Julie Ghirardi

Jérémy Jacob

Claude Le Milbeau

Hugues Bauer

Florence Quesnel

Christine Fléhoc

Johann Schnyder

Christian Di Giovanni

Biomarker assemblages associated with the Eocene-Oligocene transition in lacustrine deposits of the Rennes Basin (France)

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

BIOMARKER ASSEMBLAGES ASSOCIATED WITH THE EOCENE-OLIGOCENE TRANSITION IN LACUSTRINE DEPOSITS OF THE RENNES BASIN (FRANCE)

Julie GHIRARDI 1,2,* , Jérémy JACOB 1 , Claude LE MILBEAU 1 , Hugues BAUER 1,2 , Florence QUESNEL 1,2 , Christine FLEHOC In marine setting, it is marked by an increase in detrital input associated with an increase of primary productivity that resulted from well mixing of water masses [START_REF] Peihzen | Increase sedimentation rates and grain sizes 2-4 Myr ago due to the influence of climate change on erosion rates[END_REF]. In terrestrial setting, this cooling led to major changes in animal and plant communities. For instance, palynology data show a vegetation shift from wet to dry taxa [START_REF] Gregory | Tectonic significance of paleobotanically estimated climate and altitude of the Late Eocene Erosion Surface[END_REF] coincident with a large mammal turnover, the well-known "Grande coupure" (Stelhin, 1909). Although this transition is well documented in marine settings, the lack of data on lands -due to limited available outcrops -does not allow appreciating precisely its impacts on terrestrial ecosystems.

Here, we have analysed the sediment infill of the Rennes Basin that consists of 300 m of organic-rich lacustrine deposits (laminated and massive) that encompass the Middle Eocene to Oligocene series, including the Eocene-Oligocene transition, the whole fully cored (CINERGY project, BRGM). About 100 samples were collected through the core enabling us to precisely decipher environmental changes that occurred during the studied interval and to focus on the Eocene/Oligocene transition. All samples were subjected to Rock-Eval and EA-irMS to measure bulk δ 13 C. Four over the 100 samples were analysed for their molecular content so far. For a preliminary test, we have chosen representative facies (a laminated and a massive Upper Eocene sample and a laminated and a massive Lower Oligocene sample).

Lipids were extracted with an Automated Solvant Extractor (ASE200®, Dionex) using DCM:MeOH (9:1). The total extract was later separated into neutrals, acidic and polar compounds. The neutral compounds were further separated into aliphatic, aromatic, ethers and esters, ketones and alcohols. After addition of 5α-cholestane, each fraction was then analysed by Gas Chromatography-Mass spectrometry on a Trace GC Ultra gas chromatograph coupled to a TSQ Quantum XLS mass spectrometer equipped with an AS 3000 autosampler.

The analysis of the different fractions of each sample revealed a high diversity of compounds that could partly be related to their source organisms or inform on the depositional environment. Hydrocarbon fraction is dominated by a series of n-alkanes that display a strong odd-over-even predominance, indicative of a good preservation. The dominance of long-chain homologues shows a strong higher plants contribution. This is confirmed by the presence of abundant and diversified higher plant pentacyclic triterpenes.

These comprise angiosperms biomarkers such as oleanane, ursane and lupane, friedelin, onocerane I and diagenetic by-products (des-A-tritrepenes and aromatic derivatives). Fernenetype compounds could reveal an input from pteridophyts whereas tricyclic diterpenes indicate the presence of conifers. An algal influence is evidenced, for example, by the presence of 4methylsteranes in the four samples. C 27 to C 32 hopanoids with α,β, β,β and β,α configurations show a substantial bacterial contribution. This is confirmed by high levels of hopane ketones in some beds.

The comparison of angiosperm-derived biomarkers with gymnosperm-derived biomarkers indicates a stronger contribution of angiosperms in Upper Eocene samples than in Lower Oligocene samples (triterpenoids, aromatic, and rearranged-triterpenoids). This result is in agreement with palynological data (Bourdillon et al., 2012) and with the well-known cooling that is coeval with the Eocene/Oligocene transition. Reversely, the comparison between the two facies (massive and laminated) does not exhibit significant differences in the molecular assemblage.

Based on this preliminary inventory of biomarkers and the richness in various biomarkers, we will proceed with the identification and quantitation on the remaining samples that cover the whole section. These preliminary results are also encouraging in the perspective of performing compound-specific isotopic analyses in order to depict, at high temporal resolution, the climatic evolution and the associated changes in plant communities during the Late Eocene and Early Oligocene and, more specifically, at the Eocene/Oligocene transition.

  1,2 , Johann SCHNYDER 3 , Christian DI GIOVANNI 1 1 Institut des Sciences de la Terre d'Orléans, ISTO, UMR 7327 du CNRS/INSU, Université d'Orléans, BRGM, 1A rue de la Férollerie, 45071 Orléans, France.

	2 BRGM (French Geological Survey), 45060 Orléans Cedex 2, France.
	3 UPMC Univ Paris 06 et CNRS, UMR 7193 ISTeP, 4 place Jussieu, 75005 Paris, France.
	*) Corresponding author: julie.ghirardi@univ-orleans.fr
	The Eocene/Oligocene transition is one of the main climatic events to which more
	recent climate changes are often compared. Dated back to 34 Ma, it is remarkable because the
	Earth evolved from a greenhouse to icehouse state, permanent ice sheets appearing during the
	Late Eocene.


Nicolas Dombrowski
email: nicolas.dombrowski@helsinki.fi

AND Peter D Hislop
email: hislop@ms.uky.edu

Eric Soccorsi
email: eric.soccorsi@univ-amu.fr

Cpt

EDGE CURRENTS AND EIGENVALUE ESTIMATES FOR MAGNETIC BARRIER SCHR ÖDINGER OPERATORS

published or not. The documents may come

Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators

AMS 2000 Mathematics Subject Classification: 35J10, 81Q10, 35P20 Keywords: magnetic Schrödinger operators, snake orbits, magnetic field, magnetic edge states, edge conductance

Statement of the problem and results

We continue our analysis of the spectral and transport properties of perturbed magnetic Schrödinger operators describing electrons in the plane moving under the influence of a transverse magnetic field. In [START_REF] Hislop | Edge states induced by Iwatsuka Hamiltonians with positive magnetic fields[END_REF], two of us studied the original Iwatsuka model for which 0 < b -< b + < ∞. The basic model treated in this paper consists of a transverse magnetic field that is constant in each half plane so that it is equal to b > 0 for x > 0 and -b < 0 for x < 0. We choose a gauge so that the corresponding vector potential has the form (0, A 2 (x, y)). The second component of the vector potential A 2 (x, y) is obtained by integrating the magnetic field so that A 2 (x, y) = b|x|, independent of y. The fundamental magnetic Schrödinger operator is:

H 0 := p 2
x + (p y -b|x|) 2 , p x := -i∂/∂x, p y := -i∂/∂y, (1.1) defined on the dense domain C ∞ 0 (R 2 ) ⊂ L 2 (R 2 ). This operator extends to a nonnegative self-adjoint operator in L 2 (R 2 ).

The magnetic field is piecewise constant and equals ±b on the half-planes R * ± × R, where R * ± := R ± \{0}. The discontinuity in the magnetic field at x = 0 is called a magnetic edge. Classically, a particle moving within a distance of O(b -1/2 ) of the edge moves in a snake orbit [START_REF] Reijniers | Snake orbits and related magnetic edge states[END_REF]. Half of a snake orbit lies in the half-plane x > 0, and the other half of the orbit lies in x < 0. We prove that the quantum model has current flowing along the magnetic edge at x = 0 and that the current is localized in a small neighborhood of size O(b -1/2 ) of x = 0. 1.1. Fiber operators and reflection symmetry. Due to the translational invariance in the y-direction, the operator H 0 on L 2 (R 2 ) is unitarily equivalent to the direct integral of operators h(k), k ∈ R, acting on L 2 (R). This reduction is obtained using the partial Fourier transform with respect to the y-coordinate and defined as

(Fu)(x, k) = û(x, k) := 1 √ 2π R e -iyk u(x, y)dy, (x, k) ∈ R 2 .
Then we have FH 0 F * = H 0 where

H 0 := ⊕ R h(k)dk,
and the fiber operator h(k) acting in

H := L 2 (R) is h(k) := p 2
x + (k -b|x|) 2 , k ∈ R. Since the effective potential (k -b|x|) 2 is unbounded as |x| → ∞, the selfadjoint fiber operators h(k) have compact resolvent. Consequently, the spectrum of h(k) is discrete. We write ω j (k) for the eigenvalues listed in increasing order. They are all simple (see [12, Appendix: Proposition A.2]) and depend analytically on k. As functions of k ∈ R, these functions are called the band functions or dispersion curves and their properties play an important role. For fixed k ∈ R, we denote by ψ j (k) the L 2 -normalized eigenfunctions of h(k) with eigenvalue ω j (k). These satisfy the eigenvalue equation: h(k)ψ j (x, k) = ω j (k)ψ j (x, k), ψ j (x, k) ∈ L 2 (R), ψ j (•, k) = 1.

(1.2)

We choose all ψ j (k) to be real, and ψ 1 (x, k) > 0 for x ∈ R and k ∈ R. The rankone orthogonal projections P j (k) := •, ψ j (k) ψ j (k), j ∈ N * , depend analytically on k by standard arguments.

The full operator H 0 exhibits reflection symmetry with respect to x = 0. Let I P be the parity operator:

(I P f )(x, y) := f (-x, y), (1.3)
so that I 2 P = 1. The Hilbert space L 2 (R 2 ) has an orthogonal decomposition corresponding to the eigenspaces of I P with eigenvalue ±1. The Hamiltonian H 0 commutes with I P so each eigenspace of I P is an H 0 -invariant subspace.

This symmetry passes to the fiber decomposition. For each k ∈ R we have [h(k), I P ] = 0, where I P is the restriction to L 2 (R) of the operator defined in (1.3). Since the eigenvalues of h(k) are simple, for each k ∈ R, there is a map θ j (k) : R → {±1} so that (I P ψ j )(x, k) = θ j (k)ψ j (x, k), k ∈ R, j ∈ N * , as ψ j (x, k) is L 2 (R x )-normalized and real-valued. We show that θ j (k) is independent of k. Since the mapping k → P j (k), the orthogonal projector onto ψ j (•, k), is analytic, it follows that θ j (k) = θ j (0) for every k ∈ R. Consequently, each eigenfunction ψ j (x, k) is either even or odd in x.

We have an h(k)-invariant decomposition L 2 (R) = H -⊕ H + , according to the eigenvalues {-1, +1} of the projection (

I P f )(x) = f (-x). From this then follows that h(k) = h + (k) ⊕ h -(k), where h ± (k) := h(k) |H ± , H ± := {f ∈ H, I P f = ±f }.
We analyze the spectrum of h(k) by studying the spectrum of the restricted operators letting σ(h

± (k)) := {ω ± j (k), j ∈ N * }. Bearing in mind that ω + j (k) < ω - j (k) and σ(h(k)) = σ(h + (k)) ∪ σ(h -(k)) for every k ∈ R, we have ω + j (k) = ω 2j-1 (k), ω - j (k) = ω 2j (k), j ∈ N * . 1.2.
Effective potential. The fiber operator h(k) has an effective potential:

V ef f (x, k) := (k -b|x|) 2 , x, k ∈ R.
The properties of this potential determine those of the band functions.

Positive k > 0. There are two minima of V ef f at x ± := ±k/b. The potential consists of two parabolic potential wells centered at x ± and has value V ef f (0, k) = k 2 . As k → +∞, the potential wells separate and the barrier between the two minima grows to infinity.

Negative k < 0. The effective potential is a parabola centered at x = 0 and V ef f (0, k) = k 2 is the minimum. Consequently, as k → -∞, the minimum of this potential well goes to plus infinity.

Band functions.

The behavior of the effective potential determines the band functions. For k > 0, the symmetric double wells of V ef f indicate that there are two eigenvalues near each level of a harmonic oscillator Hamiltonian. The splitting of these eigenvalues is exponentially small in the tunneling distance in the Agmon metric between x ± . As k → +∞, this tunneling effect is suppressed and these two eigenvalues approach the harmonic oscillator eigenvalue exponentially fast. For k < 0, there is a single potential well with a minimum that goes to infinity as k → -∞. Hence, the band functions diverge to plus infinity in this limit. Several band functions along with the parabola E = k 2 are shown in Figure 1. Theorem 2.2 of [START_REF] Dombrowski | Quantization of the edge conductance for magnetic perturbation of Iwatsuka Hamiltonians[END_REF] presents the quantization of edge currents for the generalized Iwastuka model. For this model, the magnetic field b(x) is simply assumed to be monotone and to have values b ± at ±∞. The energy interval I is assumed to satisfy the following condition. There are two nonnegative integers n ± 0 for which

I ⊂ ((2n --1)|b -|, (2n -+1)|b -|)∩((2n + -1)|b + |, (2n + +1)|b + |), n ± = 0. (1.4)
If n ± = 0, the corresponding interval should be taken to be (-∞, |b ± |). Under condition (1.4), Dombrowski, Germinet, and Raikov [START_REF] Dombrowski | Quantization of the edge conductance for magnetic perturbation of Iwatsuka Hamiltonians[END_REF] We complement this result by proving in sections 3 and 4 the existence and localization of edge currents for H 0 and its perturbations. Following the notation of those sections, we prove, roughly speaking, that there is a nonempty interval ∆ E (δ 0 ) between the Landau levels (2n -1)b and (2n + 1)b and a finite constant c n > 0, so that for any state ψ = P 0 (∆ E (δ 0 ))ψ, where P 0 (∆ E (δ 0 )) is the spectral projector for H 0 and the interval ∆ E (δ 0 ), we have

ψ, v y ψ c n 2 b 1/2 ψ 2 > 0, v y := -(p y -b|x|).
This lower bound indicates that such a state ψ carries a nontrivial edge current for H 0 . We prove that this estimate is stable for a family of magnetic and electric perturbations of H 0 .

1.5. Contents. We present the properties of the band functions ω j (k) for the unperturbed fiber operator h(k) in section 2. The emphasis is on the behavior of the band functions as k → ±∞. The basic Mourre estimate for the unperturbed operator H 0 is derived in section 3 and its stability under perturbations is proven. As a consequence, this shows that there is absolutely continuous spectrum in certain energy intervals. Existence, localization, and stability of edge currents for a family of electric and magnetic perturbations is established in section 4. These edge currents and their lower bounds are valid for all times. We also prove a lower bound on the asymptotic velocity for a different class of perturbations in Theorem 4.2. In section 5, we study perturbations by negative potentials decaying at infinity. We demonstrate that such potentials create infinitely-many eigenvalues that accumulate at the bottom of the essential spectrum from below. We establish the asymptotic behavior of the eigenvalue counting function for these eigenvalues accumulating at the bottom of the essential spectrum.

1.6. Notation. We write •, • and • for the inner product and norm on L 2 (R 2 ). The functions are written with coordinates (x, y), or, after a partial Fourier transform with respect to y, we work with functions f (x, k) ∈ L 2 (R 2 ). We often view these functions f (x, k) on L 2 (R x ) as parameterized by k ∈ R. In this case, we also write f (•, k), g(•, k) and f (•, k) for the inner product and related norm on L 2 (R x ). So whenever an explicit dependance on the parameter k appears, the functions should be considered on L 2 (R x ). We indicate explicitly in the notation, such as • X , for X = L 2 (R ± ), when we work on those spaces. We write

• ∞ for • L ∞ (X) for X = R, R ± , or R 2 .
For a subset X ⊂ R, we denote by X * the set X * := X\{0}. Finally for all n ∈ N we put N n := {j ∈ N, j n} = {0, 1, . . . , n}. Remark 1. After completion of this work, we learned of a similar analysis of the band structure by Nicolas Popoff [START_REF] Popoff | Sur le spectre de l'opérateur Schrödinger magnétique dans un domaine diédral[END_REF] in his 2012 thesis at the Université Rennes I. We thank Nicolas for many discussions and for letting us use his graph in Figure 1.

Remark 2. After completing this paper, we discovered the paper "Dirichlet and Neumann eigenvalues for half-plane magnetic Hamiltonians," by V. Bruneau, P. Miranda, and G. Raikov [START_REF] Bruneau | Dirichlet and Neumann eigenvalues for halfplane Magnetic Hamiltonians[END_REF]. Their Corollary 2.4, part (i), is similar to our Theorem 5.1.

Properties of the band functions

In this section, we prove the basic properties of the band functions k ∈ R → ω j (k). We have the basic identity:

ω j (k) = ψ j (•, k), h(k)ψ j (•, k) .
According to section 1.1, the eigenfunctions of h(k) are either even and lie in H + , or odd and lie in H -, with respect to the reflection x → -x. We label the states so that the eigenfunctions ψ 2j-1 ∈ H + and ψ 2j ∈ H -, for j = 1, 2, 3, . . .. The restrictions of h(k) to H ± are denoted by h ± (k), with eigenvalues ω + j (k) = ω 2j-1 (k) and ω - j (k) = ω 2j (k), respectively. Following the qualitative description in section 1.2, we have the following asymptotics for the band functions. When k → +∞, the band function satisfies ω j (k) → (2j -1)b, whereas as k → -∞, we have ω j (k) → +∞.

Proposition 2.1. The band functions ω j (k) are differentiable and the derivative satisfies

ω j (k) = -2 b (ω j (k) -k 2 )ψ j (0, k) 2 + ψ j (0, k) 2 . (2.1)
As a consequence, we have a classification of states:

(1) Odd states: ψ 2j (0, k) = 0. The band functions satisfy: (2) Even states: ψ 2j-1 (0, k) = 0. The band functions satisfy:

(ω - j ) (k) = ω 2j (k) = -2 b ψ 2j (0, k) 2 < 0. ( 2.2)
(ω + j ) (k) = ω 2j-1 (k) = -2 b (ω 2j-1 (k) -k 2 )ψ 2j-1 (0, k) 2 . (2.3)
Proof. The Feynman-Hellmann Theorem gives us

ω j (k) = R 2(k -b|x|)ψ j (x, k) 2 dx = -1 b ∞ 0 ψ j (x, k) 2 d dx (k -bx) 2 dx + 1 b 0 -∞ ψ j (x, k) 2 d dx (k + bx) 2 dx.
Integrating by parts, and using the ordinary differential equation (1.2), we obtain (2.1). Note that lim x→±∞ x 2 ψ j (x, k) 2 = 0 since ψ j (k) is in the domain of h(k) (see [START_REF] Iwatsuka | Examples of absolutely continuous Schrödinger operators in magnetic fields[END_REF]Lemma 3.5]).

Let us note that we cannot have both ψ j (0, k) = 0 and ψ j (0, k) = 0. As consequences, the band functions for odd states are strictly monotone decreasing ω 2j (k) < 0. For even states, there is a minimum at k = κ j satisfying ω 2j-1 (κ j ) = κ 2 j . We will prove in Proposition 2.4 that this is the unique critical point of these band functions and that it is a non-degenerate minimum. This shows that there is an effective mass at this point. This is essential for the discussion in section 5.

2.1.

Absolutely continuous spectrum for H 0 . The spectrum of H 0 is the union of the ranges of the band functions σ(H 0 ) = ∪ j 1 ω j (R) = [ω 1 (κ 1 ), ∞). The band functions are analytic and nonconstant by Propositions 2.1 and 2.4. Consequently, from [START_REF] Reed | Methods of Modern Mathematical Physics. IV. Analysis of Operators[END_REF]Theorem XIII.86], the spectrum of H 0 is purely absolutely continuous.

2.2.

Band function asymptotics k → -∞. As k → -∞, we will prove that the fiber Hamiltonian h(k) is well approximated by an Airy operator

h Ai (k) := p 2 x + 2b|k||x| + k 2 , (2.4)
in the sense that the band functions of h(k) are close to the band functions of the Airy operator h Ai (k). In order to establish this, let Ai(x) be the standard Airy function whose zeros are located on the negative real axis. The Airy function satisfies the Airy ordinary differential equation:

Ai (x) = xAi(x).
By scaling and translations, it follows that the Airy function Ai(γx+σ) satisfies

(p 2 x + γ 3 x)Ai(γx + σ) = -γ 2 σAi(γx + σ), γ, σ ∈ R. (2.5)
The model Airy Hamiltonian h Ai (k) in (2.4) has discrete spectrum ωj (k) and eigenfunctions ΨAi

j (x, k) satisfying h Ai (k) ΨAi j (x, k) = ωj (k) ΨAi j (x, k). (2.6)
It follows from (2.5) that the eigenfunction ΨAi j (x, k) in (2.6) is a multiple of the scaled and translated Airy function. The non-normalized solution ΨAi j (x, k) for the eigenvalue ωj (k) is

ΨAi j (x, k) = Ai (2b|k|) 1/3 |x| + k 2 -ωj (k) (2b|k|) 2/3 , k < 0, x ∈ R, with an eigenvalue given by ωj (k) = k 2 -(2b|k|) 2/3 σ.
We determine σ as follows. The operator h Ai (k) commutes with the parity operator I P so its states are even or odd. The odd eigenfunctions

Ψ Ai,o j (x, k) of h Ai (k) must satisfy Ψ Ai,o j (0, k) = 0. Consequently, the L 2 (R)-normalized odd eigenfunctions Ψ Ai,o j (x, k) = ΨAi 2j (x, k) are given by Ψ Ai,o j (x, k) = C Ai,j (b, k)(sign x)Ai((2b|k|) 1/3 |x| + z Ai,j ), Ψ Ai,o j (0, k) = 0, (2.7)
where z Ai,j is the j th zero of Ai(x) and the corresponding eigenvalue is

ω2j (k) = k 2 -(2b|k|) 2/3 z Ai,j .
The even eigenfunctions Ψ Ai,e j (x, k) = ΨAi 2j-1 (x, k) of h Ai (k) must have a vanishing derivative at x = 0 and are given by

Ψ Ai,e j (x, k) = C Ai ,j (b, k)Ai((2b|k|) 1/3 |x| + z Ai ,j ), (Ψ Ai,e j ) (0, k) = 0, (2.8)
and the corresponding eigenvalue is

ω2j-1 (k) = k 2 -(2b|k|) 2/3 z Ai ,j ,
where z Ai ,j is the j th zero of Ai (x). The normalization constant C X,j (b, k), for X = Ai or Ai is given by

C X,j (b, k) := (2b|k|) 1/3 2c X,j 1/2
, where c X,j :=

∞ 0 Ai(v + z X,j ) 2 dv.
(2.9)

We now obtain estimates on the band functions ω j (k) as k → -∞.

Proposition 2.2. For each j ∈ N * , as k → -∞, we have

(h(k) -[k 2 -(2b|k|) 2/3 z X,j ])Ψ Ai,u j (•, k) b 4/3 (2|k|) 2/3 D X,j , (2.10)
where the constant D X,j , given in (2.11), is independent of the parameters (k, b), and (X, u) = (Ai, e) or (Ai , o), for even or odd states, respectively. This immediately implies the eigenvalue estimate

|ω j (k) -[k 2 -(2b|k|) 2/3 z X,j ]| b 4/3 (2|k|) 2/3 D X,j , k → -∞.
Proof. In order to prove (2.10), we note that

h(k) -h Ai (k) = b 2 x 2 ,
so that with the definition of Ψ Ai,u j (x, k) in (2.7) for u = o and (2.8) for u = e, and the normalization constant C X,j in (2.9), we have

[h(k) -h Ai (k)]Ψ Ai,u j (•, k) 2 = 2b 4 (2|k|b) 5/3 C 2 X,j ∞ 0 v 4 Ai(v + z X,j ) 2 dv = b 8/3 (2|k|) 4/3 D 2 X,j ,
where the constant D X,j , given by

D X,j := ∞ 0 v 4 Ai(v + z X,j ) 2 dv c X,j 1/2 , (2.11) is finite since Ai(v) ∼ e -v 3/2 as v → +∞.
2.3. Band functions asymptotics k → +∞. For k 0, the effective potential consists of two double wells that separate as k → +∞. Consequently ω + j (k) approaches ω - j (k) as k → +∞. The eigenvalues of the double well potential consists of pairs of eigenvalues whose differences are exponentially small as k → ∞. Thus, the effective Hamiltonian for k = +∞ is the harmonic oscillator Hamiltonian:

h HO (k) := - d 2 dx 2 + (bx -k) 2 .
We let e 0 (b) := 0 and e j (b) := (2j-1)b, for every j ∈ N * , denote the energy levels of the harmonic oscillator. Let Ψ HO j (k) denote the j th normalized eigenfunction of the harmonic oscillator so that h HO (k)Ψ HO j (k) = e j (b)Ψ HO j (k). It can be explicitly expressed as

Ψ HO j (x, k) := 1 (2 j j!) 1/2 b π 1/4 e -b/2(x-k/b) 2 H j (b 1/2 (x -k/b)), (2.12)
where H j is the j th Hermite polynomial.

Proposition 2.3. For each j ∈ N, there exists a constant 0 < C j < ∞, depending only on j, so that for k 0, we have,

(h(k) -e j (b))Ψ HO j (±x; k) C j be -k 2 /(4b) . (2.13)
This immediately implies the eigenvalue estimate

0 < ∓(ω ± j (k) -e j (b)) C j be -k 2 4b , k κ j , (2.14)
and the difference of the two eigenvalues is bounded as

0 ω - j (k) -ω + j (k) 2C j be -k 2 4b , k κ j . (2.15)
Proof. 1. Since Ψ HO j (k) is the eigenfunction of the harmonic oscillator Hamiltonian, we have for all x ∈ R,

(h(k) -e j (b))Ψ HO j (±x, k) = ((bx ± k) 2 -(bx ∓ k) 2 )χ R ∓ (x)Ψ HO j (±x, k), so that for any k 0, we have (h(k) -e j (b))Ψ HO j (±x, k) (bx ∓ k) 2 Ψ HO j (±x, k) L 2 (R ∓ ) . (2.16)
Here χ I stands for the characteristic function of I ⊂ R. From (2.16), the identity

(bx ∓ k) 2 Ψ HO j (±x, k) L 2 (R ∓ ) = (bx -k) 2 Ψ HO j (x, k) L 2 (R -)
, and (2.12), it follows that

(h(k) -e j (b))Ψ HO j (±x, k) c j be -k 2 /(4b) , k 0, (2.17)
for some constant c j > 0 depending only on j.

Let (Ψ HO

j ) ± (x, k) := (Ψ HO j (x, k) ± Ψ HO j (-x, k))/2 ∈ H ± . In light of (2.17) we have (h(k) -e j (b))(Ψ HO j ) ± (k) H c j be -k 2 /(4b) , k 0. (2.18) Further since (Ψ HO j ) ± (k) 2 = 1 ± R Ψ HO j (x, k)Ψ HO j (-x, k)dx /2, with R Ψ HO j (x, k)Ψ HO j (-x, k)dx cj e -k 2 /(4b) ,
for some constant cj > 0 depending only on j, we deduce from (2.18) that dist(σ(h

± (k)), e j (b)) C j be -k 2 /(4b) , k 0, (2.19)
where C j > 0 depends only on j.

3. As ω - j (k) > e j (b) for each k ∈ R, from the minimax principle, the result (2.14) for ω - j (k) follows readily from (2.19). The case of ω + is more complicated.

In the section 2.4, we prove in the derivation of Proposition 2.4 that the band function ω + j (k) has a unique absolute minimum at a value κ j ∈ (0, e 2j-1 (b) 1/2 ). Furthermore, ω + j (κ j ) ∈ (e j-1 (b), e j (b)). We also prove that (ω + j ) (k) < 0 for k < κ j and (ω + j ) (k) > 0 for k > κ j . The facts that the analytic band function is monotone increasing for k > κ j and converges to e j (b) as k → ∞ due to (2.19) imply the result (2.14) for ω + j (k).

2.4. Even band functions ω + j (k): the effective mass. We prove that the even states in H + , with band functions ω + j (k) = ω 2j-1 (k), have a unique positive minimum at κ j . We prove that the even band function ω + j (k) is concave at κ j . This convexity means that there is a positive effective mass. This positive effective mass plays an important role in the perturbation theory and creation of the discrete spectrum discussed in section 5.

Proposition 2.4. The band functions ω + j (k) = ω 2j-1 (k), corresponding to the even states of h(k), each have a unique extremum E j ∈ (e j-1 (b), e j (b)) that is a strict minimum. The minimum is attained at a single point κ j ∈ (0, e 2j-1 (b) 1/2 ). This point is the unique real solution of ω 2j-1 (k) -k 2 = 0, and E j = κ 2 j . The concavity of the band function at κ j is strictly positive and given by:

(ω + j ) (κ j ) = ω 2j-1 (κ j ) = 4κ j b ψ 2j-1 (0, κ j ) 2 > 0. (2.20)
We also have ±(ω + j ) (k) < 0 for ±(k -κ j ) < 0. Proof. 1. We first prove that there exists a unique minimum for the band function. The Feynman-Hellmann formula yields

(ω + j ) (k) = -2 R (b|x| -k)ψ + j (x, k) 2 dx, k ∈ R. (2.21)
Next, recalling (2.3), we get that

(ω + j ) (k) = 2 b f + j (k)ψ + j (0, k) 2 , f + j (k) := k 2 -ω + j (k), (2.22)
since ψ + j (0, k) = 0 and (ψ + j ) (0, k) = 0. Moreover, taking into account that h(0) = h HO (0) we see that

ω + j (k) ω + j (0) = e 2j-1 (b), k ∈ R + , (2.23) as h(k) h HO (k) in this case. Therefore we have f + j (0) = -e 2j-1 (b) < 0 and f + j (k) > 0 for all k > e 2j-1 (b) 1/2 from (2.23). The function f + j is continuous in R hence there exists κ j ∈ (0, e 2j-1 (b) 1/2 ) such that f + j (κ j ) = 0. Moreover, f + j
being real analytic, the set {t ∈ R, f + j (t) = 0} is at most discrete so we may assume without loss of generality that κ j is its smallest element. 2. We next prove that ω + j (k) is decreasing for k < κ j and increasing for k > κ j . It follows from (2.21) that (ω + j ) (k) < 2k. Integrating this inequality over the interval [κ j , k], we obtain

ω + j (k) < ω + j (κ j ) + k κ j 2tdt = ω + j (κ j ) + (k 2 -κ 2 j ), k > κ j ,
and hence f + j (k) > f + j (κ j ) for all k > κ j . This result with the fact that f + j (κ j ) = 0 and (2.22) imply that (ω + j ) (k) > 0 for k > κ j . 3. To study the concavity of the band function and establish (2.20), we differentiate (2.22) with respect to k and obtain

(ω + j ) (k) = - 2 b [(ω + j ) (k) -2k]ψ + j (0, k) 2 -2f + j (k)ψ + j (0, k)∂ k ψ + j (0, k) , k ∈ R.
(2.24) We evaluate (2.24) at κ j , recalling that f + j (κ j ) = 0 and that (ω + j ) (κ j ) = 0, in order to obtain (2.20). 4. We turn now to proving that E j (b) ∈ (e j-1 (b), e j (b)). Since (ω + j ) (k) > 0 for all k κ j from Step 2 it follows readily from (2.14) that ω + j (κ j ) < e j (b). Further it is clear that ω + 1 (κ 1 ) > 0 and we have in addition

ω + j (k) = ω 2j-1 (k) > ω 2(j-1) (k) = ω - j-1 (k) > e j-1 (b), k ∈ R, j 2,
so the result follows.

In light of Proposition 2.4, we say that there exists an effective mass at k = κ 1 , borrowing this term from the solid state physics. Further, it follows readily from Propositions 2.1 and 2.4 that the spectrum of H 0 is absolutely continuous and equal to a half-line:

σ(H 0 ) = σ ac (H 0 ) = [E 1 , +∞).
(ω - j ) (k) < 0, k ∈ R.
Proof. Let us first recall from (2.1) of Proposition 2.1 that for all k ∈ R we have the formula

(ω ± j ) (k) = - 2 b (ω ± j (k) -k 2 )ψ ± j (0, k) 2 + (ψ ± j ) (0, k) 2 . (2.25)
Bearing in mind that ψ - k (0, k) = 0 and (ψ - k ) (0, k) = 0, the result follows immediately from (2.25).

Mourre estimates, perturbations, and stability of the absolutely continuous spectrum

In this section we study the spectrum of the operator H 0 and its perturbations using a Mourre estimate. For the unperturbed operator H 0 , we prove a Mourre estimate using the fiber operator h(k). This implies a lower bound on the velocity operator for certain states proving the existence of edge currents. We prove that this estimate is stable with respect to a class of perturbations. Then there exists a constant

δ 0 = δ 0 (E) ∈ (0, d n (E)), independent of b, satisfy- ing ω -1 j (∆ E (2δ 0 )) = ∅, j 2n + 1, (3.1) and ω -1 i (∆ E (2δ 0 )) ∩ ω -1 j (∆ E (2δ 0 )) = ∅, 1 i = j 2n. (3.
2) Moreover, for every j ∈ N * 2n there is a constant c n,j = c n,j (E) > 0, independent of b, such that we have

-ω j (k) c n,j b 1/2 , k ∈ ω -1 j (∆ E (2δ 0 )). (3.3)
Proof. 1. First (3.1) follows readily from Proposition 2.4 and the fact that

∆ E (2δ 0 ) ∩ [E n+1 (b), +∞) = ∅ for all δ 0 ∈ (0, d n (E)) since E + δ 0 b < E n+1 (b).
2. Next we notice that h(k) is unitarily equivalent to the operator b ȟ(k/b 1/2 ), where ȟ(q) := -

d 2 dt 2 + (|t| -q) 2 , q ∈ R, is defined on the dense domain C ∞ 0 (R) ⊂ L 2 (R). More precisely it holds true that V b h(k)V * b = b ȟ(k/b 1/2 ), where (V b ψ)(x) := b -1/4 ψ(x/b 1/2 ), ψ ∈ L 2 (R),
is easily seen to be a unitary transform in L 2 (R). As a consequence we have

ω j (k) = bω j (k/b 1/2 ), k ∈ R, j ∈ N * , (3.4)
where {ω j } ∞ j=1 is the set of eigenvalues (arranged in increasing order) of ȟ(k). Let a n,j , j ∈ N * 2n-1 , be the unique real number obeying ωj (a n,j ) = e n (1), set a n,2n := +∞, and denote by ω-1 j the function inverse to ωj : (-∞, a n,j ) → (e n (1), +∞). As the interval [(E/b) -δ 0 , (E/b) + δ 0 ] ⊂ (e n (1), E n+1 (1)) it is in the domain of each function ω-1 j , j ∈ N * 2n , and we have ω-

1 j ([(E/b) -δ 0 , (E/b) + δ 0 ]) = [ω -1 j ((E/b) + δ 0 ), ω-1 j ((E/b) -δ 0 )], (3.5
) by Propositions 2.4 and 2.5. Further, since ω-1

j+1 (E) > ω-1 j (E) for all j ∈ N * 2n-1 , the functions ω-1 j are continuous, and 2n -1 is finite, then there is necessarily

δ 0 ∈ (0, d n (E)) such that we have ω-1 j+1 ((E/b) + δ 0 ) > ω-1 j ((E/b) -δ 0 ), j ∈ N * 2n-1 .
This and (3.4)-(3.5) yields (3.2). 3. Finally, taking into account that ȟ(q) coincides with h(q) in the particular case where b = 1, we deduce from Propositions 2.4 and 2.5 for any ∆ ⊂ (e n (1), E n+1 (1)) that inf

q∈ω -1 j (∆) (-ω j (q)) = čj (∆) > 0, j ∈ N * 2n , (3.6)
where the constant čj (∆) is independent of b. Now (3.3) follows readily from

(3.6) since [(E/b) -δ 0 , (E/b) + δ 0 )] ⊂ (e(1), E n+1 (1)) and inf k∈ω -1 j (∆ E (2δ 0 )) (-ω j (k)) = b 1/2 inf q∈ω -1 j ([(E/b)-δ 0 ,(E/b)+δ 0 )])
(-ω j (q)), according to (3.4).

Let us now introduce the operator A = A * := -y defined originally on C ∞ 0 (R 2 ). The operator A extends to a self-adjoint operator in

L 2 (R 2 ). Note that C ∞ 0 (R 2 ) is dense in Dom(H 0 ) and hence that Dom(A) ∩ Dom(H 0 ) is dense in Dom(H 0 ). Proposition 3.1. Let b > 0, n ∈ N * , E ∈ (e n (b), E n+1 (b)) and assume that δ 0 ∈ (0, d n (E)) is chosen to satisfy (3.1)-(3.2) according to Lemma 3.1. Let χ ∈ C ∞ 0 (R) with supp χ ⊂ ∆ E (2δ 0 ).
Then there exists a constant c n = c n (E) > 0, independent of b, such that we have

χ(H 0 )[H 0 , iA]χ(H 0 ) c n b 1/2 χ(H 0 ) 2 , (3.7)
as a quadratic form on Dom(A) ∩ Dom(H 0 ).

Proof. We get [H 0 , iA] = -2(p y -b|x|), (3.8)
on Dom(A) ∩ Dom(H 0 ). We recall the orthogonal projection P j (k) defined by P j (k) := ., ψ j (k) ψ(k), for all j ∈ N * . The commutator on the left in (3.8) fibers over k ∈ R, so by a direct calculation, we find that

χ(H 0 )[H 0 , iA]χ(H 0 ) = -2F *   j,m∈N * ⊕ R χ(ω j (k))χ(ω m (k))P j (k)(k -b|x|)P m (k)dk   F.
Taking into account that supp χ ⊂ ∆ E (2δ 0 ), we deduce from (3.1)-(3.2) that

χ(H 0 )[H 0 , iA]χ(H 0 ) = -2F *   2n j=1 ⊕ R χ(ω j (k)) 2 ψ j (k), (k -b|x|)ψ j (k) P j (k)dk   F, whence χ(H 0 )[H 0 , iA]χ(H 0 ) = F *   2n j=1 ⊕ R χ(ω j (k)) 2 (-ω j (k))P j (k)dk   F, (3.9)
from the Feynman-Hellmann formula. In light of (3.3), we have

-ω j (k)χ(ω j (k)) 2 c n,j b 1/2 χ(ω j (k)) 2 , j ∈ N * 2n , so (3.9) yields χ(H 0 )[H 0 , iA]χ(H 0 ) c n b 1/2 F *   2n j=1 ⊕ R χ(ω j (k)) 2 P j (k)dk   F = c n b 1/2 χ(H 0 ) 2 ,
where c n := min j∈N * 2n c n,j > 0.

Let P 0 (I) denote the spectral projection of H 0 for the Borel set I ⊂ R. Then by choosing χ in Proposition 3.1 to be equal to one on ∆ E (δ 0 ) and multiplying (3.7) from both sides by P 0 (∆ E (δ 0 )), we obtain the following Mourre estimate for H 0 : where c n is the constant defined in Proposition 3.1.

P 0 (∆ E (δ 0 ))[H 0 , iA]P 0 (∆ E (δ 0 )) c n b 1/2 P 0 (∆ E (δ 0 )). ( 3
The proof of this corollary follows directly from (3.10) since for ϕ as in the corollary, we have J y (ϕ) = ϕ, (1/2)P 0 (∆ E (δ 0 ))[H 0 , iA]P 0 (∆ E (δ 0 ))ϕ . The edge currents associated with H 0 and states ϕ as in Corollary 3.1 are also localized in a neighborhood of size roughly b -1/2 about x = 0. This follows from Proposition 4.1.

3.3.

Stability of the Mourre estimate. One of the main benefits of a local commutator estimate like (3.10) is its stability under perturbation. Namely we consider the perturbation of H 0 = (-i∇ -A 0 ), A 0 = A 0 (x, y) := (0, b|x|), by a magnetic potential a(x, y) = (a 1 (x, y), a 2 (x, y)) ∈ W 1,∞ (R 2 ) and a bounded scalar potential q(x, y) ∈ L ∞ (R 2 ). We prove that a Mourre inequality for the perturbed operator H = H(a, q) := (-i∇ -A 0 -a) 2 + q = (p x -a 1 ) 2 + (p y -b|x| -a 2 ) 2 + q, (3.12) remains true provided a W 1,∞ (R 2 ) and q ∞ are small enough relative to b. We preliminarily notice that

W = W (a) := H(a, 0) -H 0 = 2a • (-i∇ -A 0 ) -i(∇ • a) + a • a, (3.13)
with (-i∇

-A 0 )ϕ = H 0 ϕ, ϕ 1/2 λ H 0 ϕ + λ -1 ϕ 2 for all ϕ ∈ C ∞ 0 (R 2 ) and λ > 0, so we have W ϕ 2λ a ∞ H 0 ϕ + (λ -1 + ∇a ∞ + a 2 ∞ ) ϕ , λ > 0. Taking λ = 1/(4 a ∞ )
in the above inequality we find that W is H 0 -bounded with relative bound smaller than one. In light of [START_REF] Reed | Methods of Modern Mathematical Physics. IV. Analysis of Operators[END_REF][Theorem X.12] the operator H(a, 0) = H 0 + W is thus selfadjoint in L 2 (R 2 ) with same domain as H 0 , and the same is true for H = H(a, q) = H(a, 0) + q since q ∈ L ∞ (R 2 ). Proposition 3.2. Let b, n, E and δ 0 be as in Proposition 3.1. Assume that

δ = δ(E) ∈ (0, δ 0 ), a ∈ W 1,∞ (R 2 ) and q ∈ L ∞ (R 2 ) verify F n,E δ, q ∞ b , a 2 ∞ + ∇a ∞ b < 1 2 , (3.14)
1 where

F n,E (δ, a, q) := f n (δ, a, q) δ 0 2 + 2 c n a 1/2 + (2n + 1 + f n (δ, a, q)) 1/2 f n (δ, a, q) δ 0 1/2 , (3.15
) f n is given by (3.25) and c n is the constant defined in Proposition 3.1. Then we have the following Mourre estimate

P(∆ E (δ))[H, iA]P(∆ E (δ)) c n 2 b 1/2 P(∆ E (δ)), (3.16)
where P(I) denotes the spectral projection of H for the Borel set I ⊂ R.

Proof. By combining the following decomposition of ψ ∈ P(∆ E (δ))L 2 (R 2 ) into the sum

ψ = φ + ξ, φ := P 0 (∆ E (δ 0 ))ψ, ξ := P 0 (R \ ∆ E (δ 0 ))ψ, (3.17)
with the basic equality

[H, iA] = [H 0 , iA] + 2a 2 , (3.18)
obtained through standard computations, we get that

ψ, [H, iA]ψ = φ, [H 0 , iA]φ + 2 ψ, a 2 ψ + C(φ, ξ), with C(φ, ξ) := R ξ(•, k), (k -b|x|) ξ(•, k) L 2 (R) dk +2Re R φ(•, k), (k -b|x|) ξ(•, k) L 2 (R) dk .
This entails

ψ, [H, iA]ψ φ, [H 0 , iA]φ -2 ( a ∞ ψ + (p y -b|x|)ξ ) ψ , (3.19)
since

ψ 2 = φ 2 + ξ 2 ,
as can be seen from the orthogonality of φ and ξ in L 2 (R 2 ), arising from (3.17). The first term in the r.h.s of (3.19) is lower bounded by (3.10) as

φ, [H 0 , iA]φ c n b 1/2 φ 2 , (3.20)
and (p y -b|x|)ξ can be majorized with the help of the estimate

(p y -b|x|)ξ 2 ξ, H 0 ξ = ψ, H 0 ξ = H 0 ψ, ξ = (H -W -q)ψ, ξ , giving (p y -b|x|)ξ 2 ξ, H 0 ξ ((E/b) + δ + q + w) b ξ ψ , (3.21)
where q := q ∞ /b and w := W ψ /(b ψ ). Further we have

ξ δ + q + w δ 0 ψ , (3.22)
since ξ 2 = (H -E -W -q)ψ, (H 0 -E) -1 ξ . In light of the r.h.s in (3.21)-(3.22) we are thus left with the task of majorizing w. This can be done by combining the estimate (3.13). We find out that w a 1/2 a 1/2 + 2 ((E/b) + δ + q + w) 1/2 with a := ( a 2 ∞ + ∇a ∞ )/b, whence w 2a 1/2 3a 1/2 + (E/b) + δ + q 1/2 ). From this, the estimate E (2n + 1)b, arising from Proposition 2.4, and (3.21)-(3.22) then follows that

(-i∇ -A 0 )ψ = H 0 ψ, ψ 1/2 H 0 ψ 1/2 ψ 1/2 (H -W -q)ψ 1/2 ψ 1/2 , entailing (-i∇ -A 0 )ψ ((E/b) + δ + q + w) 1/2 b 1/2 ψ , with
ξ f n (δ, a, q) δ 0 ψ , (3.23)
and

(p y -b|x|)ξ (2n + 1 + f n (δ, a, q)) 1/2 f n (δ, a, q) δ 0 1/2 b 1/2 ψ , (3.24)
where 3.4. Absolutely continuous spectrum. We now apply Proposition 3.2 to prove the existence of absolutely continuous spectrum for perturbed magnetic barrier operators. Using direct computation, we deduce from (3.8) and (3.18) that [[H, iA], iA] = -2. Hence the double commutator of H with A is bounded from Dom(H) = Dom(H 0 ) to L 2 (R 2 ). Moreover, since [H, iA] extends to a bounded operator from Dom(H 0 ) to L 2 (R 2 ), the Mourre estimate (3.16) combined with [START_REF] Cycon | Schrödinger Operators with Application to Quantum Mechanics and Global Geometry[END_REF][Corollary 4.10] entails the following: Corollary 3.2. Let b, n, E and δ 0 be the same as in Proposition 3.1. Assume that δ ∈ (0, δ 0 ), q ∈ L ∞ (R 2 ) and a ∈ W 1,∞ (R 2 ) satisfy (3.14). Then the spectrum of H = H(a, q) in ∆ E (δ) is absolutely continuous.

f n (δ, a, q) := δ + q + 2a 1/2 3a 1/2 + (2n + 1 + δ + q) 1/2 . ( 3
Armed with Corollary 3.2 we turn now to proving the main result of this section.

Theorem 3.1. Let b > 0, n ∈ N * , and let ∆ be a compact subinterval of (e n (b), E n+1 (b)). Then there are two constants a * = a * (n, ∆) > 0 and q * = q * (n, ∆) > 0, both independent of b, such that for all (a, q

) ∈ W 1,∞ (R 2 )×L 2 (R 2 ) verifying a 2 ∞ + ∇a ∞ a * b and q ∞ q * b, the spectrum of H = H(a, q) in ∆ is absolutely continuous. Proof. For every E ∈ ∆ choose δ(E) ∈ (0, δ 0 (E)), a(E) > 0 and q(E) > 0 such that F n,E (δ(E), a(E), q(E)) < 1 2 , ( 3
∆ E j (δ(E j )). (3.27)
Set a * := min 1 j N a(E j ) > 0 and q * := min 1 j N q(E j ) > 0. Since F n,E j (δ(E j ), •, •), j = 1, . . . , N , is an increasing function of each of the two last variables taken separately, when the remaining one is fixed, we necessarily have

F n,E j (δ(E j ), a * , q * ) < 1/2 by (3.26). Assume that a 2 ∞ + ∇a ∞ ∈ [0, a * b) and q ∞ ∈ [0, q * b). For every j = 1, . . . , N , the spectrum of H in ∆ E j (δ(E j ))
is thus absolutely continuous by Corollary 3.2 so the result follows from this and (3.27).

Edge currents: existence, stability, localization, and asymptotic velocity

A major consequence of the Mourre estimate in Proposition 3.1 for the unperturbed operator H 0 is the lower bound on the edge current carried by certain states given in Corollary 3.1. Because of the stability result for the Mourre estimate for the perturbed operator H(a, q) in Proposition 3.2, we prove in this section that edge currents are stable under perturbations. We then prove that these currents are well-localized in a strip of width O(b -1/2 ) about x = 0. Finally, we prove that the asymptotic velocity is bounded from below demonstrating that the edge currents persist for all time.

4.1. Existence and stability of edge currents. For the perturbed operator H = H(a, q), the y-component of the velocity operator is v y,a,q := (1/2)[H, iA] = -(p y -b|x|) + a 2 , A = -y, (4.1)

according to (3.8) and (3.18). A state ϕ ∈ L 2 (R 2 ) carries an edge current if J y,a,q (ϕ) := ϕ, v y,a,q ϕ c ϕ 2 , (

for some constant c > 0. For notational simplicity we write v y (resp. J y ) instead of v y,0,0 (resp. J y,0,0 ) in the particular case of the unperturbed operator H 0 corresponding to a = 0 and q = 0. We consider states in the range of the spectral projector P 0 (•) for H 0 , and in the range of the spectral projector P a,q (•) for H = H(a, q), and energy intervals as in (3.10) for H 0 , and in (3.16) for H a,q . We then deduce from (4.1)-(4.2) the existence of edge currents for the operator H 0 and H a,q , respectively. We recall Corollary 3.1 in the first part of the following theorem. (1) Let ϕ ∈ L 2 (R 2 ) satisfy ϕ = P 0 (∆ E (δ 0 ))ϕ. Then ϕ carries an edge current obeying

J y (ϕ) c n 2 b 1/2 ϕ 2 ,
where c n is the constant defined in Proposition 3.1.

(2) Let δ ∈ (0, δ 0 ) and assume that (a, q) ∈ W 1,∞ (R 2 ) × L ∞ (R 2 ) verifies the condition (3.14), where (c n /2) is substituted for c n in the definition (3.15). Then every state ϕ ∈ P a,q (∆ E (δ))L 2 (R 2 ) carries an edge current and we have the lower bound J y,a,q (ϕ)

c n 4 b 1/2 ϕ 2 . (4.3) 4.2.
Localization of edge currents. We establish the localization of the edge currents described in Theorem 4.1 using a method introduced by Iwatsuka [14, section 3]. We refer the reader to section 3.1 for the definitions of the various quantities appearing in the following proposition.

Proposition 4.1. Let n, E, and ∆ 0 be as in Proposition 3.1 and choose δ = δ(E) ∈ (0, δ 0 ) in accordance with condition (3.14).

Let ϕ ∈ L 2 (R 2 ) satisfy ϕ = P 0 (∆ E (δ))ϕ with ϕ = 1.
Then for all ε > 0 there exists a constant b > 0, depending only on n, δ and ε such that we have

R 2 χ Iε (x)|ϕ(x, y)| 2 dxdy 1 - √ 2e -b ε , for b b.
Here χ Iε is the characteristic function of the interval

I ε := [-b -1/2+ε , b -1/2+ε ].
Proof. 1. Due to (2.14) we have max{sup ω -1 j (∆ E (δ)), j = 1, . . . , 2n} α n b 1/2 , for some constant α n > 0, depending only on n and δ. Hence there is a constant β n > 0, depending only on n and δ, such that the estimate

Q j (x, k) := V ef f (x, k) -ω j (k) b 2 (|x| -x n ) 2 > 0, (4.4)
holds for all j = 1, . . . , 2n, k ∈ ω -1 j (∆ E (δ)) and |x| x n := β n b 1/2 . 2. We will prove that an eigenfunction ψ j (k), for k ∈ ω -1 j (∆ E (δ)), decays in the region |x| x n . In particular, we will establish for j = 1, . . . , 2n that

|ψ j (x, k)| 2b π 1/4 e -b(|x|-xn) 2 /2 , |x| x n , k ∈ ω -1 j (∆ E (δ)). (4.5)
Let j ∈ N * 2n and k ∈ ω -1 j (∆ E (δ)) be fixed. In light of (4.4) and the differential equation

ψ j (x, k) = Q j (x, k)ψ j (x, k) we have ψ j (x, k)ψ j (x, k) < 0 for |x| > x n , by [14][Proposition 3.1]. This implies that ψ j (x, k) ψ j (x, k) = ψ j (x, k)ψ j (x, k) ψ j (x, k) 2 < 0, x > x n . (4.6)
Following [START_REF] Iwatsuka | Examples of absolutely continuous Schrödinger operators in magnetic fields[END_REF][Lemma 3.5], differentiating I(x, k) 2 , one finds that ∂ x I(x, k) < 0 since Q j (x, k) > 0 in the region x > x n . Since I(x, k) vanishes at infinity, due to the vanishing of ψ j (x, k) and ψ j (x, k) established by [START_REF] Iwatsuka | Examples of absolutely continuous Schrödinger operators in magnetic fields[END_REF][ Lemma 3.3], this means that I(x, k) > 0 in the region x > x n . From this we conclude that

:= ψ j (x, k) 2 - Q j (x, k)ψ j (x, k)
ψ j (x, k) 2 Q j (x, k)ψ j (x, k) 2 , x > x n . (4.7)
As a consequence of (4.4) and (4.6)-(4.7), we find that

ψ j (x, k) ψ j (x, k) -Q j (x, k) -b(x -x n ), for x > x n .
Result (4.5) follows from integrating this differential inequality over the region x > x n and arguing in the same way as above for x < -x n .

3. Choose b so large that b ε > (1+β

1/2 n ) 2 .
Then we have b -1/2+ε > x n +b (-1+ε)/2 by elementary computations, whence R\Iε

ψ j (x, k) 2 dx 2 2b π 1/2 +∞ b -1/2+ε e -b(x-xn) 2 dx √ 2e -b ε , (4.8)
from (4.5). Finally, since

R 2 χ Iε (x)|ϕ(x, y)| 2 dxdy = R 2 χ Iε (x)| φ(x, k)| 2 dxdk = 2n j=1 ω -1 j (∆ E (δ)) |β j (k)| 2 Iε ψ j (x, k) 2 dx dk,
by Lemma 3.1, where β j (k) := φ(•, k), ψ j (•, k) , the result follows readily from (4.8) and the identity 2n

j=1 ω -1 j (∆ E (δ)) |β j (k)| 2 dk = 1. 4.
3. Persistence of edge currents in time: Asymptotic velocity. We investigate the time evolution of the edge current under the unitary evolution groups generated by the Iwatsuka Hamiltonians H 0 (1.1), and by the perturbed Iwatsuka Hamiltonians H(a, q) (3.12). The general situation we address is the following. Let H be a self-adjoint Schrödinger operator on L 2 (R 2 ). This operator generates the unitary time evolution group U (t) := e -itH . Let v y := (i/2)[H, A], with A = -y, be the y-component of the velocity operator. We are interested in evaluating the asymptotic time behavior of U (t)ϕ, v y U (t)ϕ as t → ±∞ for appropriate functions ϕ.

The lower bounds on the edge currents for the unperturbed and the perturbed Iwatsuka models are valid for all times. It we replace v y in the expression J y (ϕ) = ϕ, v y ϕ in Corollary 3.1 by v y (t) := e itH 0 v y e -itH 0 , then the lower bound (3.11) remains valid since the state ϕ(t) := e -itH 0 ϕ satisfies ϕ(t) ∈ P 0 (∆ E (δ 0 ))L 2 (R 2 ) for all time. Similarly, if we replace v y,a,q in (4.2) by its time evolved current v y,a,q (t) using the operator e -itH(a,q) , then the lower bound in (4.3) remains valid for all time.

Perturbed Hamiltonians H(a, q) were treated in sections 3 and 4. Part 2 of Theorem 4.1 states that if the L ∞ -norms of a 2 j , ∇a j , for j = 1, 2, and of q are small relative to b in the sense that condition (3.14) is satisfied, then the edge current J y,a,q (ψ) is bounded from below for all ψ ∈ P a,q (∆ E (δ))L 2 (R 2 ), where ∆ E (δ) is defined at the beginning of section 3 and (E, δ) are as in Proposition 3.1 and Proposition 3.2. This relative boundedness of a j , ∇a j , and of q is rather restrictive. From the form of the current operator in (4.1), it would appear that only a 2 ∞ needs to be controlled. We prove here that if we limit the support of the perturbation (a 1 , a 2 , q) to a strip of arbitrary width R in the y-direction, and require only that a 2 ∞ be small relative to b 1/2 , then the asymptotic velocity associated with energy intervals ∆ E (δ) and the perturbed Hamiltonian H(a, q) exists and satisfies the same lower bound as in (4.3). Furthermore, the spectrum in ∆ E (δ) is absolutely continuous. This means that the edge current is stable with respect to a different class of perturbations than in Theorem 4.1.

We recall that the asymptotic velocity associated with a pair of self-adjoint operators (H 0 , H 1 ) is defined in terms of the local wave operators for the pair, see, for example [9, section 4.5-4.6]. The local wave operators Ω ± (∆) for an energy interval ∆ ⊂ R are defined as the strong limits:

Ω ± (∆) := s -lim t→±∞ e itH 1 e -itH 0 P 0,ac (∆), (4.9)
where P 0,ac (∆) is the spectral projector for the absolutely continuous subspace of H 0 associated with the interval ∆. For any ϕ, we define the asymptotic velocity

V ± y (∆) of the state ϕ by ϕ, V ± y (∆)ϕ := ϕ, Ω ± (∆)v y Ω ± (∆) * ϕ . (4.10)
In the case that H 0 commutes with v y , it is easily seen from the definition (4.9) that ϕ, V ± y (∆)ϕ = lim t→±∞ ϕ, e itH 1 P 0,ac (∆)v y P 0,ac (∆)e -itH 1 ϕ .

Our main result is the existence of the asymptotic velocity (4.10) in the ydirection for the perturbed operators H(a, q) described in section 3.3. We prove that the asymptotic velocity satisfies the lower bound given in (4.11) provided the perturbations (a, q) have compact support in the y-direction. The local wave operators appearing in the definition (4.10) are constructed from the pair (H 0 , H 1 ) where H 0 is the unperturbed Iwatsuka Hamiltonian and H 1 = H(a, q). As discussed in section 2.1, the spectrum of H 0 is purely absolutely continuous. Theorem 4.2. Let b, n, E, and δ 0 , be as in Proposition 3.1 and for any 0 < δ δ 0 , let ∆ E (δ) be as defined in section 3.1. Suppose that the perturbation a ∈ W 1,∞ (R 2 ) and q ∈ L ∞ (R 2 ) have their support in the set {(x, y) | |y| < R}, for some 0 < R < ∞. In addition, suppose that the perturbation a 2 satisfies a 2 ∞ (c n /4)b 1/2 . Then for any ϕ ∈ Ran P a,q (∆ E (δ)), we have

ϕ, V ± y (∆)ϕ c n 4 b 1/2 ϕ 2 , (4.11)
where the constant c n is defined in Proposition 3.1.

The proof of Theorem 4.2 closely follows the proof in [13, section 7] (see also [12, section 4]). We mention the main points. We first prove the existence of the local wave operators (4.9) for the pair H 0 and H 1 = H(a, q), and the interval ∆ E (δ), as in the theorem. The key point is that in the application of the method of stationary phase, we use the positivity bound (3.3). We then use the intertwining properties of the local wave operators to find

ϕ, V ± y (∆ E (δ))ϕ = ϕ, Ω ± (∆ E (δ))v y,a,q Ω ± (∆ E (δ)) * ϕ = Ω ± (∆ E (δ)) * P 1 (∆ E (δ))ϕ, v y,a,q Ω ± (∆ E (δ)) * P 1 (∆ E (δ))ϕ = P 0 (∆ E (δ))Ω ± (∆ E (δ)) * ϕ, v y,a,q P 0 (∆ E (δ))Ω ± (∆ E (δ)) * ϕ c n 4 b 1/2 P 0 (∆ E (δ))Ω ± (∆ E (δ)) * ϕ 2 ,
where we used the lower bound (3.11) of Corollary 3.1, the form of the current operator v y,a,q in (4.1), and the estimate on a 2 given in the theorem. To complete the proof, we again use the intertwining relation to write

P 0 (∆ E (δ))Ω ± (∆ E (δ)) * ϕ = Ω ± (∆ E (δ)) * P 1 (∆ E (δ))ϕ = ϕ ,
since the local wave operators are partial isometries.

5. Asymptotic behavior of the eigenvalue counting function for negative perturbations of H 0 below inf σ ess (H 0 )

In this section we apply the method introduced in [START_REF] Raikov | Eigenvalue asymptotics for the Schrödinger operator with perturbed periodic potential[END_REF] to describe the discrete spectrum of the perturbed operator H := H 0 -V near the infimum of its essential spectrum, when the scalar potential V = V (x, y) 0 decays suitably as |y| → ∞. For potentials of this type, we prove that there are an infinite number of eigenvalues accumulating at E 1 = inf σ ess (H 0 ) = inf σ ess (H) from below and we describe the behavior of the eigenvalue counting function. The only information on H 0 we use here is the local behavior of the first band function ω 1 (k) at its unique minimum k = κ 1 . Namely, we recall from Proposition 2.4 and the analyticity of k → ω 1 (k) that the asymptotic identity

ω 1 (k) -E 1 = β 1 (k -κ 1 ) 2 + O((k -κ 1 ) 3 ), k → κ 1 , holds with β 1 := ω 1 (κ 1 )/2 > 0.
5.1. Statement of the result. We first introduce the following notation. Let H be a linear self-adjoint operator acting in a given separable Hilbert space. Assume that E = inf σ ess (H) > -∞. The eigenvalue counting function N (µ; H), µ ∈ (-∞, E), denotes the number of the eigenvalues of H lying on the interval (-∞, µ), and counted with the multiplicities. We recall that ψ 1 (x, k) is the first eigenfunction of the fiber operator h(k) with band function ω 1 (k).

Theorem 5.1. Let V (x, y) ∈ L ∞ (R 2 ) satisfy the following two conditions:

i.) ∃(α, C) ∈ (0, 2) × R * + so that 0 V (x, y) C(1 + |x|) -α (1 + |y|) -α , x, y ∈ R; ii.) ∃L > 0 so that lim |y|→∞ |y| α R V (x, y)ψ 1 (x, κ 1 ) 2 dx = L. Then we have lim λ↓0 λ -1 2 + 1 α N (E 1 -λ; H 0 -V ) = 2 απ β -1/2 1 L 1/α B 3 2 , 1 α - 1 2 , (5.1)
where B(•, •) is the Euler beta function [2, section 6.2] and β 1 := ω 1 (κ 1 )/2 > 0 is the effective mass.

5.2. Some notation and auxiliary results. This subsection presents some notation and several auxiliary results needed for the proof of Theorem

Let G : L 2 (R) → L 2 (R 2 ) be a bounded operator with integral kernel g ∈ L ∞ (R 3 ). Then for every f ∈ L r (R 2 ) and h ∈ L r (R) with r ∈ [2, ∞), we have n(s; f Gh) C r (G)s -r f r L r (R 2 ) h r L r (R) , s > 0, where C r (G) := g 4/r L ∞ (R 3 ) G 2(r-2)/r .
For δ > 0 fixed, let χ = χ δ denote the characteristic function of the interval I = I δ := (κ 1 -δ, κ 1 + δ). As we shall actually apply Lemma 5.1 in §5.3 with G = Γ j , j = 0, 1, where Γ j : L 2 (R) → L 2 (R 2 ) is the integral operator with kernel

γ 0 (x, y; k) := 1 √ 2π ψ 1 (x, κ 1 )e -iyk χ(k), (x, y) ∈ R 2 , k ∈ R, (5.4)
and

γ 1 (x, y; k) := 1 √ 2π ψ 1 (x, k) -ψ 1 (x, κ 1 ) k -κ 1 e -iyk χ(k), (x, y) ∈ R 2 , k ∈ R\{κ 1 }.
(5.5) Lemma 5.2. We have γ j ∈ L ∞ (R 3 ) for j = 0, 1.

Proof. In view of (5.4)-(5.5), it suffices to prove that (

x, k) → ψ 1 (x, k) and (x, k) → (ψ 1 (x, k) -ψ 1 (x, κ 1 ))/(k -κ 1 ) are respectively bounded in R × I and R × (I \ {κ 1 }). The eigenfunction ψ 1 (•, k) is a solution to the second order ordinary differential equation -ϕ (x) + W (x; k)ϕ(x) = 0, x ∈ R, (5.6)
where

W (x; k) := (b|x| -k) 2 -ω 1 (k). The potential W (x; k) is greater than δ 2 provided |x| > x 1 := (b 1/2 + κ 1 + 2δ)/b, uniformly in k ∈ I. It follows from [12, Lemma B.3] that 0 < ψ 1 (x, k) ψ 1 (±x 1 , k)e -δ(|x|-x 1 ) , x x 1 , k ∈ I. Since (x, k) → ψ 1 (x, k) is continuous in R × I, this implies the result for j = 0. Next, bearing in mind that the L 2 (R)-valued function k → ψ 1 (•, k) is real analytic, we deduce from (5.6) that Φ(•, k) := ∂ k ψ 1 (•, k) is solution to the equation -ϕ (x) + W (x; k)ϕ(x) = -F (x; k), x ∈ R, where F (x; k) := ∂ k W (x; k)ψ 1 (x, k) = (2(k -b|x|) -ω 1 (k))ψ 1 (x, k). Therefore we get that Φ (•, k) 2 L 2 (R) + (b|x| -k)Φ(•, k) 2 L 2 (R) (C + ω 1 (k) Φ(•, k) L 2 (R) ) Φ(•, k) L 2 (R) , (5.7)
with

C := sup k∈I F (•; k) L 2 (R) < ∞, by standard computations. Since sup k∈I Φ(•, k) L 2 (R) < ∞, (5.7) thus entails that sup k∈I Φ(•, k) H 1 (R) < ∞.
From this and the estimate

Φ(x, k) 2 = 2 x -∞ Φ(x, k)Φ (x, k)dx Φ(•, k) 2 H 1 (R) , x ∈ R, k ∈ R,
then follows that sup (x,k)∈R×I |Φ(x, k)| < ∞. This yields the result for j = 1 and terminates the proof.

Finally, since the proof of Theorem 5.1 is obtained by expressing lim λ↓0 N (E 1 -λ; H 0 -V ) in terms of the asymptotics of the eigenvalue counting function for the discrete spectrum of a second-order ordinary differential operators on the real line, we recall from [5, Lemma 4.9] the following

Lemma 5.3. Assume that Q = Q ∈ L ∞ (R) satisfies the two following condi- tions: i.) ∃(α, C) ∈ (0, 2) × R * + so that |Q(x)| C(1 + |x|) -α , x ∈ R; ii.) ∃ > 0 so that lim |x|→∞ |x| α Q(x) = . For any m > 0, let H(m, Q) := -m 2 d 2 dx 2 -Q be the 1D Schrödinger operator with domain H 2 (R), self-adjoint in L 2 (R). Then we have lim λ↓0 λ 1 α -1 2 N (-λ; H(m, Q)) = 2 1 α παm B 3 2 , 1 α - 1 2 .
The proof of Lemma 5.3, which is similar to the one of [18, Theorem XIII.82], can be found in [START_REF] Levitan | Introduction to Spectral Theory. Selfadjoint Ordinary Differential Operators[END_REF]. 5.3. Proof of Theorem 5.1. The proof consists of four parts. 5.3.1. Part I: Projection on the bottom of the first band function. We define V := FV F * and recall that H 0 = FH 0 F * . The first part of the proof is to show that the asymptotics of N (E 1 -λ; H 0 -V ) as λ ↓ 0 is determined by the asymptotics of the eigenvalue counting function for a reduced operator obtained from the projection of the operator H 0 -V to the bottom of the first band function. First of all, we remark that the multiplier by V is H 0 -compact since V (x, y) goes to zero as |(x, y)| tends to infinity. As a consequence we have

inf σ ess (H 0 -V ) = inf σ ess (H 0 ), hence N (E 1 -λ; H 0 -V ) < ∞ for any λ > 0. Furthermore, since V ∈ L ∞ (R 2 ), the operator H 0 -V is lower semibounded. The operator H 0 -V is unitarily equivalent to H 0 -V, so we have N (E 1 -λ; H 0 -V ) = N (E 1 -λ; H 0 -V), λ > 0.
Let P : L 2 (R 2 ) → L 2 (R 2 ) be the orthogonal projection defined by

(P u)(x; k) := R u(t; k)ψ 1 (t, k)dt χ(k)ψ 1 (x, k), (x, k) ∈ R 2 , (5.8)
where we recall that χ denotes the characteristic function of the interval I := (κ 1 -δ, κ 1 + δ) for some fixed δ > 0.

Lemma 5.4. Let H 1 (t), t ∈ R, be the operator P (H 0 -(1 + t)V)P with domain P Dom(H 0 ). Then there is a constant N 0 0, independent of λ, such that we have

N (E 1 -λ; H 1 (0)) N (E 1 -λ; H 0 -V) N (E 1 -λ; H 1 (0)) + N 0 , λ > 0. (5.9)
Proof. Set Q := I -P . For all u ∈ L 2 (R 2 ) and ε > 0 it holds true that

| (P VQ + QVP )u, u | = 2 Re V 1/2 P u, V 1/2 Qu 2 V 1/2 P u V 1/2 Qu ε P VP u, u + ε -1 QVQu, u , which entails -εP VP -ε -1 QVQ P VQ + QVP εP VP + ε -1 QVQ,
in the sense of quadratic forms. From this and the elementary identity H 0 = P H 0 P + QH 0 Q then follows that

H 1 (ε) ⊕ H 2 (ε) H 0 -V H 1 (-ε) ⊕ H 2 (-ε), ε > 0, ( 5.10)
where H 2 (t), t ∈ R * , is the operator Q(H 0 -(1 + t -1 )V)Q with domain Q Dom(H 0 ), and the symbol ⊕ indicates an orthogonal sum. Therefore, for every λ > 0 and ε > 0 fixed, the left inequality in (5.10) implies

N (E 1 -λ; H 0 -V) N (E 1 -λ; H 1 (ε)) + N (E 1 -λ; H 2 (ε)), (5.11) while the right one yields N (E 1 -λ; H 0 -V) N (E 1 -λ; H 1 (-ε))+N (E 1 -λ; H 2 (-ε)) N (E 1 -λ; H 1 (-ε)).

(5.12) Further, the multiplier by V being H 0 -compact, QVQ is QH 0 -compact and σ ess (H 2 (ε)) = σ ess (QH 0 ), ε > 0.

(5.13)

On the other hand we have inf σ ess (QH 0 ) = min{ω 1 (κ 1 +δ), min k∈R ω 2 (k)} > E 1 hence N 0 := N (E 1 ; QH 0 ) < ∞, and (5.13) yields N (E 1 -λ; H 2 (ε)) = N (E 1 -λ; QH 0 ) N 0 , λ > 0, ε > 0.

(5.14) Putting (5.11) and (5.14) together, we get that N (E 1 -λ; H 0 -V) N (E 1 -λ; H 1 (ε)) + N 0 , λ > 0, ε > 0.

(5.15)

Letting ε ↓ 0 in (5.12) and (5.15), we obtain (5.9).

5.3.2.

Part II: Singular integral operator decomposition. This part involves relating the number of eigenvalues accumulating below the bottom of the essential spectrum of H 1 (0), to the local behavior of ω 1 (k) and ψ 1 (•, k) at κ 1 .

The main tool we use for this is the Birman-Schwinger principle, which, in this situation, implies N (E 1 -λ; H 1 (0)) = n(1; P (H 0 -E 1 + λ) -1/2 V(H 0 -E 1 + λ) -1/2 P ). (5.16) In view of (5.8) and (5.16), we set a(k; λ) := (ω 1 (k) -E 1 + λ) -1/2 , k ∈ R, λ > 0,

(5.17 For every λ > 0 the operator χa(λ)Γ * V Γa(λ)χ is self-adjoint and nonnegative in L 2 (R k ). Furthermore we get Due to (5.9) and (5.20), we are left with the task of computing the asymptotics of n(1; W Γa(λ)χ) as λ ↓ 0. In this subsection, we shall prove that Γ and a(λ)χ may be replaced by, respectively, Γ 0 and a(λ)χ, in the above expression. The operator Γ 0 : L 2 (R) → L 2 (R 2 ) is the operator with integral kernel γ 0 (x, y; k) given by (5.4). We obtain a(λ) from a(λ) in (5.17 hold for all λ > 0 and ε ∈ (0, 1).

P (H 0 -E 1 + λ) -1/2 V(H 0 -E 1 + λ) -
Proof. 1. We use the decomposition Γa(λ)χ = 1 j=0 Γ j a j (λ)χ, where Γ 1 : L 2 (R) → L 2 (R 2 ) is the operator with integral kernel γ 1 (x, y; k) defined in (5.5), and a j (k; λ) := (k -κ 1 ) j a(k; λ), j = 0, 1.

Since γ j ∈ L ∞ (R 3 ), j = 0, 1, by Lemma 5.2, the operators Γ j are bounded with Γ 0 = 1 and Γ 1 2

sup k∈I R ψ 1 (x, k) -ψ 1 (x, κ 1 ) k -κ 1 2 dx.
We notice from (5.3) that n(1 + ε; W Γ 0 a 0 (λ)χ) -n(ε; W Γ 1 a 1 (λ)χ) n(1; W Γa(λ)χ) n(1 -ε; W Γ 0 a 0 (λ)χ) + n(ε; W Γ 1 a 1 (λ)χ), λ > 0, ε ∈ (0, 1). (5.23)

2. We obtain an upper bound for n(ε; W Γ 1 a 1 (λ)χ) in (5.23) from Lemma 5.1 taking G = Γ 1 , f = W , and h = a 1 (λ)χ. We get that

n(ε; W Γ 1 a 1 (λ)χ) C r (Γ 1 )ε -r W r L r (R 2 ) a 1 (λ)χ r L r (R)
n r ε -r , λ > 0, ε ∈ (0, 1), with n r := C r (Γ 1 ) W r L r (R 2 ) a 1 (0)χ r L r (R) . From this and (5.23) then follows that n(1 + ε; W Γ 0 a 0 (λ)χ) -n r ε -r n(1; W Γa(λ)χ) n(1 -ε; W Γ 0 a 0 (λ)χ) + n r ε -r , (5.24) for λ > 0, ε ∈ (0, 1). 3. Next, recalling that χ is the characteristic function of the interval (κ 1δ, κ 1 + δ), for ε ∈ (0, 1) fixed, we choose δ > 0 so small that (1 + ε) -1 a(k; λ)χ(k) a 0 (k; λ)χ(k) (1 -ε) -1 a(k; λ)χ(k), k ∈ R, λ > 0, where a(λ) is defined in (5.21). It follows from this and the simple identity that n(s, tH) = n(t -1 s, H), for s, t > 0, that we have n(s(1 + ε); W Γ 0 a(λ)χ) n(s; W Γ 0 a 0 (λ)χ) n(s(1 -ε); W Γ 0 a(λ)χ), s > 0, (5.25) Moreover (5.3) and the minimax principle yield n(s(1 + ε); W Γ 0 a(λ)) -n(sε; W Γ 0 a(λ)(1 -χ)) n(s; W Γ 0 a(λ)χ) n(s; W Γ 0 a(λ)), (5.26) for s > 0, ε ∈ (0, 1), as we have a(λ)Γ * 0 V Γ 0 a(λ) χa(λ)Γ * 0 V Γ 0 a(λ)χ in the sense of quadratic forms. Combining the second inequality of (5.25) with s = 1 -ε with the second inequality of (5.26) with s = (1 -ε) 2 , we obtain n(1 -ε; W Γ 0 a 0 (λ)χ) n((1 -ε) 2 ; W Γ 0 a(λ)χ) n((1 -ε) 2 ; W Γ 0 a(λ)), λ > 0.

(5.27)

Similarly, combining the first inequality of (5.26) with s = (1 + ε) 2 with the first inequality of (5.25) for s = 1 + ε, we find that n((1 + ε) 3 ; W Γ 0 a(λ)) -n(ε(1 + ε) 2 ; W Γ 0 a(λ)(1 -χ))

n((1 + ε) 2 ; W Γ 0 a(λ)χ) n(1 + ε; W Γ 0 a 0 (λ)χ), λ > 0.

(5.28)

1. 4 .

 4 Relation to edge conductance. Dombrowski, Germinet, and Raikov[START_REF] Dombrowski | Quantization of the edge conductance for magnetic perturbation of Iwatsuka Hamiltonians[END_REF] studied the quantization of the Hall edge conductance for a generalized family of Iwatsuka models including the model discussed here. Let us recall that the Hall edge conductance is defined as follows. We consider the situation where the edge lies along the y-axis as discussed above. Let I := [a, b] ⊂ R be a compact energy interval. We choose a smooth decreasing function g so that supp g ⊂ [a, b]. Let χ = χ(y) be an x-translation invariant smooth function with supp χ ⊂ [-1/2, 1/2]. The edge Hall conductance is defined by σ I e (H) := -2πtr (g (H)i[H, χ]), whenever it exists. The edge conductance measures the current across the axis y = 0 with energies below the energy interval I.

  proved σ I e (H) = (sign b -)n --(sign b + )n + . Applied to the model studied here where b + > 0 and b -= -b + < 0, and under condition (1.4), we have σ I e (H) = -(n -+ n + ). In particular, if b + = b > 0, and I ⊂ ((2n -1)b, (2n + 1)b), we have σ I e (H) = -2n.

Figure 1 .

 1 Figure 1. Approximate shape of the band functions k → ω j (k), for j = 1, . . . , 8, of the Iwatsuka Hamiltonian with -b < 0 < b and b = 1. The dotted curve is E = k 2 . Graph courtesy of N. Popoff.

2. 5 .

 5 Odd band functions ω - j (k): strict monotonicity. The behavior of the odd band functions is much simpler. Proposition 2.5. The odd band functions ω - j (k) = ω 2j (k) are strictly monotone decreasing functions of k ∈ R:

3. 1 .Lemma 3 . 1 .

 131 Mourre estimate for H 0 . For all E ∈ R and all δ > 0 we note ∆ E (δ) := [E -(δ/2)b, E + (δ/2)b]. Let n ∈ N * , E ∈ (e n (b), E n+1 ) and d n (E) be the distance between E/b and the set {e n (1), E n+1 (1)}), i.e. d n (E) := max ((E/b) -e n (1), E n+1 (1) -(E/b)) .

  .25) Putting (3.19)-(3.20) and (3.23)-(3.24) together and recalling (3.15) we end up getting that ψ, [H, iA]ψ c n (1 -F n,E (δ, q, a)) b 1/2 ψ 2 , so (3.16) follows readily from this and (3.14).

Theorem 4 . 1 .

 41 Let b, n, E, and δ 0 be as in Proposition 3.1.

√ 2π ψ 1

 1 ) and denote by Γ :L 2 (R k ) → L 2 (R 2x,y ) the operator with integral kernelγ(x, y; k) := 1 (x, k)e iyk χ(k), (x, y) ∈ R 2 , k ∈ R.

5 . 3 . 3 .

 533 1/2 P = U * χa(λ)Γ * V Γa(λ)χU,(5.18) by direct calculation, where U : Ran P → L 2 (I) is the unitary transform(Uf )(k) := R f (x, k)ψ 1 (x, k)dx χ(k), k ∈ R.From (5.16)-(5.18) then follows thatN (E 1 -λ; H 1 (0)) = n(1; χa(λ)Γ * V Γa(λ)χ), λ > 0.(5.19)Putting W := V 1/2 we deduce from (5.2) and (5.19) thatN (E 1 -λ; H 1 (0)) = n(1; W Γa(λ)χ), λ > 0.(5.20) Part III: Reduction to the quadratic leading term of the first band function.

  ) by replacing ω 1 (k) by the first two terms of the expansion of ω 1 (k) about κ 1 :a(k; λ) := β 1 (k -κ 1 ) 2 + λ -1/2 , k ∈ R, λ > 0.(5.21) Lemma 5.5. Let r 2 fulfill r > 2/α. Then there exists a constant N r 0 such that the estimatesn((1 + ε) 3 ; W Γ 0 a(λ)) -N r ε -r n(1; W Γa(λ)χ)n((1 -ε) 2 ; W Γ 0 a(λ)) + N r ε -r ,(5.22)

  Edge currents for H 0 . We can prove the existence of edge currents for the unperturbed Hamiltonian H 0 based on the Mourre estimate (3.10). A state ϕ ∈ L 2 (R 2 ) carries an edge current of the Hamiltonian H if J y (ϕ) := ϕ, v y ϕ is strictly positive, where the velocity operator is v y = (i/2)[H, A]. Corollary 3.1. Let b, n, E, and δ 0 be as in Proposition 3.1. Let ϕ ∈ L 2 (R 2 ) satisfy ϕ = P 0 (∆ E (δ 0 ))ϕ. Then ϕ carries an edge current and the edge current

					.10)
	3.2. is bounded below by	J y (ϕ)	c n 2	b 1/2 ϕ 2 ,	(3.11)

  .26) where F n,E is defined in(3.15).Since ∆ is compact and ∆ ⊂ ∪ E∈∆ ∆ E (δ(E)), there exists a finite set {E j } N

	j=1
	of energies in ∆ such that
	N
	∆ ⊂
	j=1

  I (H) denotes the spectral projection of H associated with the interval I ⊂ R. Let X 1 and X 2 be two separable Hilbert spaces. For a linear compact operator H : X 1 → X 2 , we set

	where P n(s; H) := n(s 2 ; H * H), s > 0.	(5.2)
	If H j : X 1 → X 2 , j = 1, 2, are two linear compact operators, we will use Ky
	Fan inequality
	n(s 1 + s 2 , H 1 + H 2 ) n(s 1 , H 1 ) + n(s 2 , H 2 ),	(5.3)
	which holds for s 1 > 0 and s 2 > 0 according to [3, Chapter I, Eq. (1.31)] and
	[11, Chapter II, Section 2, Corollary 2.2].
	For further reference, we recall from [17, Eq. (2.1) & Lemma 2.3] the following
	technical result.
	Lemma 5.1. [17, Lemma 2.3]
		5.1,
	which is presented in §5.3.
	For a linear compact self-adjoint operator H acting in a separable Hilbert
	space, we define
	n(s; H) := rank P (s,∞) (H), s > 0,

Notice that the function Fn,E depends on E through δ0 = δ0(E).

4. In order to evaluate n(ε(1 + ε) 2 ; W Γ 0 a(λ)(1 -χ)) in (5.28), we use Lemma 5.1 with G = Γ 0 , f = W and h = a(λ) (1 -χ). We obtain that

. Finally, (5.22) follows from (5.24) and (5.27)-(5.29) upon setting N r := n r + n r .

Summing up (5.9) and (5.20)-(5.22), we have so far derived the following upper bound: and

To obtain a lower bound on N (E 1 -λ; H 0 -V ) from the first inequality in (5.30), we take s = (1 + ε) 3 in (5.31) and obtain lim inf

The upper bound is obtained in a similar manner taking s = (1 -ε) 2 in (5.31), lim sup

α , ε ∈ (0, 1). (5.33) Letting ε ↓ 0 in (5.32)-(5.33), we obtain (5.1). This completes the proof of Theorem 5.1.