr (h x h y ) 3/4+δ 1 -P ⊗2 γ (2)
Ψ N 1 -P ⊗2 1/2 (49)
× Tr ρ -1 h x h y + ρ 3+4δ 1-4δ γ (2)
Ψ N + P ⊗2 γ (2)
Ψ N P ⊗2 1/2 .
Let us deal with the second factor in (49), we have

Tr ρ -1 h x h y + ρ 3+4δ 1-4δ γ (2)
Ψ N + P ⊗2 γ (2)
Ψ N P ⊗2 ≤ C ρ -1 Tr h 1 h 2 γ (2)
Ψ N + ρ 3+4δ 1-4δ
and by optimizing over ρ > 0, we obtain

Tr ρ -1 0 h x h y + ρ 3+4δ 1-4δ 0 γ (2)
Ψ N + P ⊗2 γ (2)
Ψ N P ⊗2 ≤ C Tr h 1 h 2 γ (2) Ψ N 3/4+δ . (50)
Now, let us find an upper bound of the first factor in (49). We define Q := 1 -P and we notice that

1 -P ⊗2 = 1 -(1 -Q) ⊗ (1 -Q) = Q ⊗ 1 + 1 ⊗ Q -Q ⊗ Q ≤ Q ⊗ 1 + 1 ⊗ Q. (51)
This allows us to write

1 -P ⊗2 (h ⊗ h) 3/4+δ 1 -P ⊗2 ≤ C Qh 3/4+δ ⊗ h 3/4+δ + h 3/4+δ ⊗ Qh 3/4+δ (52)
≤ C 1 L 1/4-δ h ⊗ h 3/4+δ + h 3/4+δ ⊗ h (53) ≤ C 1 L 1/4-δ η -1 h ⊗ h + η 3+4δ 1-4δ (h ⊗ 1 + 1 ⊗ h) , (54)
for all η > 0. In (52), we used (51), whereas in (53) we used that Q ≤ L -1 h. Finally, in (54) we used (47). Now, taking the trace of (54) against γ

Ψ N and optimizing over η > 0 we get

Tr (h x h y ) 3/4+δ 1 -P ⊗2 γ (2)
Ψ N 1 -P ⊗2 ≤ 1 L 1 4 -δ Tr h ⊗ hγ (2) Ψ N 3/4+δ
Tr hγ

(1) Ψ N 1/4-δ . (55)
Finally, from (50) and (55) we get

Tr w N γ (2)
Ψ N -P ⊗2 γ (2)
Ψ N P ⊗2 ≥ - C δ L 1/8-δ/2 Tr h ⊗ hγ (2) Ψ N 3/4+δ/2 Tr hγ (1) Ψ N 1/8-δ/2
. (56) Now, using the moment estimates [START_REF] Santos | Roton-maxon spectrum and stability of trapped dipolar Bose-Einstein condensates[END_REF] and [START_REF] Seiringer | Gross-Pitaevskii theory of the rotating Bose gas[END_REF] of Lemma 13 the result follows.

3.2.3.

Final energy estimates and stability of the second kind. We follow the bootstrap argument of [START_REF]A note on 2d focusing many-boson systems[END_REF]. From Lemma 15 we have

e H,N ≥ e N ≥ e H,N -C (1 + |e N,ε |) εL -C δ L 3(1+s -1 )+2δ N + (1 + |e N,ε |) 13/8+3δ/2 ε 13/8+3δ/2 L 1/8-δ/2 . ( 57
)
We want to prove that e N,ε is bounded, otherwise, since we take L ≥ 1 we have

(1 + |e N,ε |) εL ≤ (1 + |e N,ε |) 13/8+3δ/2 ε 13/8+3δ/2 L 1/8-δ/2 . ( 58
)
We can therefore omit the left side of (58) in inequality (57). Now, let us rewrite (57), replacing w by (1ε) -1 w, we get

e ε H,N ≥ e N,ε ≥ e ε H,N -C δ   L 3(1+ 1 s )+2δ N + (1 + |e N,ε ′ |) 13 8 + 3δ 2 (ε ′ -ε) 13 8 + 3δ 2 L 1 8 - δ 2   , (59)
for all 1 > ε ′ > ε > 0. Note as well that e ε H,N ≥ 0. Thanks to the classical stability of w (13), we know that for all 0 ≤ ε ≤ 1 we have e N,ε ≥ -CN 3β-1 . Now, as in [START_REF]A note on 2d focusing many-boson systems[END_REF], this leads us to make the following induction hypothesis, denoted I η , for η ≥ 0, lim sup

N →∞ |e N,ε | 1 + N η < ∞, ∀ε ∈ (0, 1).
It is clear that I η holds for η = 3β -1, and we would like to prove I 0 . By choosing L = N τ , for τ > 0, we see that if I η holds, then I η ′ also holds as soon as

η ′ > max 3τ (1 + s -1 ) -1, 13η -τ 8 . (60)
Optimizing over τ , (60) becomes

η ′ > η - s -η(15 + 14s) 24 + 25s .
In order to choose some η ′ < η, the condition η < s/(15 + 14s) must be fulfilled. In term of β (in order to start the induction) this means

β < 1 3 + s 45 + 42s . (61)
If condition ( 61) is fulfilled, we can show that (I 0 ) holds, after applying (59) finitely many times. Namely, we have convergence of the energy per particle with the following estimate on the rate of convergence

e N ≥ e H,N -C δ N - 1 (25+24/s)+12δ . (62)
3.3. From Hartree theory to Gross-Pitaevskii. We have shown above that the many-body ground state energy e N is well approximated by Hartree's energy e H,N . It remains to show that e H,N is a good approximation of the Gross-Pitaevskii energy e GP (a, b). This is easier than the approximation of the many body energy by the Hartree energy. The case w ∈ L 1 (R 3 ) follows from standard arguments [START_REF] Lewin | The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases[END_REF][START_REF]A note on 2d focusing many-boson systems[END_REF] and it remains to adapt the proof to deal with K.

Lemma 16 (From Hartree to Gross-Pitaevskii).

Let w 0 ∈ L 1 (R 3 ) and K(x) = Ω(x)/|x| 3 satisfying the assumptions of Theorem 5. Let b ∈ R and a = R 3 w 0 , let us define w = w 0 + b1 |x|>R K for some R > 0. Then

lim N →∞ sup u∈H 1 u =0 u -4 H 1 w N (x -y)|u(x)| 2 |u(y)| 2 dxdy -a |u(x)| 4 dx -b K(x -y)|u(x)| 2 |u(y)| 2 dxdy = 0 (63)
We have not stated here any rate of convergence as it depends on the properties of the short range potential w 0 . The dipolar part of the Hartree energy converges with an error of O(N -β ) towards the dipolar part of the Gross-Pitaevskii energy. As an example, under the extra assumption |x|w 0 (x) ∈ L 1 (R 3 ), the short range part of the energy converges also with an error of O(N -β ).

Lemma [START_REF] Harrow | The church of the symmetric subspace[END_REF] shows that e H,N = e GP + o(1) as N → ∞ and ends the proof of Theorem 5.

Proof. For the L 1 part of the potential, the proof of convergence is the same as in [24, Lemma 7] except that the integration domain is R 3 and not R 2 . We quickly recall it here: let w 0 ∈ L 1 (R 3 ) and A > 0. We have

w 0,N (x -y)|u(x)| 2 |u(y)| 2 dxdy -a |u(y)| 4 dy = w 0 (x)|u(y)| 2 |u(N -β x + y)| 2 -|u(y)| 2 dxdy ≤ 2 1 |x|<A |w 0 (x)||u(y)| 2 1 0 |u∇u(tN -β x + y) • x|N -β dtdxdy + 2 1 |x|>A |w 0 (x)|dx u 4 L 4 ≤ 2 AN -β + 2 1 |x|>A |w 0 (x)|dx u 4 H 1 .
Taking A = N β/2 gives the desired result. For the dipolar part, the same proof will not work since the potential is not integrable both at the origin and at infinity. Nevertheless, as we saw in Lemma 10, the convolution by K extends to a bounded operator on L 2 (R 3 ) and coincide in Fourier space with the multiplication by a bounded function K ∈ L ∞ (R 3 ). By definition, for f ∈ L p (R 3 ),

K ⋆ f := lim ǫ→0 η→∞ K η ε ⋆ f where K η ε (x) = K(x)1 ε≤|x|≤η (x), ∀x ∈ R 3 .
As computed in Lemma 10

K η ε = S 2 η ε cos(rp • ω) -1 Ω(ω) dr r dσ(ω). (64)
In order to prove (63), we seek to find a bound on the error (K RN -β 0 ⋆ |u| 2 )|u| 2 . First, let us notice that for p ∈ R 3

K RN -β 0 (p) ≤ C S 2 RN -β 0 |cos (rp • ω) -1| dr r dσ(ω) ≤ C|p|N -β . (65)
Then,

R 3 (K N 1 |x|≥R 0 N -β ⋆ |u| 2 )|u| 2 - R 3 (K ⋆ |u| 2 )|u| 2 ≤ R 3 K R 0 N -β 0 |u| 2 2 ≤ CN -β R 3 |p| |u| 2 (p) 2 . ( 66
)
By a density argument we can assume that u is smooth and use that |u| 2 = |u| ⋆ |u|, we obtain

R 3 |p| |u| 2 (p) 2 ≤ |p| |u|(p -k) |u|(k) |u|(p -q) |u|(q)dqdkdp ≤ 2 |p -k| |u|(p -k) |u|(k) |u|(p -q) |u|(q)dqdkdp,
where we used that |p| ≤ |p -k| + |k| and the symmetry of the integrand. We now define f = F -1 |u| and g = F -1 |p| |u| such that f H 1 = u H 1 and g L 2 ≤ ∇u L 2 . And finally,

R 3 |p| |u| 2 2 ≤ |g(x)||f (x)| 3 dx ≤ g L 2 f L 6 ≤ g L 2 f H 1 ≤ u 4 H 1 (67)
Inequality (67) inserted in (66) concludes the proof of Lemma 16.

Appendices

Appendix A. Proof of Proposition 8

Here, we show that in the dipolar case K = K dip , any admissible (a, b), i.e satisfying the condition [START_REF] Combes | Classical bounds and limits for energy distributions of hamilton operators in electromagnetic fields[END_REF] of Theorem 1, comes from a pair potential in the many-body problem. The proof uses two well chosen potentials w dip and w dip which give the case of equality in [START_REF] Combes | Classical bounds and limits for energy distributions of hamilton operators in electromagnetic fields[END_REF] respectively for b > 0 and b < 0. Higher a's are then achieved by adding a non negative pair function having the correct mass. First, we need the following lemma in order to build a classically stable potential behaving like the dipolar one at large distances.

Lemma 17 ([20], Dipolar approximation).

There is a constant C > 0 such that for all x, h ∈ R 3 with R + h = 0,

1 |x + h| - 1 |x| - e x • h |x| 2 + 3(e x • h) 2 -|h| 2 ) 2|x| 3 ≤ C|h| 3 |x| 3 |x + h| with e x = x/|x|.
We now recall the definition of w dip ,

w dip (x) = 2W (x) -W (x + dn) -W (x -dn) with W (x) = 1 -e -|x| |x| . ( 68
)
The potential w dip represents the total interaction of two dipoles given by four charged particles interacting via the smeared Coulomb potential W whose position and charge are (O, 1), (dn, -1) (for the first dipole) and (x, 1), (x + dn, -1) (for the second dipole). The dipolar moment d is a characteristic of the system and is parallel to n, the common direction of the dipoles. In particular, by Lemma 17, if we fix some R > 0, the potential w dip can be written

w dip = w 0 + d 2 1 |x|≥R K (69)
with w 0 ∈ L 1 ∩ L ∞ . Furthermore, w dip is of positive type since

w dip (k) = 2 W (k) -W (• + dn)(k) -W (• -dn)(k) = 8π |k| 2 (1 + |k| 2 ) (1 -cos(k • dn)) ≥ 0. ( 70
)
Computation of the short range parameter a. Now, let us compute the short range strength a = w 0 , defined in (69). Recall that this quantity is independent of the choice of R > 0. We have

R 3 w 0 = R 3 w dip -d 2 1 |x|>R K = lim A→∞ B(0,A) w dip ,
thanks to the cancellation property (4).

Lemma 18 (Computation of the short range strength).

The short range parameter of w dip is given by

R 3 w 0 = 4π 3 d 2 . Proof. Recall that w -d 2 1 |x|>R K ∈ L 1 , we can consider F w dip -d 2 1 |x|>R K = w dip -d 2 K + d 2 K -K1 |x|>R (71)
which is a continuous function and satisfies F w dipd 2 1 |x|>R K (0) = w 0 . Now, we compute the limit when p → 0 of the right side of (71). The third term is an error term and according to (65), it is a O(p) as p → 0. On the other hand, for any p ∈ R 3 , we can write the first term as

F(w dip )(p) -d 2 F(K)(p) = 4π 2 (1 + |p| 2 )|p| 2 (1 -cos(p • dn)) - (p • dn) 2 |p| 2 + d 2 3 ,
which tends to 4π 3 d 2 as p → 0. Now, using w dip , we will show that any a and b satisfying assumptions [START_REF] Combes | Classical bounds and limits for energy distributions of hamilton operators in electromagnetic fields[END_REF] are reached for some potential w. For this purpose, let us distinguish two cases.

f (p) = 4π (1 + p 2 ) d -2 + p 2 1 +p 2 2 4
.

We have f ∈ L 1 , since

R 3 1 (1 + p 2 ) 1 + p 2 1 + p 2 2 dp 1 dp 2 dp 3 = R dp 3 1 + p 2 3 R 2 dp 1 dp 2 1 + p 2 1 + p 2 2 3/2 < ∞.
One can verify that (1 -∆) 2 f ∈ L 1 and therefore f ∈ L 1 . Furthermore, the inequality

2 (1 -cos(p 3 d)) d -2 + p 2 1 + p 2 2 4 ≤ p 2 , implies w dip = f -w dip ≥ 0. Moreover, since f ∈ L 1 , f ∈ L 1 ∩ L 2
the potential w dip satisfies the assumptions of Theorem 5 and its short range parameter is given by a = (fw 0 )(0) = -8πb/3. This concludes the case of equality of Proposition 8, the other cases are obtained by adding positive functions to the potential w dip .

Appendix B. Stabilization of the long range potential

Here we show that given a potential K = Ω(x/|x|)|x| -3 , with Ω an even function satisfying the cancellation property (4) and some regularity condition, there always exists a classically stable potential w stab behaving like K at infinity (Proposition 19). Moreover, if some potential w has the long range behavior of K, we show that one can always make it classically stable by increasing its value in a fixed neighborhood of the origin (Proposition 20).

Proposition 19 (Existence of stable potential).

Let K = Ω(x/|x|)|x| -3 with Ω ∈ W 4,q (S 2 ), an even function, satisfying the cancellation property (4), for some q > 1. Then, for any R > 0, there exists some continuous function w stab ∈ L p (R 3 ), for 1 < p ≤ ∞, of positive type (hence classically stable), such that w stab -1 |x|>R K ∈ L 1 (R 3 ). Moreover, we can choose w stab such that |w stab (x) -1 |x|>R K(x)| = O(e -µ|x| ), for some µ > 0, as x tends to infinity.

Proposition 20 (Long range stability).

Let K be as in Proposition 19, R > 0 and w ∈ L 1 (R 3 ) with (w) -∈ L ∞ (R 3 ) and such that w(x) -1 |x|>R K(x) ∈ L 1 (R 3 ). For all δ > 0 there exists η > 0 such that w + η1 |x|<δ is classically stable.

To show the properties above we need the following lemma whose proof will be postponed.

0 ∈ L 1 loc (R 3 ), w 0 > η > 0 and w 1 ∈ L ∞ (R 3 ) such that |w 1 (x)| ≤ C 0 1/|x| 3+ǫ (72)
for all x ∈ R 3 , where η, ǫ, C 0 > 0. Let us define for R > 0

φ R = w 0 1 |x|≤R + w 1 1 |x|≥R .
Then, there is come constant C ε , given by (81), such that, if

C ε ≤ R 2+ǫ η C 0 (1 + R -6 1+ 2 ε ) , (73)
then, there exists ψ ∈ L 1 (R 3 ) such that φ R 0 ≥ ψ, ψ ≥ 0 and ψ ∈ L Proof of Lemma 21. The strategy of the proof is to increase the function φ R , by adding a function ψ 1 whose Fourier transform is well controlled, and such that φ R + ψ 1 is positive. We will then look for conditions on R, η and C 0 to bound φ R + ψ 1 from below by a function of positive type ψ 2 , such that ψ := ψ 2ψ 1 is classically stable. Let α be a smooth positive function with support in B(0, 1) and such that α = 1. Let us denote α R (x) = (R/2) -3 α(2x/R). For |x| > R, we have

B(0,R/2) 1 |x -x ′ | 3+ǫ α R (x ′ )dx ′ ≥ 1 (|x| + R/2) 3+ǫ ≥ R 3+ǫ (R + R/2) 3+ǫ 1 |x| 3+ǫ ≥ 2 3 3+ε 1 |x| 3+ε . ( 75
)
Let us define 72) and (75) we deduce

ψ 1 (x) = C 0 3 2 3+ε 1 |x ′ | 3+ǫ 1 |x ′ |≥R/2 ⋆ α R (x), from (
|w 1 (x)|1 |x|>R ≤ C 0 1 |x| 3+ǫ 1 |x|>R ≤ ψ 1 (x)
and in particular we obtain

ψ 1 ≤ w 1 1 |x|>R + 2ψ 1 . (76)
We now look for a bound on ψ 1 , we have

(1 + p 2 ) 3 ψ 1 ∞ ≤ (1 -∆) 3 ψ 1 L 1
from which we obtain

| ψ 1 (p)| ≤ C 0 3 3+ǫ (1 -∆) 3 α R L 1 1 |x|≥R 1/|x| 3+ǫ L 1 2 3+ǫ (1 + p 2 ) 3 = 4πC 0 3 3+ǫ (1 -∆) 3 α R L 1 2 3+ǫ R ε (1 + p 2 ) 3 . ( 77
)
Let us define for µ > 0, λ > 1, ψ µ λ (x) = µ e -λ|x|/2e -λ|x| |x| .

We have (78)

ψ µ λ (p) = µ 1 |p| 2 + λ 2 /4 - 1 |p| 2 + λ 2 =
We want to find a condition on λ and µ such that ψ µ λ (x) ≤ ψ 1 (x) for |x| > R, and since it is sufficient to have

ψ µ λ (x) = µ e -λ|x|/2 -e -λ|x| |x| ≤ µ e -λ|x|/2 |x| ≤ C 0 1 |x| 3+ǫ ≤ ψ 1 (x) ∀ |x| > R,
it is then sufficient to assume µ ≤ C 0 C ε λ 2+ε where C ε = (4 + 2ε) 2+ε e -(2+ε) . Therefore, with (76) we obtain

ψ µ λ (x) -2ψ 1 (x) ≤ w 1 (x), ∀ |x| > R. ( 79
)
On the other hand for µ ≤ η, we have ψ µ λ (x) -2ψ 1 (x) ≤ µ ≤ η ≤ w 0 (x), ∀ |x| < R. From (77) and (78) we deduce that if µR ǫ ≥ 3 4 πλ 2 C 0 (1 -∆) 3 α R L 1 and λ > 1, then 0 ≤ ψ µ λ -2 ψ 1 . Gathering the conditions, we deduce that if

3 4 π (1 -∆) 3 α R 1+ 2 ε L 1 (4 + 2ε) 4+2ε ǫ e -4+2ε ǫ < R 2+ε η C 0 (80)
then we can find some λ, µ > 0 such that ψ := ψ µ λ -2ψ 1 ≤ φ R where ψ fulfills the assumptions of the lemma. We define

C ε,0 = 3 4 πe 4+2ε ǫ (4 + 2ε) 4+2ε ǫ sup R>0 inf α (1 -∆) 3 α R 1+ 2 ε L 1 1 + R -6 1+ 2 ε . ( 81
)
Proof of Proposition 19. Let χ be a smooth function such that χ(x) = 1 for |x| ≤ 1/2 and χ(x) = 0 for |x| ≥ 1. Let us denote by χ R (x) = χ(x/R) and g = (1χ R )K. One can verify that ∂ α Ω still satisfies the cancellation property (4) for any multi-index |α| ≤ 4. One can also notice that ∂ α g ∈ L q (R 3 ) is a sum of terms in L 1 (R 3 ) and terms satisfying the assumptions of Lemma 10 and hence of bounded Fourier transform. Therefore (1 -∆) 2 g = (1 + |p| 2 ) 2 g and

(1 + |p| 2 ) 2 | g(p)| ≤ (1 -∆) 2 g L 1 ≤ C R Ω W 4,q .
Now, for µ > 0, consider ψ µ 1 ∈ L 1 (R 3 ) ∩ L ∞ (R 3 ) as defined in the proof of Lemma 21. It satisfies

ψ µ 1 (p) ≥ 3 4 µ (1 + |p| 2 ) 2 .
Taking µ ≥ 4C R Ω q /3, we define w stab = ψ µ 1 + (1χ R )K. Thanks to Sobolev embeddings W 4,q (S 2 ) ⊂ C 1 (R 3 ) and we have w stab -1 |x|>R K ∈ L ∞ (R 3 ). Moreover ψ µ 1 (x) = O(e -µ|x| ) as x → ∞, this concludes the proof of Proposition 19.

Proof of 2). This result is stated for the case K = 0 in a remark of [START_REF] Lieb | The mathematics of the Bose gas and its condensation[END_REF]Ch. 7]. Here, we give a proof working for any K satisfying the assumption of Lemma (10) using the fact that the interaction term is convex in ρ = |u| 2 . We write the Fourier decomposition of u in cylindrical coordinates u(r, θ, z) = where we used that if ra ∞ ≤ 1/2 then |n/r + a| ≥ |a| for any n ∈ Z. From inequality (84), we know that taking |u| 2 1/2 θ , where • θ stands for the θ-average, can only lower the θ-component of kinetic energy. We know that it also lowers the other components of the kinetic energy which are convex in ρ = |u| 2 since they are not affected by A [25, Theorem 7.8] and proves that minimizers are non negative up to a constant phase. Besides, the interaction term is strictly convex in ρ, indeed using that a + b K ≥ 0 and denote by < F (ρ) > ρ = (F (ρ 1 ) + F (ρ 2 ))/2 for some ρ 1 , ρ 2 and any fonction F , we have

a ρ 2 + b (K ⋆ ρ)ρ ρ = a + b K | ρ| 2 ρ ≥ a + b K | ρ ρ | 2 = a + b K | ρ ρ | 2 = a ρ 2 ρ + b K ⋆ ρ ρ ρ ρ .
The strict convexity of the functional gives the uniqueness of the minimizer by standard means.

R 3 Ω

 3 (y/|y|) |y| 3 |u(xy)| 2 dy = lim ε→0 |y|>ε Ω(y/|y|) |y| 3 |u(xy)| 2 dy

  since H N Ψ N = N e N Ψ N , then by taking the trace of (32) against γ = |Ψ N Ψ N | we get e Nε Tr(hγ) ≥ e N,ε

Case 1 :

 1 b > 0. Let (a, b) satisfy condition (10) with b > 0. Let us take d 2 = b and a ≥ 4πb/3. Definew = f + w dip , with f ∈ L 1 (R 3 ) ∩ L 2 (R 3) of positive type, such that f = a -4πb/3. Then, the potential w is classically stable, since w ≥ w dip and w dip is. It satisfies the assumptions of Theorem 5, and so the result holds for b > 0. Case 2: b < 0. Without loss of generality, let us assume n = e 3 and let us take d 2 = -b. Define w dip = fw dip with f being the inverse Fourier transform of

Lemma 21 (

 21 Stabilisation by a short range potential). Let w

3 4 λ 2 µ 3 4 λ 2 µ

 3232 (|p| 2 + λ 2 /4)(|p| 2 + λ 2 ) ≥ (|p| 2 + λ 2 ) 2 .

n∈Z c n

 n (r, z)e inθ , ∀r, z > 0, θ ∈ [0, 2π[, where c n (r, z) , θ, z)e -inθ dθ, ∀n ∈ Z.

  1 . In particular, φ R 0 is classically stable. If Lemma 21 and Proposition 19 hold, then Proposition 20 is easily verified. Proof of Proposition 20. Let us consider w stab as in Proposition 19. Let us write w = w 0 1 |x|≤R + 1 |x|>R K and rewrite w as,w = w 0 1 |x|≤Rw stab 1 |x|<R + (Kw stab )1 |x|>R + w stab . (74)The second term in (74) is smaller than w stab L ∞ , the third term is a O(e -|x| ) and the last term is of positive type according to Proposition 19. Then, by increasing w 0 sufficiently in some fixed neighborhood of the origin, one can make w classically stable by using Lemma 21. The idea of the proof is based on a result of Ruelle [29, Proposition 3.2.8].

Now we prove Lemma 21 and Proposition 19.

Appendix C. Proof of Theorem 4: uniqueness Proof of 1). We use the implicit function theorem to prove the uniqueness of the ground state of the Gross-Pitaevskii energy in some neighborhood of ϕ = 0. This method does not allow to give an explicit value of ϕ 0 in the lemma.

Let us fix some admissible parameters (a, b), i.e. satisfying condition [START_REF] Combes | Classical bounds and limits for energy distributions of hamilton operators in electromagnetic fields[END_REF], and let us define

We assume without loss of generality that V ≥ 0. A minimizer of E(•, ϕ) satisfies the Euler-Lagrange equation

where u = u 1 + iu 2 with u 1 , u 2 are real valued,

We want to apply the implicit function on Φ in a neighborhood of (u 0 , 0). For that, we first need the following lemma.

Lemma 22.

Define the operator

. Then L -is a non-negative self-adjoint operator with compact resolvent. Moreover, its ground state is u 0 and it is non-degenerate.

Proof. Thanks to Lemma 10 and Sobolev embeddings we have

). The self-adjointness and the compactness of the resolvent of H 0 follow from Sobolev inequality. The uniqueness and the non-negativity of the ground state up to a constant phase follow from [START_REF] Lieb | [END_REF]Theorem 7.8]. Now, since u 0 minimizes E(•, 0), we know that H 0 u 0 = µ 0 u 0 and also that for any v ∈ D(h

. Taking iv with v real in the last inequality shows that L -≥ 0 (since we know the ground state of L -is real up to a constant phase). Taking v = u 0 shows that u 0 is the ground state.

To verify the assumptions of the implicit function theorem we must compute

and denote by {u 0 } ⊥ the orthogonal space in L 2 (R 3 ) to u 0 with respect to the L 2 scalar product. We emphasize that the latter are spaces composed of real valued functions. Using Sobolev inequality and Lemma 10, it is elementary that Φ, d 1 Φ and d 2 Φ are continuous functions. In order to apply the implicit function theorem, we want to prove that

are bounded linear operators. Since, L ± have compact resolvants, it is sufficient to prove that 0 is not en eigenvalue of L + and that ker(L -) = span{u 0 }. We prove this by contradiction, let ξ ∈ ker(L + ), writing ξ = λu 0 + ξ ⊥ with ξ ⊥ ∈ {u 0 } ⊥ , we have

From (83), and since µ(u 0 , 0) > 0 because we assumed V ≥ 0, we deduce that λ = 0 and then from (82) together with Lemma 22 and the admissibility of (a, b) we obtain that ξ ⊥ = 0 which is the contradiction we seek. Similar arguments gives that ker(L -) = span(u 0 ). We in fact only proved that

Hence, there is uniqueness of the minimizer in X. Now we must prove that there is some other neighborhood of 0 which we will still denote by ]ϕ 0 , ϕ 0 [ such that for all |ϕ| < ϕ 0 , any minimizer of E(•, ϕ) (restricted to the unit sphere of L 2 (R 3 )) belongs to X (up to a constant phase). We prove it by contradiction, assume there is a sequence

It suffices to take θ =tan -1 ( ℑ(u), u 0 / ℜ(u), u 0 ). Hence, we can assume that u n ∈ H for all n. Then, notice that (u n ) is bounded in H it is thus precompact in L 2 (R 3 ) and we can assume without loss of generality that it converges to some function u ∞ ∈ L 2 (R 3 ). Besides, for any u and |ϕ| ≤ C we have

And since E(•, 0) is lower semi-continuous, by passing to the limit above we obtain E(u ∞ , 0) ≤ E(u 0 , 0) hence u ∞ = u 0 by uniqueness. We recall that X is a H-neighborhood of u 0 so we must prove the convergence of u n in the topology of H. We know that (h

thus, without loss of generality, we assume it converges weakly to some limit we denote by v ∞ . Then, for any ξ ∈ V ⊕ V we have

). Now, let us prove that the convergence is strong. For that, it suffices to prove the conservation of mass in the limit: h

as n → ∞. We already know that the first term tends to 0, for the two other, it comes from Sobolev embeddings and Lemma 10. Hence u n → u 0 in H and this is the contradiction we seek : there is some ϕ 0 > 0 such that if u is a minimizer of E(•, ϕ) then u ∈ X. This ends the proof.


Rafael Angarita
email: rafael.angarita@lamsade.dauphine.fr

Maude Manouvrier
email: manouvrier@lamsade.dauphine.fr

Marta Rukoz
email: marta.rukoz@lamsade.dauphine.fr

An Agent Architecture to Enable Self-healing and Context-aware Web of Things Applications

Keywords: Internet of Things, Web of Things, Self-healing, Fault-tolerance, Application-context-aware

The Internet of Things paradigm promises to connect billions of objects in an Internet-like structure. Applications composed from connected objects in the Internet of Things are expected to have a huge impact in the transportation and logistics, healthcare, smart environments, and personal and social domains. The world of things is much more complex, dynamic, mobile, and failure prone than the world of computers, with contexts changing rapidly and in unpredictable ways. The growing complexity of Internet of Things applications will be unmanageable, and will hamper the creation of new services and applications, unless the systems will show "self-*" functionality such as self-management, self-healing and self-configuration. The Web of Things builds on top of the Internet of Things to create applications composed of smart things relying on standard and well-known Web technologies. In this paper, we present a new agent architecture to enable self-healing and context-aware Web of Things applications. Our agents are the representation of physical objects, Web services, or humans in the Web.

INTRODUCTION

The Internet of Things paradigm has gained ground, both in the industry and in research worlds [START_REF] Atzori | The Internet of Things: A survey[END_REF]. It was also included by the US National Intelligence Council in the "Disruptive Civil Technologies -Six Technologies With Potential Impacts on US Interests Out to 2025" conference report (National Intelligence Council, 2008). The European Union has invested more than 100 million euros in projects related to the Internet of Things, and the government of China released the 12th Five-Year Plan for Internet of Things development [START_REF] Chen | A Vision of IoT: Applications, Challenges, and Opportunities With China Perspective[END_REF]. Failures in Internet of Things applications may lead to loss of production time, equipment damage, environmental catastrophes, or loss of human life [START_REF] Alho | Service-oriented Approach to Fault Tolerance in CPSs[END_REF].

The world of things is much more dynamic, mobile, and failure prone than the world of computers, with contexts changing rapidly and in unpredictable ways [START_REF] Mattern | From the Internet of Computers to the Internet of Things[END_REF]. In the Internet of Things Strategic Research Roadmap (Verme- san et al., 2011), Vermesan and his coauthors place autonomous and responsible behavior of resources as one of the fourth macro trends that will shape the future of the Internet of Things in the years to come. We extract the following paragraph: " ... the trend is towards the autonomous and responsible behaviour of resources. The ever growing complexity of systems, possibly including mobile devices, will be unmanageable, and will hamper the creation of new services and applications, unless the systems will show "self-*" functionality such as self-management, self-healing and selfconfiguration."

In another Internet of Things research di-rections paper, Stankovic stated that the areas of distributed and adaptive control are not developed well enough to support the open, dynamic environment of the Internet of Things [START_REF] Stankovic | Research Directions for the Internet of Things[END_REF]. Petersen and his coauthors argue that fault-tolerance and survivability play a key role in the designing of Internet of Things applications [START_REF] Petersen | The Role of the Internet of Things in Network Resilience[END_REF].

They propose a mandatory "disaster mode" for Internet of Things devices, allowing applications to continue working only with vital functionalities even in the presence of failures. Athreya and his coauthors suggest that the natural direction for Internet of Things devices is to manage themselves in terms of software, hardware, and resource consumption [START_REF] Athreya | Designing for self-configuration and selfadaptation in the Internet of Things[END_REF]. Cherrier and his coauthors also underline the importance of fault tolerance, recovery, and coherence mechanism in the Internet of Things applications [START_REF] Cherrier | Fault-recovery and coherence in Internet of Things choreographies[END_REF]. Cirani and his coauthors present a self-configuring architecture for service discovery in the Internet of Things where they identify the resilience to changes as a crucial issue to be addressed [START_REF] Cirani | A Scalable and Self-Configuring Architecture for Service Discovery in the Internet of Things. Internet of Things Journal[END_REF].

It is clear that fault-tolerance, resilience, self-healing, and other self-* research are very active areas since they face new challenges in the Internet of Things context due to its promise of connecting billions of devices in an Internet-like structure.

The Web of Things (WoT) [START_REF] Guinard | Architecting the Internet of Things, chapter From the Internet of Things to the Web of Things: Resource-oriented Architecture and Best Practices[END_REF] builds on top of the Internet of Things to create applications composed of smart things using standard and well-known Web technologies. In this paper, we present an agent architecture to enable self-healing and contextaware WoT applications. Our agents are the representation of physical objects, Web services, or humans in the Web, and they may be hosted and run inside physical objects or in a cloud infrastructure. In our architecture, an agent may also be responsible of managing and monitoring WoT applications, which includes the communication with other participant agents.

The rest of this paper is organized as follows. We provide an overview of our proposed architecture in Section 2. We present a case study to illustrate our proposal in Section 3. This is followed by related work, and conclu-sions and future work directions in Sections 4 and 5, respectively.

OUR AGENT ARCHITECTURE

In this section, we present the main aspects of our agent architecture for self-healing and context-aware WoT applications. First, we describe the building block of our approach: the WoT Agent. Then, we present the WoT Application Manager, which is in charge of WoT applications. Before going into further detail, we present our definition of a WoT application. Definition 1. WoT Application. A WoT application is a composition of things linked by data or control dependency, where things may be physical objects, Web services, or human beings. A WoT application has an associated QoS (e.g., execution time) which is the aggregation of the participating things QoS.

In our approach, all things participating in a WoT application, independently of their nature, are represented virtually by a WoT agent, which we define in the following paragraphs.

WoT Agent. Inspired from our previous work [START_REF] Angarita | Modeling dynamic recovery strategy for composite web services execution[END_REF], we introduce the WoT application component agent depicted in Fig. 1. It is in charge of executing an operation in a WoT application, communicating with peers, and applying fault-tolerance mechanisms if necessary. It may be implemented using Node.js, and its main components are the following:

• API : an interface to communicate with the WoT agent via JSON objects indicating the id of the WoT application, the sender, and data types and their corresponding values.

In case the id of the WoT application is empty, the WoT agent acts in a context-free way.

• Core: it contains the basic execution control elements of WoT application component agents; that is, waiting for inputs, invoking its corresponding operation, and sending produced outputs, if any.

• Autonomic Component : it represents the control loop of the autonomic computing [START_REF] Psaier | A Survey on Selfhealing Systems: Approaches and Systems[END_REF], and it de-tects local or global degradations on the application behavior, selects an appropriate action, and applies it.

• Context Manager and Knowledge Base: It handles information regarding the WoT applications the WoT agents is participating in. Its main components are:

-Knowledge Base: it contains information described in RDF regarding WoT applications: rules indicating normal behavior and collaborators, requirement verification, self-healing actions, replacement operations, and replacement WoT agents.

-Inference Engine: it infers logical consequences from a set of asserted facts in the knowledge base. The Apache Jena framework may be used to reason over a RDF graph and query them using SPARQL.

• WoT Application Manager: stores and monitors WoT applications.

• Object Interface: it is the module in charge of the communication with its corresponding object.

Figure 1: WoT Agent Architecture

We present in detail the autonomic component of the WoT agent architecture in Figure 2.

The detection component (Figure 2 (a)) takes into account one external and two internal data sources. The external information regards the expected QoS; for example, the WoT application manager may allow a certain QoS degradation. The internal information refers to the QoS degradation of its corresponding operation (e.g., it is taking more time than expected), and to operation failure.

The diagnosis component (Figure 2 (b)) analyzes the current degradation and computes a solution. The three possible diagnosis correspond to the three states of a self-healing system: normal; degraded; and broken. The choice of the recovery mechanism is influenced by available options (e.g., retry or replacement), and constraints imposed by the WoT application manager (e.g., expected QoS).

The recovery component (Figure 2 (c)) is in charge of applying the selected fault tolerance mechanisms: retry and replacement. A WoT agent may retry or replace its own operation, retry the communication with another WoT agent, or replace another WoT agent in the current WoT application.

Figure 2: Autonomic Component

WoT Application Manager. We present an overview of the WoT Application Manager in Figure 3. Besides performing its corresponding operation, a WoT agent may manage a set of WoT applications consisting of its participants components and their data relationship, a set rules and requirements of the WoT application, a triggering event indicating when the WoT application starts its execution, a disaster mode specifying what to do in case of irrepara- ble failures, its historical executions, and a deployment protocol.

The deployment protocol analyzes participating components, defines the WoT application execution mode, creates the necessary components, and sends them the required information. A WoT application manager may need to create agents to control participating components or communicate with existing WoT agents. The deployment protocol contacts participating components and verifies their capacity to manage application context. A WoT application has the following execution modes:

• Locally-hosted: the WoT application component agents in charge of the operations of the WoT application are hosted and run in the component managing the WoT application. This may happen when dealing with agentless components of a WoT application such as resourcesless objects or RESTful services, in the absence of a cloud infrastructure, and when the host component has enough resources to manage the WoT application.

• Cloud-based: the WoT application component agent in charge of the operations of the WoT application are hosted and run in a cloud infrastructure. This also may be the case when dealing with agentless components. Also, WoT agents may be already provided by their corresponding objects but hosted in a a cloud infrastructure.

• Distributed: the WoT agents may be managed and hosted by their respective objects.

• any combination of the previous modes. Figure 4 shows a deployed WoT application. The WoT Application Manager sends the required information to all participant WoT agents, which may then communicate between them during the WoT application execution.

WoT Agent Failure. The corresponding operation of a WoT agent may fail; in this case, the failure may be fixed by operation retry or replacement, similarly as showed in our previous work [START_REF] Angarita | The Semantic Web: ESWC 2012 Satellite Events: ESWC 2012 Satellite Events, Heraklion[END_REF][START_REF] Angarita | Modeling dynamic recovery strategy for composite web services execution[END_REF]. In the WoT application context, WoT agents may fail themselves, specially if they are hosted in mobile physical objects. In this case, the failed agent must be detected and the WoT application must be reconfigured. We propose two WoT agent failure detection mechanisms:

1. Predecessor detection: when a WoT agent sends a message to another WoT agent, it waits for the status code OK. If it receives another status code, or it gives timeout, the WoT agent may retry or replace its successor WoT agent. It may also execute the disaster mode.

2. WoT application timeout: if a WoT agent crashes during the executing of its operation, the WoT application manager eventually gives timeout, check which WoT agents are not alive, and performs retry, replacement, or executes the disaster mode. For further details about application timeout detection see [START_REF] Angarita Arocha | An approach for Self-healing Transactional Composite Services[END_REF].

CASE STUDY

Figure 5 shows a Petri net representing a fictional e-Health application we adapted from [START_REF] Angarita Arocha | An approach for Self-healing Transactional Composite Services[END_REF] to illustrate our proposition.

This application is built from 9 components, and it is installed in the mobile phone of a patient called Jenny; that is, the phone has a WoT agent which also plays the role of the WoT application manager, represented by phone ♦ and phone . The WoT application triggering event specifies that it runs every 30 minutes. Jenny wears the SugarImplant and VitalSignsImplant smarts devices that gather information about her health.

When the WoT application starts, phone ♦ tells both devices to send their data to SugarAnalysis and VitalSignsAnalysis, which send their conclusions to Diagnoser. Diagnoser sends its results to the appropriate components depending on the necessary actions.

Table 1 shows the manager and loca- To understand context-awareness in this WoT application, suppose that the application started its execution, SugarImplant, VitalSignsImplant, SugarAnalysis, and VitalSignsAnalysis analysis were successfully invoked. SugarAnalysis and VitalSignsAnalysis send their messages to Diagnoser. We show these messages in Listings 1 and 2. Note that both messages indicate the id of the WoT application they belong to. In this case, the id is "e427b92cfb07e68215110b6e8f00357b", which refers to Jenny's e-Health application.

Remember that Diagnoser may be receiving also data from other applications it is participating in, or processing context-free requests.

{"wot-app": { "id": "e427b92cfb07e68215110b6e8f0035 7b", "sender": "SugarAnalysis", "data": [ {"name": "sugarResult1", "value": " someValue1"}, {"name": "sugarResult2", "value": " someValue2"} ] }} Listing 1: SugarAnalysis Message {"wot-app": { "id": "e427b92cfb07e68215110b6e8f0035 7b", "sender": "VitalSignsAnalysis", "data": [ {"name": "vitalSignsResult1", " value": "vitalSignsValue"} ] }} Listing 2: VitalSignsAnalysis Message When Diagnoser receives Jenny's data from SugarAnalysis and VitalSignsAnalysis, it performs its corresponding operation which may produce the following output: NORMAL, WARNING, or EMERGENCY .

Diagnoser also contains the monitoring rules associated to Jenny's application showed in Listing 3. These rules are stored in the knowledge base of Diagnoser and verified by its autonomic component (see the WoT agent architecture depicted in Figure 1). The rules showed in Listing 3 are all post-condition rules, but other kind may exist. Also, rules may reflect QoS requirements and self-healing actions. The first three rules specify which component should be invoked. The last rule states that the disaster mode must be executed in

RELATED WORK

In 2001, IBM published the Autonomic Computing (IBM, 2001) manifesto expressing their concerns about the inevitable increasing of the size and complexity of computer systems. For them, it was clear that such complexity of heterogeneous and distributed systems will minimize the benefits of future technology; therefore, solving the increasing com-plexity problem was the "next Grand Challenge". Two years later, we had the Vision of Autonomic Computing (Kephart and Chess, 2003) where the authors reaffirmed that the only solution to the software complexity crisis was through computing systems that can manage themselves. They presented the concept of self-management as the building block of autonomic computing. The self-management concept includes four main aspects: selfconfiguration, self-optimization, self-healing, and self-protection.

Kephart and Chess described the selfhealing property of autonomic systems as the system's ability to automatically detect, diagnose, and repair software and hardware problems. In a survey published in 2007, Ghosh and his coauthors presented the now well-known concepts of self-healing states and properties [START_REF] Ghosh | Self-healing Systems -Survey and Synthesis[END_REF].

They explained that the vision of large scale systems was already a reality and that self-healing research was active. In 2011, Psaiser and Dustar published a survey showing the advancements on self-healing research [START_REF] Psaier | A Survey on Selfhealing Systems: Approaches and Systems[END_REF]. The collected self-healing research areas in this survey included embedded systems, operating systems, architecture based, cross/multi-layer-based, multi agentbased, reflective-middleware, legacy application and Aspect Oriented Programming, discovery systems, and Web services and QoSbased.

In [START_REF] Mrissa | An Avatar Architecture for the Web of Things[END_REF], the authors propose a software component called avatar for the WoT. The idea is to give objects a virtual representation in the Web so the may rely on Web languages, protocols, and semantic annotations. These avatars collaborate together in composed applications; however, they exhibit neither fault-tolerance nor self-healing properties. Also, they do not provide mechanisms to handle application context; that is, avatars function in a request-response way, and WoT applications need to be managed by a central coordinator. Moreover, they are not able to act differently depending on the application the are participating in. RESEARCH DIRECTIONS

We have presented a proposal to enable selfhealing and context-aware WoT applications. The building block of our approach is a software component called WoT agent, which are the representation of physical objects, Web services, or humans in WoT applications. These agents may be provided by their respective components or may be created by our system to manage agentless components. They are also equipped with a knowledge base to handle application specific information and requirements, and with an autonomic component to exhibit self-healing properties. Our next steps are the formal definition of our approach, the implementation of its main functionalities, and its experimental evaluation.

Figure

  Figure 3: WoT Application Manager

Table 1 :

 1 WoT agents managers and hosts

		Figure 5: Case Study: the e-Health WoT application
	tion of each of the WoT application com-
	ponents according to the deployment pro-
	tocol described in Section 2. SugarImplant
	and VitalSignsImplant are powerful
	jects hosting WoT agents.	SugarAnalysis,
	VitalSignsAnalysis, and Diagnoser are not phys-
	ical objects; however, they are provided as
	WoT agents hosted in a cloud infrastructure.
	CallEmergency, Noti f yContact, Noti f yDoctor,
	and DisplayMessage are RESTful services
	which do not have WoT agents. Their cor-
	responding WoT agents are created, hosted,
	and managed by the WoT application manager
	which is hosted and runs in Jenny's mobile
	phone (Phone).
	Component	Manager Host
	SugarImplant	Itsel f	Itsel f
	VitalSignsImplant Itsel f	Itsel f
	SugarAnalysis	Itsel f	Cloud
	VitalSignsAnalysis Itsel f	Cloud
	Diagnoser	Itsel f	Cloud
	CallEmergency	Phone	Phone
	Noti f yContact	Phone	Phone
	Noti f yDoctor	Phone	Phone
	DisplayMessage	Phone	Phone