Monique Dauge
email: monique.dauge@univ-rennes1.fr

Jean-Philippe Miqueu
email: jean-philippe.miqueu@laposte.net

Nicolas Raymond
email: nicolas.raymond@univ-rennes1.fr

ON THE SEMICLASSICAL LAPLACIAN WITH MAGNETIC FIELD HAVING SELF-INTERSECTING ZERO SET

Keywords: 2010 Mathematics Subject Classification. 35P20, 35Q40, 65M60 Magnetic Laplacian, Semiclassical asymptotics, Exponential concentration of eigenvectors

This paper is devoted to the spectral analysis of the Neumann realization of the 2D magnetic Laplacian with semiclassical parameter h > 0 in the case when the magnetic field vanishes along a smooth curve which crosses itself inside a bounded domain. We investigate the behavior of its eigenpairs in the limit h → 0. We show that each crossing point acts as a potential well, generating a new decay scale of h 3/2 for the lowest eigenvalues, as well as exponential concentration for eigenvectors around the set of crossing points. These properties are consequences of the nature of associated model problems in R 2 for which the zero set of the magnetic field is the union of two straight lines. In this paper we also analyze the spectrum of model problems when the angle between the two straight lines tends to 0.

INTRODUCTION

1.1. The magnetic Laplacian. Let Ω be a bounded, smooth and simply connected open set of R 2 , and A ∈ C ∞ (Ω, R 2 ) be a regular potential vector. For h > 0, we consider the self-adjoint operator P A h,Ω = (-ih∇ + A) 2 , with domain Dom(P A h,Ω ) = {u ∈ H 2 (Ω), (-ih∇ + A)u • ν = 0 on ∂Ω}, where ν is the outward pointing normal at the boundary of Ω.

The operator P A h,Ω has compact resolvent and is associated with the quadratic form Q A h,Ω defined on the form domain Dom(Q A h,Ω ) = H 1 (Ω) by (1.1)

Q A h,Ω (u) = Ω |(-ih∇ + A)u(x)| 2 dx .
Here x = (x 1 , x 2 ) denotes Cartesian coordinates in R 2 . We have the gauge invariance

(1.
2) e -iφ/h (-ih∇ + A) 2 e iφ/h = (-ih∇ + A + ∇φ) 2 , for any φ ∈ H 2 (Ω). Therefore, the spectrum of P A h,Ω only depends on the magnetic field (1.3)

B = ∇ × A = ∂ 2 A 1 -∂ 1 A 2 .
Notation 1.1. We denote by λ n (h) the n-th eigenvalue (with multiplicity) of P A h,Ω . In all of the paper, S(P) denotes the spectrum of any operator P.

We are interested in the behavior of the eigenvalues λ n (h) and their associated eigenfunctions in the semiclassical limit h → 0 for special configurations of the magnetic field B.

1.2. Motivations and context. The spectral analysis of the magnetic Laplacian comes from the theory of superconductivity in which the magnetic Laplacian appears in the study of the third critical field of the Ginzburg-Landau functional (see for instance [START_REF] Saint-James | Type II Superconductivity, International series of monographs in natural philosophy[END_REF] and also the books [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF] and [START_REF]Bound states of the magnetic Schrödinger operator[END_REF], and the references therein). The regime when h goes to 0 (called the semiclassical limit) is equivalent to the strong magnetic field limit which is often involved in applications. In this paper, we restrict to dimension two. 1.2.1. Overview of the literature. In the past two decades, most of the contributions dealt with non-vanishing magnetic fields. We can refer for instance to the works by Bolley & Helffer [START_REF] Bolley | The Ginzburg-Landau equations in a semi-infinite superconducting film in the large κ limit[END_REF], Bauman, Phillips & Tang [START_REF] Bauman | Stable nucleation for the Ginzburg-Landau system with an applied magnetic field[END_REF], del Pino, Felmer & Sternberg [START_REF] Pino | Boundary concentration for eigenvalue problems related to the onset of superconductivity[END_REF], Helffer & Morame [START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF], Bonnaillie-Noël [START_REF] Bonnaillie | On the fundamental state energy for a Schrödinger operator with magnetic field in domains with corners[END_REF], Lu & Pan [START_REF] Pan | Eigenvalue problems of Ginzburg-Landau operator in bounded domains[END_REF], Raymond [START_REF] Raymond | Sharp asymptotics for the Neumann Laplacian with variable magnetic field: case of dimension 2[END_REF], Bonnaillie-Noël & Dauge [START_REF] Ël | Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners[END_REF], Bonnaillie-Noël & Fournais [START_REF] Ël | Superconductivity in domains with corners[END_REF], Raymond & Vu-Ngoc [START_REF] Raymond | Geometry and spectrum in 2D magnetic wells[END_REF].

The present paper is devoted to the case of vanishing magnetic fields. Such an investigation was initially motivated by a paper of Montgomery [START_REF] Montgomery | Hearing the zero locus of a magnetic field[END_REF], followed by the contributions of Helffer & Morame [START_REF]Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells[END_REF], Helffer & Kordyukov [START_REF] Helffer | Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: analysis near the bottom[END_REF][START_REF] Helffer | Aspects théoriques et appliqués de quelques EDP issues de la géométrie ou de la physique[END_REF], and Dombrowski & Raymond [START_REF] Dombrowski | Semiclassical analysis with vanishing magnetic fields[END_REF]. The aforementioned papers do not investigate the case when the zero set of the magnetic field B intersects the boundary. This was the purpose of the work by Pan & Kwek [START_REF] Kwek | Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains[END_REF] and [START_REF] Miqueu | Eigenstates of the Neumann magnetic Laplacian with vanishing magnetic field[END_REF]. We can find in [START_REF] Kwek | Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains[END_REF] a one term asymptotics of the first eigenvalue λ 1 (h). The paper [START_REF] Miqueu | Eigenstates of the Neumann magnetic Laplacian with vanishing magnetic field[END_REF] establishes a sharper result by giving an explicit control of the remainder as well as expansions of all the eigenvalues and eigenfunctions, as the semiclassical parameter h goes to 0, under suitable assumptions when the zero set of B does not self-intersect. 1.2.2. When the zero set of B self-intersects. In the present paper, we want to include nondegenerate quadratic cancellations inside the domain, which is a new configuration in the investigations about vanishing magnetic fields. Assumption 1.2. Let Γ = {x ∈ Ω : B(x) = 0} , and assume that Γ = ∅. We work under the following assumptions i) The set Σ = {x ∈ Γ, ∇B(x) = 0} is non-empty, finite and such that ∂Ω ∩ Σ = ∅. ii) For any x ∈ Σ, the Hessian matrix Hess B(x) of the magnetic field at the point x has two non-zero eigenvalues with opposite signs. iii) The set Γ ∩ ∂Ω is finite, and in each of these intersection points, Γ is non tangent to ∂Ω.

Note that assumptions i)-ii) imply that the set Γ is a simple curve in a neighborhood of each of its intersection points with ∂Ω. Moreover, the set Σ is made of isolated points which are locally the intersection point of two smooth curves. Notation 1.3. Choose x 0 ∈ Σ.

i) B x 0 denotes the second order Taylor expansion of the magnetic field at x 0 . So B x 0 (x) = 1 2 (x -x 0 ) Hess B(x 0 )(x -x 0 ) . ii) A x 0 denotes the Taylor expansion of the magnetic potential A to the third order at the point x 0 . Thus B x 0 = ∇ × A x 0 . iii) Let α(x 0 ), β(x 0 ) be the eigenvalues of 1 2 Hess B(x 0 ), agreeing that |α(x 0 )| ≤ |β(x 0 )|. Set ε(x 0 ) = |α(x 0 )|/|β(x 0 )| and Ξ(x 0 ) = |β(x 0 )| , so that in a suitable local system of orthogonal coordinates y = (s, t) centered at x 0

B x 0 (x) = α(x 0 )s 2 + β(x 0 )t 2 = -β(x 0 ) ε 2 (x 0 )s 2 -t 2 .
Hence the zero set of B x 0 has the equation ε 2 (x 0 )s 2 -t 2 = 0: It is the union of the two lines {t = ±ε(x 0 )s}. These lines are the two tangents to the set Γ at the point x 0 .

The main novelty in this paper is related to the presence of Σ and to the role of the following family of model operators indexed by ε and acting on L 2 (R 2 ), defined as (1.4)

X ε = D σ - τ 3 3 + ε 2 σ 2 τ 2 + D 2 τ , (σ, τ ) =: Y ∈ R 2 ,
with D σ = -i∂ σ and D τ = -i∂ τ . Note that the magnetic field associated with the operator

X ε is B(σ, τ ) = -τ 2 + ε 2 σ 2 .
That is why the operator X ε for ε = ε(x 0 ) will serve as a model magnetic operator at point x 0 . As a consequence of [START_REF] Helffer | Caractérisation du spectre essentiel de l'opérateur de Schrödinger avec un champ magnétique[END_REF][START_REF] Helffer | Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs[END_REF], the operator X ε has a compact resolvent in R 2 . Then it follows that the eigenfunctions of X ε have an exponential decay (see [START_REF] Agmon | Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N -body Schrödinger operators[END_REF], [START_REF]Bounds on exponential decay of eigenfunctions of Schrödinger operators[END_REF], [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF]Theorem B.5.1] and the example in [30, p. 100]). To sum up: Proposition 1.4. Let ε > 0. The spectrum of the operator X ε is formed by a non-decreasing unbounded sequence of positive eigenvalues denoted by (κ n (ε)) n≥1 . Moreover, for any eigenfunction

Ψ ε of X ε , there exists c > 0 such that e c|Y | Ψ ε ∈ L 2 (R 2 ).
1.3. Semiclassical expansions of the magnetic eigenvalues. The model operators X ε have the following homogeneity property, due to their magnetic potential of degree 3. By rescaling, we find immediately: Lemma 1.5. Set A ε := (-τ 3 3 + ε 2 σ 2 τ, 0) the magnetic potential of X ε . Let (κ, Ψ) be a normalized eigenpair of X ε . Setting for h > 0 and Ξ > 0

ψ h (y) = Ξ 1/4 h -1/4 Ψ(Ξ 1/4 h -1/4 Y)
we obtain that (h 3/2 Ξ 1/2 κ, ψ h ) is a normalized eigenpair for the semiclassical magnetic operator (-ih∇ + ΞA ε ) 2 on R 2 .

After the scale h for non-vanishing magnetic fields, the scale h 4/3 for magnetic field vanishing at order 1, we note the apparition of the new scale h 3/2 .

Since for each crossing point x 0 ∈ Σ the operator (-ih∇ + Ξ(x 0 )A ε(x 0 ) ) 2 is unitarily equivalent to the "tangent" magnetic operator (-ih∇ + A x 0 ) 2 , we can guess that the behavior of the low lying spectrum of the operator P A h,Ω corresponds to the low lying spectrum of the operator

(1.5) P A,Σ h := x∈Σ -ih∇ + Ξ(x)A ε(x) 2 , on L 2 (R 2 ) Σ
, where Σ is the cardinal of the finite set Σ. Lemma 1.5 then gives that

S(P A,Σ h ) = x∈Σ h 3/2 Ξ(x) 1/2 S(X ε(x) ).
This leads to introduce, for all x ∈ Σ and all n ∈ N * , the enumeration of the eigenvalues

(1.6a) Λ x n = Ξ(x) 1/2 κ n (ε(x)
) , and the ordered set

(1.6b) Λ B n , n ∈ N * = x∈Σ {Λ x n , n ∈ N * } ,
for which the same value can possibly appear several times (for instance, if we have

Λ x n = Λ x n for (n, x) = (n , x ))
. Note in particular that the smallest element of this set is given by

(1.7) Λ B 1 = min x∈Σ Ξ(x) 1/2 κ 1 (ε(x)) ,
where we recall that κ 1 (ε) is the first eigenvalue of X ε and ε(x) is defined in Notation 1.3. We are ready to state the main two results relating to the low lying spectrum of the operator P A h,Ω . The first result provides a localized estimate from below of the energy functional

Q A h,Ω
(defined in (1.1)) and exponential decay estimates for the eigenvectors of P A h,Ω . The second result states an asymptotic expansion for the eigenvalues of P A h,Ω . 1.3.1. Estimate from below of the energy and Agmon estimates. Let the function x → d(x, Σ) be the Euclidean distance between the point x ∈ Ω and the set Σ.

Theorem 1.6. Under Assumption 1.2, for any exponent d ∈ ( 3 16 , 1 4 ), there exist c d , C d > 0, h d > 0, such that for all u ∈ Dom(Q A h,Ω
) and all h ∈ (0, h d ),

(1.8a) Q A h,Ω (u) ≥ Ω I Σ h,d (x) |u(x)| 2 dx, with (1.8b) I Σ h,d (x) = Λ B 1 h 3/2 -C d h min{2-2d, 3/4+4d} if d(x, Σ) ≤ h d c d h 4/3+2d/3 if d(x, Σ) > h d ,
where [START_REF] Ël | Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners[END_REF]. Note that the exponents present in the definition of I Σ h,d satisfy (1.9) min{2 -2d, ). There exist C, h d > 0 such that for all h ∈ (0, h d ), and all eigenpair (λ(h), ψ h ) of P A h,Ω with λ(h) ≤ Lh 3/2 , we have

Λ B 1 is defined in (1.
(1.10) Ω e 2d(x,Σ)/h d |ψ h (x)| 2 dx + h -3/2 Q A h,Ω e d(•,Σ)/h d ψ h ≤ C ψ h 2 .
In fact estimate (1.10) would also hold for the relaxed condition λ(h) ≤ Lh 4/3+2d/3 if L < c d .

1.3.2. Expansion of lowest eigenvalues. Let us recall that

λ 1 (h) ≤ λ 2 (h) ≤ . . . ≤ λ n (h) . . .
denote the increasing sequence of the eigenvalues of the operator

P A h,Ω while Λ B 1 ≤ Λ B 2 ≤ . . . ≤ Λ B n .
. . denote the elements defined in (1.6a)-(1.6b) from the eigenvalues of the model operators X ε .

Theorem 1.8. Under Assumption 1.2, for all N ∈ N * , there exist C N > 0 and h 0 > 0 such that, for all h ∈ (0, h 0 ) and all n = 1, . . . , N

|λ n (h) -h 3/2 Λ B n | ≤ C N h 7/4
. In Section 3, an extended version of this theorem is proved, providing full expansions of all lowest eigenvalues λ n (h), see Theorem 3.6.

1.4. Low lying eigenvalues of the magnetic cross in the small angle limit. When ε tends to 0, the angle between the two lines {τ = ±εσ} tends to 0. It is interesting to understand the behavior of κ n (ε) in such a limit. One could naively expect that, when ε goes to 0, inf S(X ε ) goes to inf S(M [2] ) where

M [2] = D 2 τ + D σ - τ 3 3 2 acting on L 2 (R 2 ).
The operator M [2] is sometimes called Montgomery operator of order two. By Fourier transform, we have inf S(M [2] ) = inf

ξ∈R inf S(M [2] ξ ) , where M [2] ξ = D 2 τ + ξ - τ 3 3 2 acting on L 2 (R).
The quantity inf S(M [2] ) has been numerically estimated, see [23, Table 1]. Actually, the limit ε → 0 is singular: The operator X ε is partially semiclassical. Indeed, after the scaling σ = ε -1 s, τ = t, the operator X ε becomes (1.11)

D 2 t + εD s - t 3 3 + s 2 t 2 with D t = -i∂ t , D s = -i∂ s .
The operator (1.11) is the Weyl quantization Op w ε (X α,ξ ) of the symbol X α,ξ where (1.12)

X α,ξ = D 2 t + ξ - t 3 3 + α 2 t
2 is a self-adjoint operator acting on L 2 (R) depending on the two real parameters α and ξ. In the small angle regime, the spectral analysis of the operator X ε is related to the spectral analysis of the family (X α,ξ ) (α,ξ)∈R 2 . The asymptotic expansions of the first eigenvalues of X ε is related to the"band function" 1 defined by the ground state energy of X α,ξ 1 (α, ξ) = min S(X α,ξ ) , (α, ξ) ∈ R 2 , see for instance [START_REF]Breaking a magnetic zero locus: asymptotic analysis[END_REF] and [START_REF] Ël | Magnetic WKB Constructions[END_REF], where such operators and reductions are considered. One of the requirements to apply the theory developed in [START_REF] Ël | Magnetic WKB Constructions[END_REF] (and that relates 1 to the eigenvalue asymptotic expansions when ε → 0 for the operator X ε ) is that 1 has a minimum. Theorem 1.9. The function 1 reaches its infimum S 0 in R 2 , so

S 0 = min (α,ξ)∈R 2 1 (α, ξ),
and this minimum is reached on the set

(α, ξ) ∈ R 2 , |ξ| ≤ 2 3 |α| 3 .
By construction, S 0 ≤ inf S(M [2] ). The numerical simulations that we have performed provide the upper bound (1.13) S 0 ≤ 1 (α 0 , 0) 0.4941 for α 0 = 0.786.

We note that the value given in [23, Table 1] for inf S(M [2] ) is 0.66, corresponding to the value ξ = 0 of the Fourier parameter. Hence we have obtained the strict inequality S 0 < inf S(M [2] ).

Our numerical results for the band function 1 , see Figures 1 and2, suggest that the minimum S 0 is attained at the sole two values ±(α 0 , 0) of (α, ξ).

The following result gives the convergence of the eigenvalues κ n (ε) of the model operator X ε as ε → 0. Theorem 1.10. For all n ≥ 1, there exist C > 0 and ε 0 > 0 such that for all ε ∈ (0, ε 0 )

|κ n (ε) -S 0 | ≤ Cε.
1.5. Organization of the paper. In Section 2, we prove the preliminary Theorems 1.6 and 1.7. In Section 3, we establish the full asymptotic expansions of the eigenvalues and eigenfunctions. Section 4 is devoted to the proofs of Theorems 1.9 and 1.10 and to the presentation of numerical simulations concerning the band function 1 and the eigenpairs of X ε as ε → 0.

BOUNDS FROM BELOW AND EXPONENTIAL DECAY

In this section we prove the preliminary bounds from below (1.8a)-(1.8b) and the Agmon estimates (1.10). We will use several times a perturbation formula for the magnetic Laplacian which we first state.

2.1. Perturbation of the magnetic potential. Let A be another smooth magnetic potential defined on Ω. In practice A will be the Taylor expansion of A at some point x 0 and to various orders (2, 3 or 4). By expanding the square, we get (2.1)

Q A h,Ω (u) = Q A h,Ω (u) + 2 Re (-ih∇ + A )u, (A -A )u + (A -A )u 2 .
This yields

Q A h,Ω (u) ≥ Q A h,Ω (u) -2Q A h,Ω (u) 1/2 (A -A )u by Cauchy-Schwarz inequality, leading to the parametric estimate (based on the inequality 2ab ≤ ηa 2 + η -1 b 2 , for all η > 0) (2.2) ∀η > 0, Q A h,Ω (u) ≥ (1 -η)Q A h,Ω (u) -η -1 (A -A )u 2 .
Such a lower bound is used, for instance, in the seminal paper [20, p. 51], and in the book [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF]Chap. 8].

2.2.

Lower bound. The proof of (1.8a) is based on a quadratic partition of unity. Let us recall that such a partition is given for each relevant h > 0 by a finite collection of smooth cutoff functions (χ h j ) satisfying on Ω

(2.3) j |χ h j | 2 = 1 .
Then we have the following well-known localization formula (see [START_REF] Cycon | Schrödinger operators with application to quantum mechanics and global geometry[END_REF])

(2.4) Q A h,Ω (u) = j Q A h,Ω (χ h j u) -h 2 j u |∇(χ h j )| 2 L 2 (Ω) .
We introduce three sets Σ [1] , Σ [2] (h) and Σ [3] (h) covering Ω.

Notation 2.1. Let B(x 0 , r) denote the open ball of center x 0 and radius r and [START_REF] Berezin | Wick and anti-Wick symbols of operators[END_REF] ) and introduce (i) Σ [1] = Ω \ B(Σ, R 1 ). (ii) Σ [2]

B(Σ, r) = x∈Σ B(x, r). Let R 1 > 0 such that B(Σ, 3R 1 ) ⊂ Ω. Choose d ∈ ( 3 16 , 1
(h) = B(Σ, 2R 1 ) \ B(Σ, 1 2 h d ). (iii) Σ [3] (h) = B(Σ, h d ).
Then we consider a partition of unity composed of three cutoff functions (X [1] , X h, [2] , X h, [3] ) associated with this covering of Ω in the sense that

|X [1] | 2 + |X h,[2] | 2 + |X h,[3] | 2 = 1,

and

supp X [1] ⊂ Σ [1] , supp X h, [2] ⊂ Σ [2] (h), supp X h, [3] ⊂ Σ [3] (h). The distance between ∂Σ [2] (h) ∩ Σ [1] (h) and ∂Σ [1] (h) ∩ Σ [2] (h) is R 1 . The distance between ∂Σ [3] (h) and ∂Σ [2] (h) ∩ Σ [3] (h) is 1 2 h d . Hence we can choose the cutoff functions so that (2.5)

|∇X [1] | 2 ≤ K loc , |∇X h,[2] | 2 ≤ K loc h -2d , |∇X h,[3] | 2 ≤ K loc h -2d .
with a constant K loc independent of h and d. Combined with the localization formula (2.4) associated with the partition (X [1] , X h, [2] , X h, [3] ), the estimates (2.5) yield for all u ∈ H 1 (Ω), [2] u and u 3 = X h, [3] u, and are going to bound each

(2.6) Q A h,Ω (u) ≥ Q A h,Ω (X [1] u) + Q A h,Ω (X h,[2] u) + Q A h,Ω (X h,[3] u) -K loc h 2-2d u 2 L 2 (Ω) , We set u 1 = X [1] u, u 2 = X h,
Q A h,Ω (u k ) from below (for k = 1, 2, 3).
Lemma 2.2 (Lower bound on Σ [1] ). There exist c 1 and h 0 > 0 such that for all 0 < h < h 0

Q A h,Ω (u 1 ) ≥ c 1 h 4/3 u 1 2 .
Proof. This result is a consequence of [26, Theorem 1.9].

Lemma 2.3 (Lower bound on Σ [2] (h)). There exist c 2 and h 0 > 0 such that for all 0 < h < h 0

Q A h,Ω (u 2 ) ≥ c 2 h 4/3+2d/3 u 2 2 .
Proof. Set ρ = 1 4 . We introduce a second partition of unity (χ h j ) j∈J on Σ [2] (h) associated with a family of balls B(x j , d j h ρ ) for some constants d j . We first cover the zero set Γ ∩ Σ [2] (h) and choose the centers x j ∈ Γ with d j = 1 so that the distances between consecutive points along Γ is 1 2 h ρ . Hence, setting Σ [2] (h) = Σ [2] (h) \

x j ∈Γ B(x j , h ρ )
we obtain that the distance between Γ and Σ [2] (h) is larger than 1 2 h ρ . We then cover Σ [2] (h) with balls B(x j , d j h ρ ) choosing d j = 1 4 and the mutual distances between the centers bounded from below by 1 8 h ρ . Finally we can choose the functions χ h j so that |∇χ h j | 2 ≤ C 2,loc h 2ρ and the localization formula (2.4) yields

(2.7) Q A h,Ω (u 2 ) ≥ j Q A h,Ω (χ h j u 2 ) -C 2,loc h 2-2ρ u 2 2 .
We have, by [START_REF] Miqueu | Eigenstates of the Neumann magnetic Laplacian with vanishing magnetic field[END_REF]Lemma 2.3] (for the case

( ) = (3) of Table 2.1), (2.8) Q A h,Ω (χ h j u 2 ) ≥ 1 2 M 0 |∇B(x j )| 2/3 h 4/3 -2Ch 6ρ χ h j u 2 2 (x j ∈ Γ).
As a consequence of the non degeneracy of ∇B on Γ outside Σ (namely ∇B| = 0 on Γ\Σ) and the non degeneracy of Hess B on Σ (meaning that the eigenvalues of the Hessian matrix are not equal to 0, according to Assumption 1.2) there exists a positive constant C(B) such that

|∇B(x)| ≥ C(B) d(x, Σ), ∀x ∈ Ω .
Since d(x j , Σ) is larger than 1 2 h d by construction, from (2.8) we get

Q A h,Ω (χ h j u 2 ) ≥ 1 4 M 0 C(B) 2/3 h 4/3+2d/3 -2Ch 6ρ χ h j u 2 2 .
We note that, with ρ = 1 4 and d < 1 4 , the exponent 6ρ = 3 2 is (strictly) larger than 4 3 + 2 3 d. Therefore, for h small enough (2.9)

Q A h,Ω (χ h j u 2 ) ≥ 1 8 M 0 C(B) 2/3 h 4/3+2d/3 χ h j u 2 2 (x j ∈ Γ).
When x j does not belong to Γ, in the ball B(x j , d j h ρ ) we use the lower bound (see [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF]Lemma 1.4.1])

(2.10)

Q A h,Ω (χ h j u 2 ) ≥ inf x∈B(x j ,d j h ρ ) |B(x)| h χ h j u 2 2 (x j ∈ Γ).
For any x ∈ B(x j , d j h ρ ), its distance to Γ is larger than

1 2 h ρ by construction. Let g ∈ Γ be such that d(x, Γ) = d(x, g). Then d(g, Σ) is larger than 1 2 h d . As a consequence of the Morse lemma |B(x)| ≥ C (B) d(x, g) d(g, Σ) ≥ Ch ρ+d .
Hence, with (2.10) we obtain (2.11)

Q A h,Ω (χ h j u 2 ) ≥ Ch 1+ρ+d χ h j u 2 2 (x j ∈ Γ). With ρ = 1 4 and d < 1 4 , the exponent 1 + ρ + d is < 4 3 + 2 3 d.
Therefore, for h small enough, as a result of (2.9) and (2.11) we have, for a positive constant c independent of h

Q A h,Ω (χ h j u 2 ) ≥ ch 4/3+2d/3 χ h j u 2 2 (∀j ∈ J).

With (2.7) this yields

Q A h,Ω (u 2 ) ≥ ch 4/3+2d/3 j χ h j u 2 2 -C 2,loc h 2-2ρ u 2 2 ≥ c 2 h 4/3+2d/3 u 2 2 ,
since 2 -2ρ = 3 2 > 4/3 + 2d/3 for any d < 1 4 . The lemma is proved. Lemma 2.4 (Lower bound on Σ [3] ). There exist C 3 and h 0 > 0 such that for all 0 < h < h 0

Q A h,Ω (u 3 ) ≥ Λ B 1 h 3/2 -C 3 h 3/4+4d u 3 2 .
Proof. For h small enough the set Σ [3] is the union of the balls B(x 0 , h d ), with x 0 spanning Σ. It suffices to consider each point x 0 separately. We denote by u x 0 3 the restriction of u 3 to B(x 0 , h d ) and use the perturbative lower bound (2.2) with A = A x 0 the third order Taylor expansion of

A at x 0 : ∀η > 0, Q A h,Ω (u x 0 3 ) ≥ (1 -η)Q A x 0 h,Ω (u x 0 3 ) -η -1 (A -A x 0 )u x 0 3 2 .
Recall from the introduction that the magnetic field associated with A x 0 is B x 0 , and that the magnetic operator

(-ih∇ + A x 0 ) 2 is unitarily equivalent to (-ih∇ + Ξ(x 0 )A ε(x 0 ) ) 2 whose first eigenvalue is h 3/2 Λ x 0 1 . Besides, |A(x) -A x 0 (x)| ≤ Ch 4d , ∀x ∈ B(x 0 , h d ). So we have ∀η > 0, Q A h,Ω (u x 0 3 ) ≥ (1 -η)h 3/2 Λ x 0 1 -η -1 Ch 8d u x 0 3 2 .
Choosing η = h 4d-3/4 to equalize the remainders, we deduce the inequality

Q A h,Ω (u x 0 3 ) ≥ h 3/2 Λ x 0 1 -Ch 3/4+4d u x 0 3 2 .
Taking the infimum over all the points of the finite set Σ gives the result.

We can now conclude with the proof of Theorem 1.6.

Proof of Theorem 1.6. We gather the estimates provided by Lemmas 2.2-2.4 and combine them with the localization estimate (2.6) and obtain that

Q A h,Ω (u) is bounded from below by c 1 h 4/3 u 1 2 + c 2 h 4/3+2d/3 u 2 2 + Λ B 1 h 3/2 -C 3 h 3/4+4d u 3 2 -K loc h 2-2d u 2 .
Since d < 1 4 , then h 2-2d h 4/3+2d/3 and the lower bound above can be replaced by

ch 4/3+2d/3 ( u 1 2 + u 2 2 ) + Λ B 1 h 3/2 -C 3 h 3/4+4d -K loc h 2-2d u 3 2 ≥ ch 4/3+2d/3 (1 -χ)u 2 + Λ B 1 h 3/2 -Ch min{3/4+4d,2-2d} χu 2 with χ the characteristic function of the set {x ∈ Ω, d(x, Σ) < h d }. The theorem is proved. 2.3. Agmon estimates. Let d ∈ ( 3 16 , 1 4
) and L > 0. We consider an eigenpair (λ(h), ψ h ) of

P A h,Ω such that λ(h) ≤ Lh 3/2 . For all function Φ ∈ W 1,∞ (Ω), we have e Φ ψ h ∈ Dom(Q A h,Ω ) and
(2.12)

Q A h,Ω (e Φ ψ h ) = λ(h) e Φ ψ h 2 + h 2 |∇Φ| e Φ ψ h 2 . Defining Φ(x) = d(x, Σ)h -d , we have |∇Φ(x)| 2 = h -2d . Hence (2.13) h 2 |∇Φ| e Φ ψ h 2 ≤ h 2-2d e Φ ψ h 2 .
We denote

Z near h = {x ∈ Ω, d(x, Σ) ≤ h d } and Z far h = {x ∈ Ω, d(x, Σ) > h d } . We introduce δ such that δ = min{3/4 + 4d, 2 -2d} -3 2 . Since d ∈ ( 3 16 , 1 4
), the number δ is positive. Using (1.8a)-(1.8b) we obtain

Q A h,Ω (e Φ ψ h ) ≥ c d h 4/3+2d/3 e Φ ψ h 2 L 2 (Z far h ) + Λ B 1 h 3/2 -C d h 3/2+δ ) e Φ ψ h 2 L 2 (Z near h ) ≥ c d h 4/3+2d/3 e Φ ψ h 2 L 2 (Z far h ) -C d h 3/2+δ e Φ ψ h 2 L 2 (Z near h ) .
Combining the latter inequality with (2.12) and (2.13), we obtain (using the fact that λ(h) ≤ Lh 3/2 )

c d h 4/3+2d/3 -Lh 3/2 -h 2-2d e Φ ψ h 2 L 2 (Z far h ) ≤ (Lh 3/2 + h 2-2d + C d h 3/2+δ ) e Φ ψ h 2 L 2 (Z near h )
from which we immediately deduce that for h small enough

e Φ ψ h 2 L 2 (Z far h ) ≤ e Φ ψ h 2 L 2 (Z near h ) . Since by construction |Φ| is bounded by 1 on Z near h , there holds e Φ ψ h 2 L 2 (Z near h ) ≤ ψ h 2 L 2 (Z near h ) , we finally get e Φ ψ h 2 L 2 (Ω) ≤ ψ h 2 L 2 (Ω)
, which implies the Agmon estimates (1.10).

ASYMPTOTIC EXPANSIONS OF EIGENVALUES

In this section, we prove Theorem 1.8 in two steps. First, the localization around each crossing point x 0 ∈ Σ, and, second, a perturbation argument for the magnetic potential around each crossing point. We conclude the section by stating a full asymptotic expansion for eigenvalues and a sketch of the proof.

3.1.

Preliminaries. We will use several times an argument based on reciprocal quasimodes between two operators. We state this in a general form. Lemma 3.1. Let P and P two positive operators with discrete spectra associated with sesquilinear forms a and a , respectively. Let (µ n ) and (µ n ) be the increasing sequences of their eigenvalues (counted with multiplicity). Let (ϕ n ) be an associated orthonormal basis of eigenvectors for P. Let N be a positive integer and assume that for each n = 1, . . . , N , there exists ϕ n ∈ Dom(a ) such that

(3.1a) ϕ n , ϕ m = ϕ n , ϕ m + ν n,m and a (ϕ n , ϕ m ) = a(ϕ n , ϕ m ) + µ n,m
and set

(3.1b) ν = max n,m |ν n,m | and µ = max n,m |µ n,m | Assume that ν < 1 N . Then (3.2) µ n ≤ µ n + nµ 1 -nν , n = 1, . . . , N.
Proof. Let M ≤ N . By the min-max formula,

µ M ≤ max ϕ∈span{ϕ n , n=1,...,M } a (ϕ, ϕ) ϕ, ϕ
We write ϕ as a sum n γ n ϕ n . Then

a (ϕ, ϕ) = n,m γ n γm a (ϕ n , ϕ m ) = n |γ n | 2 µ n + n,m γ n γm µ n,m ≤ (µ M + M µ) n |γ n | 2 . Likewise ϕ, ϕ = n,m γ n γm ϕ n , ϕ m = n |γ n | 2 + n,m γ n γm ν n,m ≥ (1 -M ν) n |γ n | 2 .
Whence formula (3.2)

We will also need a simple but useful consequence of Agmon estimates.

Lemma 3.2. Assume that the family of function (ψ h ) h>0 satisfies (for some γ and δ > 0) the estimate

R 2 e γ|y|/h δ |ψ h (y)| 2 dy ≤ C ψ h 2 ,
for h small enough and C independent of h. Then for all m > 0, there exists C m such that for h > 0 small enough

R 2 |y| m |ψ h (y)| 2 dy ≤ C m h mδ ψ h 2 .
Proof. It suffices to write

R 2 |y| m |ψ h (y)| 2 dy ≤ max ρ>0 ρ m e -γρ/h δ R 2 e γ|y|/h δ |ψ h (y)| 2 dy
and notice that max ρ>0 ρ m e -γρ/h δ = max ρ>0 (h δ ρ) m e -γρ = h mδ max ρ>0 ρ m e -γρ .

3.2.

Localization. Introduce a positive radius r 0 such that the collection of balls B(x 0 , r 0 ), x 0 ∈ Σ, are pairwise disjoint. Then consider the collection of operators P A h,B(x 0 ,r 0 ) for x 0 ∈ Σ. The main result of this section is: Lemma 3.3. Recall that λ n (h) is the increasing sequence of eigenvalues of P A h,Ω (counted with multiplicity). Denote by λ loc n (h) the increasing sequence of eigenvalues of

x 0 ∈Σ P A h,B(x 0 ,r 0 ) . Let d ∈ ( 3 16 , 1 4
) and let N be a positive integer. If λ loc N (h) ≤ Lh 3/2 for h > 0 small enough, then there exist C N and h N > 0 such that for all h ∈ (0, h N )

|λ n (h) -λ loc n (h)| ≤ C N e -r 0 /h d , ∀ n = 1, .
. . , N. Proof. Choose for each x 0 ∈ Σ a smooth cut-off function χ x 0 with support in B(x 0 , r 0 ) and equal to 1 on B(x 0 , 1 2 r 0 ). i) Use Lemma 3.1 with P = P A h,Ω and P = ⊕ x 0 ∈Σ P A h,B(x 0 ,r 0 ) (acting on ⊕ x 0 ∈Σ L 2 (B(x 0 , r 0 ))): For each eigenvector ϕ n of P, one crossing point x 0 ∈ Σ is selected and we set

ϕ n = χ x 0 ϕ n defined on Ω.
Relying on the assumption that λ loc n (h) ≤ Lh 3/2 for n = 1, . . . , N , we may apply the Agmon estimates (1.10) to the operator P A h,B(x 0 ,r 0 ) . This yields conditions (3.1a)

-(3.1b) considering ν = µ = Ce -r 0 /h d . Hence λ n (h) ≤ λ loc n (h) + C N e -r 0 /h d .
ii) Conversely, we swap the roles of the two operators: P = P A h,Ω and P = ⊕ x 0 ∈Σ P A h,B(x 0 ,r 0 ) . For each eigenvector ϕ n of P, we consider

ϕ n := χ x 0 ϕ n x 0 ∈Σ defined on x 0 ∈Σ B(x 0 , r 0 ) .
Note that, as a consequence of the previous step of the proof, we have that λ n (h) ≤ 2Lh 3/2 for all n = 1, . . . , N . As above, we conclude with the help of Agmon estimates for the operator

P A h,Ω that λ loc n (h) ≤ λ n (h) + C N e -r 0 /h d .
The lemma is proved.

3.3.

Taylor approximation of a localized operator. With Lemma 3.3 at hand, we can assume that Ω = B(x 0 , r 0 ). Thus Σ is reduced to one element, x 0 . Recall that A x 0 denotes the third order Taylor expansion of A at the point x 0 . We are going to consider the operator P A x 0 h := (-ih∇ + A x 0 ) 2 posed on R 2 . We have seen in the introduction that the eigenvalues of P A x 0 h are the h 3/2 Λ x 0 n , see (1.6a), and that its eigenvectors ψ x 0 h,n are scaled from the eigenvectors of X ε with ε = ε(x 0 ) and Ξ = Ξ(x 0 ). As a consequence of the exponential decay of the eigenvectors of X ε (Proposition 1.4) and the scaling provided by Lemma 1.5, we find that, for some positive constant γ

(3.3) R 2 e 2γ|y|/h 1/4 |ψ x 0 h,n (y)| 2 dy + h -3/2 Q A x 0 h e γ|y|/h 1/4 ψ x 0 h,n ≤ C ψ x 0 h,n 2 .
Lemma 3.4. With Ω = B(x 0 , r 0 ), we denote by λ x 0 n (h) the eigenvalues of P A h,Ω . The eigenvalues of P A x 0 h are given by h 3/2 Λ x 0 n . For any d 0 < 1 4 and any positive integer N , there exist C N and h N > 0 such that for all h ∈ (0, h N )

(3.4) h 3/2 Λ x 0 n -C N h 3/2+d 0 ≤ λ x 0 n (h) ≤ h 3/2 Λ x 0 n + C N h 7/4 , n = 1, . . . , N.
Proof. The proof combines Lemma 3.1 with the perturbation identity (2.1). We still use the cut-off function χ x 0 as in the proof of Lemma 3.3.

i) Use Lemma 3.1 with P = P A x 0 h and P = P A h,Ω . For each eigenvector ϕ n of P, we consider the quasimode ϕ n = χ x 0 ϕ n for P A x 0 h . The localization error is exponentially decaying thanks to (3.3) but the principal part of the discrepancy for the quasimodes arises from the difference between A and A x 0 . Thus the bound ν in (3.1b) satisfies (for some r 0 > 0)

ν ≤ Ce -r 0 /h 1/4 .
For estimate the diagonal terms µ n,n in (3.1a), we use the identity (2.1) for A = A x 0 . Then

Q A h,Ω (χ x 0 ψ x 0 h,n ) = Q A x 0 h (χ x 0 ψ x 0 h,n ) + 2 Re (-ih∇ + A x 0 )χ x 0 ψ x 0 h,n , (A -A x 0 )χ x 0 ψ x 0 h,n + (A -A x 0 )χ x 0 ψ x 0 h,n 2 .
Hence the difference µ n,n :

= Q A h,Ω (χ x 0 ψ x 0 h,n ) -Q A x 0 h (χ x 0 ψ x 0 h,n ) is estimated by |µ n,n | ≤ 2 Q A x 0 h (χ x 0 ψ x 0 h,n ) 1/2 (A -A x 0 )χ x 0 ψ x 0 h,n + (A -A x 0 )χ x 0 ψ x 0 h,n 2
Using the Agmon estimates (3.3) and Lemma 3.2 with δ = 1 4 and m = 8, we find

|µ n,n | ≤ C(h 3/4 h + h 2 ).
The reasonning is similar for µ n,m , n = m. Hence the right part of inequalities (3.4).

ii) For the left part of (3.4), we swap the roles of P A 4 and any positive integer N , there exist C N > 0 and h 0 > 0 such that, for all h ∈ (0, h 0 ) and all n = 1, . . . , N

(3.5) h 3/2 Λ B n -C N h 3/2+d ≤ λ n (h) ≤ h 3/2 Λ B n + C N h 7/4
, n = 1, . . . , N. To go further, we are going to exhibit, for each n, series expansions of eigenpairs. Owing to the exponential localization given by Lemma 3.3, it suffices to restrict the construction to any chosen localized operator P A h,B(x 0 ,r 0 ) . In order to alleviate notations, we will remove the mention of x 0 in general, and work in the Cartesian coordinates y for which the crossing point is at the origin. Then the domain Ω is the ball B(0, r 0 ) and the magnetic field cancels to the order 2 at 0. After a possible change of gauge, we can assume that the magnetic potential A cancels to the order 3 at 0. We write its Taylor formal series as A ∼ j≥0 A j where A j is polynomial and homogeneous of degree 3 + j.

The first nonzero term is A 0 formerly denoted by A x 0 . We retrieve the principal part of P A h,Ω at 0, and its natural expansion in powers of h 1/4 by considering, via the change of variables Y = y/h 1/4 :

h -3/2 P A h,Ω [y, ∇ y ] = h -3/2 P A h,Ω/h 1/4 [h 1/4 Y, h -1/4 ∇ Y ] .
We expand the right hand side as a formal series of operators j h j/4 L j [Y, ∇ Y ] defined on R 2 . We have

j h j/4 L j [Y, ∇ Y ] = h -3/2 -ih 3/4 ∇ Y + j≥0 h (3+ )/4 A j (Y) 2 .
Hence the series j h j/4 L j starts with L 0 given by

L 0 = -i∇ Y + A 0 (Y) 2
and the other terms L j for j ≥ 1 are partial differential operators of degree 1 with polynomial coefficients. The main term L 0 is isospectral to (-i∇ + Ξ(x 0 )A ε (x 0 )) 2 , see (1.5), and its eigenvalues are given by the Λ x 0 n (n ∈ N * ), see (1.6a). Choose a normalized eigenpair of L 0 , which we denote by ( 0 , Ψ 0 ). We look for j ∈ R and Ψ j ∈ Dom(L 0 ), j = 1, 2, . . .

solving j h j/4 L j k h k/4 Ψ k = j h j/4 j k h k/4
Ψ k in the sense of formal series, i.e., solving the sequence of equations,

m j=0 L j Ψ m-j = m j=0 j Ψ m-j m = 1, 2, . . . .
We write the first equation (for m = 1) as

(L 0 -0 )Ψ 1 = 1 Ψ 0 -L 1 Ψ 0 ,
and the next ones as

(L 0 -0 )Ψ m = m Ψ 0 -L m Ψ 0 + m-1 j=1 ( j -L j )Ψ m-j .
If 0 is a simple eigenvalue of L 0 , the solution of such a sequence of equations is classical, resulting from the Fredholm alternative for the self-adjoint operator L 0 . For instance, we get

(3.6) 1 = L 1 Ψ 0 , Ψ 0 if m = 1.
If 0 is a multiple eigenvalue of L 0 , we cannot choose a priori an associated eigenvector, but have to work in the whole associated eigenspace E 0 . Then identity (3.6) is replaced by an eigen-equation for a finite dimensional hermitian matrix acting on the eigenspace E 0 . The process can be pursed as well, see [START_REF] Dauge | Eigenmode asymptotics in thin elastic plates[END_REF] for details on this procedure. The terms Ψ m belong to the domain of L 0 and have, furthermore, exponential decay. Setting for m ≥ 1

[m] (h) = h 3/2 m j=0 h j/4 j and ψ [m] h (y) = χ x 0 (y) m j=0 h j/4 Ψ j (y/h 1/4 )
we have constructed a quasimode for P A h,Ω to the order h 3/2+(m+1)/4 , i.e. P A h,Ω (ψ

[m] h ) = [m] (h)ψ [m] h + ρ [m+1] h with ρ [m+1] h ≤ C m h 3/2+(m+1)/4 ψ [m] h .
Combining this with (3.5), we deduce by the spectral theorem that there holds Theorem 3.6. Under Assumption 1.2, for all integers N ≥ 1 and M ≥ 0, there exist coefficients

Λ B n,m ∈ R + for 1 ≤ n ≤ N, 0 ≤ m ≤ M, with Λ B n,0 = Λ B n
and constants C N,M > 0 and h 0 > 0 such that, for all h ∈ (0, h 0 ) and all n = 1, . . . , N

λ n (h) -h 3/2 M m=0 h m/4 Λ B n,m ≤ C N h 3/2+(M +1)/4 .
Of course, Theorem 1.8 is a particular case of the above statement if we choose M = 0.

SMALL ANGLE LIMIT

The first part of this section is devoted to theoretical results on the band function (α, ξ) → 1 (α, ξ) and to their numerical illustration. In the second part, we rely on these results to prove the convergence of eigenvalues of X ε in the small angle limit and present the computations of their first eigenstates for a set of small values of ε.

4.1.

Operator symbol and band function. Here we study the behavior of the first eigenvalue

1 (α, ξ) of the operator symbol X α,ξ , for (α, ξ) ∈ R 2 X α,ξ = D 2 t + ξ + α 2 t - t 3 3 2 ,
acting on L 2 (R), see (1.12).

4.1.1. Preliminaries. Let us introduce the potential V α,ξ of X α,ξ and its generating polynomial P α,ξ :

P α,ξ (t) = ξ + α 2 t - t 3 3 and V α,ξ (t) = P α,ξ (t) 2 . V α,ξ (t) = ξ 2 + 2ξα 2 t -2ξ t 3 3 + α 4 t 2 -2α 2 t 4 3 + t 6 9
The potential V α,ξ depends smoothly on the parameters (α, ξ) and is confining for each value of (α, ξ). So there holds Proposition 4.1. For all (α, ξ) ∈ R 2 , the operator X α,ξ has a compact resolvent and the family (X α,ξ ) with (α, ξ) ∈ R 2 is analytic (of type (B) according to Kato theory, see [START_REF] Kato | Perturbation theory for linear operators[END_REF]).

Since the X α,ξ are Sturm-Liouville operators, we obtain Corollary 4.2. For all (α, ξ), the eigenvalue 1 (α, ξ) is simple and depends analytically of α and ξ. The associated eigenfunctions do not vanish and the unique normalized and positive eigenfunction u α,ξ associated with 1 (α, ξ) depends analytically of (α, ξ).

As a consequence we have the following "Feynman-Hellmann" formulas. Corollary 4.3. For all (α, ξ) we have the following identities

(∂ α 1 )(α, ξ) = 4α R ξ + α 2 t - t 3 3 tu 2 α,ξ (t) dt, (∂ ξ 1 )(α, ξ) = 2 R ξ + α 2 t - t 3 3 u 2 α,ξ (t) dt.
The potential V α,ξ has the following obvious symmetry properties: V α,ξ (t) = V -α,ξ (t) and V α,-ξ (t) = V α,ξ (-t). Hence the band function 1 (α, ξ) is even with respect to each of the two variables α and ξ, so its analysis can be restricted to the first quadrant {α ≥ 0, ξ ≥ 0}. The following lemma gives an expression of the roots of the generating polynomial P α,ξ , depending on the sign of its discriminant. Lemma 4.4. For all α ≥ 0 and all ξ ≥ 0, denote by t k (α, ξ) the three roots of the polynomial P α,ξ , agreeing that Re t 1 (α, ξ) ≤ Re t 2 (α, ξ) ≤ Re t 3 (α, ξ). Then, if (α, ξ) = (0, 0), t 3 (α, ξ) is a positive simple real root. More precisely we have • For ξ < 2 3 α 3 , the polynomial P α,ξ admits three distinct real roots (that we denote t 1 (α, ξ) < t 2 (α, ξ) < t 3 (α, ξ)) given (for all k ∈ {1, 2, 3}) by

(4.1a) t k (α, ξ) = j 3-k 3 1 2 3ξ + i 4α 6 -9ξ 2 + j k-3 3 1 2 3ξ -i 4α 6 -9ξ 2 ,
where j is the complex number defined by j = e 2iπ/3 = -

1 2 + i √ 3
2 .

• For ξ = 2 3 α 3 , the polynomial P α,ξ admits a simple real root and a double real root respectively given by

(4.1b) t 3 (α, ξ) = 2α and t 2 (α, ξ) = t 1 (α, ξ) = -α.
• For ξ > 2 3 α 3 , the polynomial P α,ξ admits a unique real root given by

(4.1c) t 3 (α, ξ) = 3 1 2 3ξ + -4α 6 + 9ξ 2 + 3 1 2 3ξ --4α 6 + 9ξ 2 .
The next result shows that the minimum cannot be reached on the set {ξ ≥ 2 3 α 3 > 0}. Its proof uses Corollary 4.3 and the fact that, if ξ ≥ 2 3 α 3 > 0, the polynomial t → (t -t 3 )P α,ξ (t) is negative. Proposition 4.5. For all (α, ξ) such that ξ ≥ 2 3 α 3 > 0, we have

(∂ α 1 )(α, ξ) -2αt 3 (α, ξ)(∂ ξ 1 )(α, ξ) < 0.
In particular, there is no critical point on the set {ξ ≥ 2 3 α 3 > 0}. 4.1.2. Behavior of the band function at infinity. Now, the remaining part of this section is devoted to prove that 1 (α, ξ) tends to infinity as |α| + |ξ| tends to infinity, namely

(4.2) lim |α|+|ξ|→∞ 1 (α, ξ) = ∞ .
Note that Proposition 4.5 combined with (4.2) implies Theorem 1.9.

To prove (4.2), we split (for each R > 1) the region

A R := {(α, ξ) ∈ R 2 , α ≥ 0, ξ ≥ 0, and α + ξ > R}
into the three subregions

A • R := A R ∩ {(α, ξ) ∈ R 2 , α ∈ [0, 1] ξ > 2 3 α 3 } (4.3a) A R := A R ∩ {(α, ξ) ∈ R 2 , α ∈ [1, ∞) ξ > 2 3 α 3 } (4.3b) A R := A R ∩ {(α, ξ) ∈ R 2 , α ∈ [1, ∞) ξ ≤ 2 3 α 3 } (4.3c)
and are going to prove the next lemma. Lemma 4.6. We denote Q α,ξ the quadratic form associated with the operator X α,ξ . There exists constants R > 1 and B > 0 such that for all ψ ∈ C ∞ 0 (R) the following lower bounds hold

Q α,ξ (ψ) ≥ B ξ 2/3 ψ 2 L 2 (R) , ∀(α, ξ) ∈ A • R (4.4a) Q α,ξ (ψ) ≥ B ξ 2/9 ψ 2 L 2 (R) , ∀(α, ξ) ∈ A R (4.4b) Q α,ξ (ψ) ≥ B α 2/3 ψ 2 L 2 (R) , ∀(α, ξ) ∈ A R (4.4c)
We can see that

ξ 2/9 > 1 2 ξ 2/9 + 1 3 α 2/3 on A • R ∪ A R and α 2/3 > 1 2 (α 2/3 + ξ 2/9
) on A R . Therefore the bounds (4.4) imply (4.2).

We let Ω (1) = (-∞, α + ) and Ω (2) = (α -, +∞) and denote ψ 1 = χ 1 ψ and ψ 2 = χ 2 ψ. We will work out a lower bound of the quadratic form on each of these subdomains.

On both subdomains, using the factorization (4.5), we start from the expression of X α,ξ as (4.9)

X α,ξ = D 2 t + V α,ξ with V α,ξ = N α,ξ (t) 2 t -t 3 (α, ξ) 2 .
On Ω (1) , we bound from below the potential V α,ξ as

(4.10) V α,ξ ≥ B 1 (α, ξ) N α,ξ (t) 2 with B 1 (α, ξ) = min t≤α + t -t 3 (α, ξ) 2 .
Therefore, the quotient

Q α,ξ (ψ 1 )/ ψ 1 2 L 2 (R)
is bounded from below by the ground state energy of (4.11)

D 2 t + B 1 (α, ξ) N α,ξ (t) 2 on R.
Using the canonical form of the factor N α,ξ (4.12)

N α,ξ (t) = 1 3 t + t 3 (α, ξ) 2 - t 3 (α, ξ) 2 12 + ξ t 3 (α, ξ)
.

we find by translation and scaling that the operator (4.11) is isospectral to the operator (4.13)

( 4 9 ) 1/3 B 1 (α, ξ) 1/3 D 2 τ + τ 2 2 -η 2 2
on R, for a suitable real number η. We know from [START_REF]The Montgomery model revisited[END_REF] that the ground state energy γ(η) of the operator D 2 τ + τ 2 2 -η 2 as a function of η ∈ R reaches its minimum (for a positive value η 0 of η). Therefore this minimum is positive. We denote it by M 0 . Finally we bound Q α,ξ (ψ 1 ) from below as follows

(4.14) Q α,ξ (ψ 1 ) ≥ ( 4 9 ) 1/3 B 1 (α, ξ) 1/3 M 0 ψ 1 2 L 2 (R) .
On Ω (2) , we swap the roles of the two factors in P α,ξ and obtain the lower bound:

(4.15) V α,ξ ≥ B 2 (α, ξ) t -t 3 (α, ξ) 2 with B 2 (α, ξ) = min t≥α - N α,ξ (t) 2 .
By translation and homogeneity, we get via the harmonic oscillator:

(4.16) Q α,ξ (ψ 2 ) ≥ B 2 (α, ξ) 1/2 ψ 2 2 L 2 (R) .
Finally, combining (4.14) and (4.16) with (4.8) we find

(4.17) Q α,ξ (ψ) ≥ min ( 4 9 ) 1/3 B 1 (α, ξ) 1/3 M 0 , B 2 (α, ξ) 1/2 - K 2 loc (α + -α -) 2 ψ 2 L 2 (R) .
It remains to choose α + and α -so that we can find suitable lower bounds for the constants B 1 (α, ξ) and B 2 (α, ξ). This will be done finding upper and lower bounds for t 3 (α, ξ).

Proof of (4.4b). In the region A , we have ξ > 2 3 α 3 (and α > 1). With the aim of finding bounds for t 3 (α, ξ) we calculate the derivative of expression (4.1c) with respect to α:

∂ α t 3 (α, ξ) = - 4α 5 -4α 6 + 9ξ 2 1 2 3ξ + -4α 6 + 9ξ 2 -2/3 + 4α 5 -4α 6 + 9ξ 2 1 2 3ξ --4α 6 + 9ξ 2 -2/3
. 4.1.3. Numerical simulations. We have computed an approximation of the band function 1 on a grid Σ of values of (α, ξ) covering the square (-2, 2) 2 . The grid points of Σ are (α k , ξ l ) with α k = -2 + k/100 and ξ l = -2 + l/100, for k, l ∈ {0, . . . , 400}. In Figure 1, we plot the level lines of 1 above the grid Σ and in Figure 2, we plot the same band function restricted on the axis ξ = 0. The computations for Figures 1 and2 are performed by the finite element method 1 and are based on a Galerkin projection on the interval (-5, 5) t with natural boundary conditions at the ends, discretized by 10 elements with polynomial degree 10. With this number of elements the degree 10 saturates the double precision, see Table 1. Enlarging the domain yields numbers 1 All our computations are preformed with the FEM library XLiFE++ under a GNU GPL licence. with the same first 13 digits, which proves that (-5, 5) is large enough to capture the numerical support of the first eigenvectors when (α, ξ) belongs to [-2, 2] 2 .

According to the analysis of [START_REF] Helffer | Spectral properties of higher order anharmonic oscillators[END_REF], the bottom of the spectrum inf S(M [2] ) of the Montgomery operator of order two coincides with 1 (0, 0). We can see in Figure 2, that α = 0 is a local maximum of the function α → 1 (α, 0). Table 1 provides the value 0.660952004868639 for 1 (0, 0) (with presumably 13 correct digits), and the value 0.786 for α 0 with 3 correct digits. Refining the sampling of α by 101 values in the interval [0.786, 0.787] yields α 0 = 0.78628 and 1 (α 0 , 0) = 0.49410921120. 1. Computed values of 1 (α 0 , 0) and 1 (α 0 , 0) with polynomial degree Q on 10 elements in (-5, 5). The numerical value of α 0 is also provided. 4.2. Asymptotic analysis in the small angle limit. In this section, we prove Theorem 1.10. The presentation is mostly inspired by [9, Section 2] and we just highlight the most important steps and differences. 4.2.1. Changes of variables. We are interested in the behavior of the first eigenpair of X ε (defined in (1.4)) as ε → 0. To investigate this, we perform two changes of variables. First, the scaling (4.18) (σ, τ ) -→ (s, t) so that s = εσ, t = τ, brings the spectral analysis of the operator X ε to the following unitarily equivalent operator (4. [START_REF] Helffer | Caractérisation du spectre essentiel de l'opérateur de Schrödinger avec un champ magnétique[END_REF])

Q 1 (0, 0) α 0 1 (α 0 , 0) 1 0.
L ε = D 2 t + εD s + s 2 t - t 3 3 2 .
Second, we localize in s around a point α 0 such that there exists a value ξ 0 for which 1 reaches its minimum S 0 in (α 0 , ξ 0 ). By the new change of variable (4.20) (s, t) -→ (s, t) so that s = α 0 + ε 1/2 s, t = t, and a gauge transform, the operator L ε becomes (4.21)

L ε = D 2 t + ξ 0 + ε 1/2 D s + εs 2 t + 2ε 1/2 α 0 st + α 2 0 t - t 3 3 2 .
Thus the above three operators have the same eigenvalues

S(X ε ) = S(L ε ) = S(L ε ).
Proposition 4.7. For all N ≥ 1, there exist C N > 0 and ε N > 0 such that for all ε ∈ (0, ε N ) there exist at least N eigenvalues (counted with multiplicity) of the operator X ε contained in the ball of radius C N ε centered at S 0 .

Proof. We can write

L ε = L 0 + ε 1/2 L 1 + O(ε), with L 0 = X α 0 ,ξ 0 and L 1 = 2 ξ 0 + α 2 0 t - t 3 3 (D s + 2α 0 st) .
We are looking for quasimodes in the form

ψ = ψ 0 + ε 1 2 ψ 1 and κ = κ 0 + ε 1 2 κ 1 , such that L ε ψ = κψ + O(ε) is satisfied.
Gathering the terms in ε 0 we get the equation

X α 0 ,ξ 0 ψ 0 = κ 0 ψ 0 .
Thus κ 0 is in the spectrum of X α 0 ,ξ 0 and that ψ 0 is an associated eigenfunction. We choose κ 0 = S 0 , and we take ψ 0 (unitary) in the following form (4.22) ψ 0 (s, t) = f 0 (s) u 0 (t) , with u 0 = u α 0 ,ξ 0 and f 0 in the Schwartz class. Gathering the terms in ε 1/2 , we get the equation

(4.23) (L 0 -S 0 )ψ 1 = -(L 1 -κ 1 )ψ 0 .
A careful check of the proof of [19, Théorème (1.1)], involving only commutators of analytic vector fields with respect to ε, shows that C 2 does not depend on ε ∈ (0, 1). The extension of this estimate to the form domain follows by density.

We can reformulate Lemma 4.8 in terms of the operator L ε and deduce the following.

Proposition 4.9. There exists C 2 > 0 such that, for all ε ∈ (0, 1), and all ψ in the form domain of L ε ,

(4.27) |s 2 -t 2 | + 2|t| 1/4 ψ 2 ≤ C 2 L ε ψ, ψ + ψ 2 .
In particular, for all S * 0 > S 0 there exists R * > 0 such that, for all ε ∈ (0, 1), L Dir,R * ε ≥ S * 0 , where L Dir,R * ε is the Dirichlet realization of the operator L ε outside the ball of center 0 and radius R * .

Proof. To deduce the second assertion, it suffices to note that lim

|s|+|t|→+∞ |s 2 -t 2 | + 2|t| = +∞.
It is classical to deduce from Proposition 4.9 that the eigenfunctions associated with the low lying eigenvalues satisfy Agmon estimates with respect to (s, t) (see, for instance, [9, Section 2.2]). By taking derivatives of the eigenvalue equation, we finally get the following corollary.

Corollary 4.10. Let C 0 and k, l, d ∈ N. There exist ε 0 > 0, C > 0, and c 0 > 0 such that for all ε ∈ (0, ε 0 ) and all eigenpairs (κ, ψ) of the operator L ε with κ ≤ S 0 + C 0 ε, we have

t k s l ψ ≤ C ψ , Q ε (t k s l ψ) ≤ C ψ 2 , (D t ) d s l t k ψ ≤ C ψ , (εD s ) d s l t k ψ) ≤ C ψ .
4.2.3. Coherent states. Following a classical formalism, see [START_REF] Berezin | Wick and anti-Wick symbols of operators[END_REF][START_REF] Combescure | Coherent states and applications in mathematical physics[END_REF] for instance, we introduce the annihilation operator a and the creation operator a

* a = 1 √ 2 (s + ∂ s ) , a * = 1 √ 2 (s -∂ s ) .
We have the following identities

(4.28) s = a + a * √ 2 , ∂ s = a -a * √ 2
, and [a, a * ] = 1.

Setting g 0 (s) = π -1/4 e -s 2 /2 , we introduce the coherent states for any u ∈ R and p ∈ R,

f u,p (s) = e ips g 0 (s -u) ,
and for all ψ ∈ L 2 (R), the associated projection defined by

Π u,p ψ = ψ, f u,p L 2 (R,ds) f u,p .
We have the resolution of the identity

ψ = R 2 Π u,p ψ dudp ,
and the Parseval formula

ψ 2 = R R 2
|Π u,p ψ| 2 dudp ds .

Lemma 4.11 ([4]

). For all n, m ∈ N, we have

(a) m (a * ) n = R 2 u + ip √ 2 m u -ip √ 2 n Π u,p dudp.
Now we have all tools at hand to end the proof of Theorem 1.10. Taking Proposition 4.7 into account, it remains to prove the following lemma: Lemma 4.12. Consider an eigenfunction Ψ ε of X ε associated with an eigenvalue κ(ε) ≤ S 0 + C 0 ε. Then there exists C only depending on C 0 such that

(4.29) κ(ε) Ψ ε 2 = X ε Ψ ε , Ψ ε ≥ (S 0 -Cε) Ψ ε 2 .
Proof. In view of (4.19), it suffices to prove (4.29) for an eigenpair (κ(ε), ψ ε ) of the operator L ε . By the scaling s = ε 1 2 s, L ε becomes the unitarily equivalent operator

L ε = D 2 t + ε 1/2 D s + εs 2 t - t 3 3
2 , which we expand as

L ε = L 0 + ε 1/2 L 1 + ε L 2 + ε 3/2 L 3 + ε 2 L 4 , where
L 0 = D 2 t + t 6 9 , L 1 = - 2t 3 3 D s , L 2 = - 2 3 t 4 s 2 + D 2 s , L 3 = t s 2 D s + D s s 2 , L 4 = s 4 t 2 .
We use (4.28) and we commute a and a * to put all the a * on the right. With Lemma 4.11, we get that there exist (γ, δ) ∈ R 2 , a homogeneous polynomial of order 1, L(X, Y ), and a non-commutative homogeneous polynomial of order 2, P (X, Y ), such that (4.30)

L ε = L W ε + εL 2,rem + ε 3/2 L 3,rem + ε 2 L 4,rem , where L W ε = R 2 X u √ ε, p √ ε Π u,p dudp ,
and L 2,rem = γt 4 + δ, L 3,rem = tL(a, a * ), L 4,rem = t 2 P (a, a * ) .

Consider an eigenfunction ψ ε of L ε associated with an eigenvalue κ(ε) ≤ S 0 + C 0 ε. From (4.30) and using the quadratic forms

Q ε of L ε and Q α,ξ of X α,ξ , we get (4.31) Q ε ( ψ ε ) ≥ R R 2 Q u √ ε, p √ ε (Π u,p ψ ε ) dudp ds -ε (γt 4 + δ) ψ ε ψ ε -ε 3/2 L(a, a * ) ψ ε t ψ ε -ε 2 P (a, a * ) ψ ε t 2 ψ ε .
Since S 0 is a minimum for the band function of X α,ξ , we have, for each (u, p) ∈ R 2 and s ∈ R, the bound from below

Q u √ ε, p √ ε (Π u,p ψ ε )(s, •) ≥ S 0 Π u,p ψ ε (s, •) 2 L 2 (R,dt) . Thus by Parseval formula (4.32) R R 2 Q u √ ε, p √ ε (Π u,p ψ ε ) dudp ds ≥ S 0 ψ ε 2 L 2 (R 2 ,dsdt) .
Let us estimate the remainder terms in (4.31). The homogeneity of the polynomials P and L yields ε 3/2 L(a, a * ) ψ ε = εL(ε 1/2 a, ε 1/2 a * ) ψ ε and ε 2 P (a, a * ) ψ ε = εP (ε 1/2 a, ε 1/2 a * ) ψ ε .

Thus, using the scaling s = ε 1/2 s and Corollary 4.10, we find ε (γt 4 + δ) ψ ε ψ ε + ε 3/2 L(a, a * ) ψ ε t ψ ε + ε 2 P (a, a * ) ψ ε t 2 ψ ε ≤ Cε ψ ε 2 .

From this, combined with (4.31) and (4.32), follows that

κ(ε) ψ ε 2 = Q ε ( ψ ε ) ≥ (S 0 -Cε) ψ ε 2 .
Coming back to the operator L ε and finally to the operator X ε , this ends the proof of Lemma 4.12, and thus of Theorem 1.10.

4.2.4.

Computation of the ground states of X ε . In the last section of this paper, we present computations of the first eigenpair of the operator X ε for a decreasing sequence of values of ε.

Let us agree that ε = 2 -1-/2 , ≥ 0 so that ε 0 = 1/2, ε 1 = √ 2/4, ε 2 = 1/4, etc. . . We have computed the first eigenpair of X ε for ε = ε , ∈ {0, 1, . . . , 12} with natural boundary conditions on the domain [-a , a ] × [-8, 8] (with a = 4/ε = 8 • 2 /2 ) by a finite element discretization on a uniform rectangular grid of 48 × 6 elements of partial degree 10 in each variable.

Theorem 1.10 yields the convergence of κ 1 (ε) to S 0 at a rate of O(ε). Our computations (see Figures 1 and2) suggest that S 0 0.4941. In Figure 3 we plot the difference κ 1 (ε ) -0.4941 versus ε (here we use a log -log scale in base 2). Figure 4 gives the modulus of the first eigenvector of the operator X ε and a numerical value of the eigenvalue κ 1 (ε), for ε = ε with = 0, . . . , 5. For ε = ε , the horizontal scale (variable σ) is [-a , a ] with a = 5 • 2 /2 and the vertical scale (variable τ ) is always [-5, 5]. The proportion between the two scales is kept, which makes the plot of the vertical scale shrink as increases.

In Figure 5, we zoom some region around the point α 0 /ε of the plot of the first eigenvector of X ε when ε = ε with = 3, . . . , 10. When = 6, . . . , 10, we have computed the eigenvector on the smaller region [-a 2 , a 2 ] × [-4, 4] on a uniform rectangular grid of 48 × 6 elements of partial degree 10 in each variable. We represent the modulus of the eigenvector on the region

α 0 ε - 2 √ ε , α 0 ε + 2 √ ε × [-3, 3] .
This choice is driven by the structure of the first term (s, t) → ψ 0 (s, t) = f 0 (s)u 0 (t) (4.22) of a possible eigenvector asymptotics for the operator L ε (4.21). We note that with ξ 0 = 0, we do not need any gauge transform to go from L ε to L ε . Thus, using the change of variables (4.20), we find that ψ 0 (s, t) = f 0 s -α 0 √ ε u 0 (t).

Going back to the "physical variables" σ, τ with (4.18), we find ψ 0 (s, t) = f 0 εσ -α 0 √ ε u 0 (τ ).

Taking σ ∈ [ α 0 ε -2 √ ε , α 0 ε + 2 √ ε ],
we see that εσ-α 0 √ ε spans the fixed interval [-2, 2]. Hence we expect that the zoom can yield the image of a convergence as ε → 0. Indeed, this is exactly what we can detect from Figure 5.

Thus, in the small angle limit ε → 0, there are two centers of localization that are spread at the scale ε -1/2 and go away at the order 1 ε . The "area of localization" of the first eigenfunctions goes to infinity. This fact is quite understandable. Considering that the limit ε = 0 is singular (the two lines of cancellation become identical and we recover the Montgomery operator M [2] whose essential spectrum is non-empty); it is rather natural to observe a "loss of mass at infinity", characteristic of Weyl sequences.

x 0 h, 1 4 ) 4 .

 044 and P A h,Ω . The sole difference consists in Agmon estimates for the truncated eigenvectors of P A h,Ω . Thanks to the right part of (3.4), Agmon estimates (1.10) hold and then Lemma 3.2 with δ = d ∈ ( 3 16 and m = 8, from which we deduce |µ n,n | ≤ C(h 3/4 h 4d + h 8d ). Choosing d = 3 16 + d 0 4 , we obtain the left part of (3.4). 3.Expansion of eigenvalues. Putting together Lemmas 3.3 and 3.4, we can see that Lemma 3.4 provides the bound λ loc n (h) ≤ Lh 3/2 which validates the application of Lemma 3.3. Therefore, we have now proved: Lemma 3.5. Under Assumption 1.2, for any d < 1

8 FIGURE 1 .

 81 FIGURE 1. Contour lines for the numerical values of 1 (α, ξ) on the grid Σ, with a zoom in a neighborhood of the two points where the minimum is attained. The curves {α → ξ = ± 2 3 α 3 } are also represented.

4 FIGURE 2 .

 42 FIGURE 2. Numerical values for 1 (α, ξ) on the axis ξ = 0.

1 FIGURE 3 .

 13 FIGURE 3. Plot of log 2 (κ 1 (ε ) -0.495) versus log 2 (ε ).

κ 1

 1 = 0.7039 ( = 0) κ 1 = 0.6563 ( = 1) κ 1 = 0.6266 ( = 2) κ 1 = 0.6063 ( = 3) κ 1 = 0.5892 ( = 4) κ 1 = 0.5712 ( = 5)

FIGURE 4 .

 4 FIGURE 4. Modulus of the first eigenvector of X ε for = 0, . . . , 5. The white cross represents the two lines of cancellation of the magnetic field and the black dots mark the two points (±α 0 /ε , 0). The rectangular boxes in the second row indicate the zoom region displayed in Figure 5.

κ 1 FIGURE 5 .

 15 FIGURE 5. Modulus of the first eigenvector of X ε for = 3, . . . , 10. The white cross represents the two lines of cancellation of the magnetic field and the black dot marks the point (α 0 /ε , 0).

  Agmon estimates are an a priori result of exponential decay of the eigenfunctions. The following theorem states that the eigenfunctions associated with eigenvalues of order h 4/3+2d/3 are localized near Σ.

	3 4 + 4d} > 3 2	and 4 3 + 2d 3 < 3 2 .
	Theorem 1.7. Let L > 0 and d ∈ ( 3 16 , 1 4

Proof of (4.4a). We recall that, on the set {ξ > 2 3 α 3 ≥ 0}, the polynomial P α,ξ admits a unique real root, denoted t 3 (α, ξ). Using that P α,ξ (t 3 (α, ξ)) = 0, we immediately check that we have the factorization (4.5) P α,ξ (t) = -N α,ξ (t) t -t 3 (α, ξ) with N α,ξ (t) = 1 3 t 2 + t 3 (α, ξ) 3 t + ξ t 3 (α, ξ) .

The factor N α,ξ (t) is positive for all t when ξ > 2 3 α 3 and we have ∂ t N α,ξ (t) = 0 for t = -t 3 (α,ξ) 2 . Therefore, for all ξ > 2 3 α 3 , we have the lower bound

.

is bounded from below by the ground state energy of the operator

By considering the expression of t 3 (α, ξ) given in (4.1c), we get, uniformly in α ∈ [0, 1],

Hence, there exist constants B > 0 and K > 0 such that for all ξ > K (with ξ > 2 3 α 3 and 0 ≤ α ≤ 1)

For all ξ > K,

is bounded from below by the ground state energy of D 2 t + B 2 ξ 4/3 (t -t 3 (α, ξ)) 2 on R. By translation and homogeneity, we get (using the harmonic oscillator):

. This concludes the proof of the estimate (4.4a).

Preliminaries for the proof of (4.4b) and (4.4c). For the proof of estimates (4.4b) and (4.4c), we use a quadratic partition of unity (χ 1 , χ 2 ) on R in order to isolate the root t 3 (α, ξ) from the other two roots of P α,ξ . For this we choose two real numbers α -and α + such that

and we take the two functions χ 1 and χ 2 such that χ 2 1 + χ 2 2 = 1 on R and

with the control of their derivatives

The localization formula (see (2.4)) gives, for all function ψ in the form domain,

We can see that the modulus of the second term is larger than the modulus of the first term. Hence the positivity of the derivative ∂ α t 3 (α, ξ). Therefore

, ξ , ∀ξ > 0. With (4.1c) (and using again that ξ > 2 3 α 3 ), we deduce

We choose

α -= 0 and α + = α.

Thus, we get for the constants B 1 (α, ξ) and B 2 (α, ξ) appearing in (4.10) and (4.15):

Then (4.17) yields

Proof of (4.4c). In the region A , 2 3 α 3 ≥ ξ and the polynomial P α,ξ has three real roots. We note that P α,ξ (t) → -∞ as t → +∞ and P α,ξ (t) → +∞ as t → -∞. We check that

Now we choose α -= 1 2 α and α + = α, and we get for the constants B 1 (α, ξ) and B 2 (α, ξ):

Then (4.17) yields

The proof of Lemma 4.6 is now achieved, hence Theorem 1.9 is proved.

We have

At this point, we introduce the notation

Taking the derivative of the equation X α,ξ u α,ξ = 1 (α, ξ) u α,ξ with respect to α and ξ, we obtain

In particular, for (α, ξ) equal to the critical point (α 0 , ξ 0 ) we find that

3 u α 0 ,ξ 0 . Thus we have obtained explicit solutions of equation (4.23) as:

Therefore, for any function ψ in the form

where ψ 0 and ψ 1 are respectively given by in (4.22) and (4.25), we have

In the construction procedure of ψ 0 and ψ 1 , the function f 0 is left undetermined. Therefore the above estimate holds on finite dimensional spaces of arbitrary dimensions, which ends the proof of the proposition.

where m ε (σ, τ ) = |ε 2 σ 2 -τ 2 | + 2|σ|ε 2 + 2|τ | .

Proof. The lemma follows from [START_REF] Helffer | Caractérisation du spectre essentiel de l'opérateur de Schrödinger avec un champ magnétique[END_REF]Théorème (1.1)]. We only have to check the assumptions of the theorem and the uniformity with respect to ε. Here the magnetic field has one component B = ε 2 σ 2 -τ 2 . We take the integer r used there as 1. The condition [19, (1.9)] is trivially satisfied for C 1 = 8 2ε 2 + 2 ≤ C 1 (m(σ, τ ) + 1) . Then [19, (1.11)] tells us that ∀ψ ∈ C ∞ 0 (R 2 ), m(σ, τ ) 1/4 ψ 2 ≤ C 2 X ε ψ, ψ + ψ 2 .


Panagiota Perlepe

Itziar Oyarzabal
email: oyarzabal@crpp-bordeaux.cnrs.fr

S Kasper

Philippe Pedersen

Denise Négrier

Mathieu Mondieig

Elizabeth Rouzieres

Fabrice Hillard

Andrei Wilhelm

Elizaveta Rogalev

Suturina

Kasper S Pedersen

Philippe Negrier

Denise Mondieig

Mathieu Rouzières

Elizabeth A Hillard

Fabrice Wilhelm

Andrei Rogalev

Elizaveta A Suturina

Corine Mathonière
email: corine.mathoniere@icmcb.cnrs.fr

Rodolphe Clérac
email: clerac@crpp-bordeaux.cnrs.fr

Cr(pyrazine)2(OSO2CH3)2: A two-dimensional coordination polymer with an antiferromagnetic ground state

Keywords: coordination polymers, pyrazine, magnetic order, X-ray spectroscopy

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

Introduction

Transition metal pyrazine-based coordination polymers have been extensively studied, with many examples of various dimensionalities already reported in the literature [1].

Specific attention has been given to {M II (pyrazine)2L2-nX -n}Y -2-n (n = 0, 1 or 2) and {M II (pyrazine)2X n-}Y -2-n (n = 1 or 2) systems (M = metal ion, L = neutral coligand, X = anionic coligand, Y = counteranion), which all feature parallel sheets containing a periodic square or quasi-square array of metal ions linked by pyrazine ligands. The final dimensionality of these coordination compounds is conditioned by the denticity of the coligands (L or X); monodentate ligands lead to two-dimensional (2D) networks [2], while bidentate ligands result in three-dimensional (3D) systems [3]. In the latter case, the counterions (Y) are located inside the voids and ensure the neutrality of the system when necessary. From the point of view of the magnetic properties, the exchange interactions between the metal centers through the pyrazine ligands are usually weak, often giving rise to a 3D ordered antiferromagnetic ground state at low temperatures [2,3,4].

Very recently, our group reported the synthesis and characterization of an air-stable 2D coordination polymer based on chromium metal ions that belongs to this abovementioned family of compounds [5]. The reaction of CrCl2 with a large excess of pyrazine at 200 °C affords Cr(pyrazine)2Cl2 which incorporates Cr III metal ions and a singly reduced pyrazine scaffold, thanks to the redox non-innocence of the pyrazine ligand and the reducing ability of the chromium(II) used in the synthesis [6]. This unique electronic structure and the resulting strong degree of p-d conjugation in the Cr III /pyrazine layers induce an enormous Cr 3+ -radical exchange interaction (around -3000 K with the -2J convention) that stabilizes a 3D ferrimagnetic order at relatively high temperature (55 K).

Concomitantly, the remarkable electronic delocalization within the chromium/pyrazine layer leads to a high electrical conductivity of 32 mS cm -1 at room temperature, which constitutes one of the highest conductivity values reported so far for coordination solids [7,8]. This unique material, in which a ferrimagnetic order and high electrical conductivity coexist [5], encouraged us to develop an intense research activity in order to modulate and eventually improve the physical properties of these 2D materials. With the aim to study the effect of the axial ligands on the magnetic and electronic properties of analogous chromium/pyrazine 2D systems, we report herein the synthesis and characterization of Cr(pyrazine)2(OSO2CH3)2 for which the chloride coligands of Cr(pyrazine)2Cl2 have been replaced by methanesulfonate anions.

Experimental section

Materials and physical measurements

All reactions were performed under a dry argon or nitrogen atmosphere using Schlenk or glovebox techniques. Acetonitrile was purified with an Innovative Technologies solventpurification system. All chemicals were purchased from Sigma-Aldrich (chromium powder, 99.5%, ~100 mesh and pyrazine, ≥ 99 %) or Alfa Aesar (methanesulfonic acid, ≥ 98 %) and used as received.

CHNS elemental analyses were performed by the Service d'Analyse Elémentaire of the University of Lorraine, Nancy. The FT-IR spectra were recorded in the range 600-4000 cm -1 on a Thermo Scientific Nicolet TM 6700 ATR (attenuated total reflection) spectrometer equipped with a Smart iTR diamond window (Figure S2). Raman spectroscopic characterization was carried out on a Horiba Jobin Yvon Xplora microscope equipped with a cooled Andor CCD detector (Figure S3). X-ray absorption (XAS) spectra were obtained at the ID12 beamline of the European Synchrotron Radiation Facility (ESRF). The fundamental harmonic of an Apple-II type undulator was used for the experiments at the Cr K-edge. All XAS spectra were recorded using total fluorescence yield detection mode and were subsequently corrected for reabsorption effects. Thermogravimetric analysis (TGA) of Cr(OSO2CH3)2(H2O)4 was carried out under an argon flow at a rate of 5 °C/min on a TA Q50 thermobalance (Figure S4). Magnetic measurements were performed on a Quantum Design SQUID magnetometer MPMS-XL operating between 1.8 and 300 K for applied dc fields ranging from -7 to 7

T. An M vs H measurement was carried out at 100 K to confirm the absence of ferromagnetic impurities. The measurements were performed on a microcrystalline sample (12.90 mg) sealed in polypropylene bag (size and mass: 3 × 0.5 × 0.02 cm; 27.6 mg). The magnetic data were corrected for the sample holder and intrinsic diamagnetic contributions of the sample. Heat capacity was measured on a manually pressed pellet of Cr(pyrazine)2(OSO2CH3)2 (7.0 mg) between 298 and 2 K by a thermal relaxation technique with a Quantum Design Physical Property Measurement System (PPMS-9) in zero-dc field. A blank including a small amount of Apiezon N grease (1.5 mg) used for thermal contact was measured prior to sample mounting and subtracted from the measured total heat capacity.

Synthesis of anhydrous Cr(OSO2CH3)2

A mixture of chromium powder (2.0 g, 38 mmol) and methanesulfonic acid (6.0 g, 62 mmol) in 7 mL of water was stirred under reflux overnight, to yield a bright blue solution.

While still hot, the solution was filtered under argon to remove the unreacted Cr powder and allowed to cool to room temperature. The solvent was then removed under reduced pressure and the obtained powder was heated under dynamic vacuum at 180 °C for 7 hours. Yield: 55%. Elemental analysis calc. for C2H6CrO6S2 (242.19 g mol -1 ) C: 9.92%, H: 2.50%, S: 26.48%; found C: 9.52%, H: 2.42%, S: 26.76%. FT-IR (𝑣̅ , cm -1 ): 3034(w), 2944(w), 1419(w), 1334(m), 1270(m), 1236(w), 1170(w), 1146(m), 1122(m), 1068(s), 981(m), 785(s).

Synthesis of Cr(pyrazine)2(OSO2CH3)2

Anhydrous Cr(OSO2CH3)2 (0.20 g, 0.83 mmol) and pyrazine (1.0 g, 12.5 mmol) were placed inside a 30 mL Teflon-lined solvothermal reactor and allowed to react for 15 hours at 200 °C in an oven. The reactor was cooled to room temperature over about 2 hours and the brown microcrystalline product was washed with 20 mL of acetonitrile and dried in vacuo. Yield: 60-65%. Elemental analysis calc. for C10H14CrN4O6S2 (402.37 g mol -1 ) C: 29.85%, H: 3.51%, N: 13.92%, S: 15.94%; found C: 29.59%, H: 3.49%, N: 13.90%, S: 16.09%. FT-IR (𝑣̅ , cm -1 ): 3108(w), 3019(w), 2930(w), 1490(w), 1421(s), 1343(w), 1251(s), 1160(s), 1139(s), 1115(s), 1071(m), 1033(s), 837(m), 805(m), 775(s), 559(s).

Raman (l = 785 nm) (𝑣 $ , cm -1 ): 671,708,785,872,1034,1053,1166,1244,1431,1525.

X-ray powder diffraction

High-resolution X-ray powder diffraction measurements using the Debye-Scherrer geometry and transmission mode were performed with a horizontally mounted INEL cylindrical position sensitive detector (CPS-120) made of 4096 channels [9].

Monochromatic Cu-Kα1 (l = 1.54056 Å) radiation was selected by means of an asymmetrically focusing incident beam curved quartz monochromator. The generator power was set to 1.0 kW (40 kV and 25 mA). External calibration was performed by means of cubic phase NAC (Na2Ca2Al2F14) mixed with silver behenate to convert the channels into angles by means of cubic spline fittings, providing an angular step of 0.029° (2q) between 0° and 120°. The powder sample was introduced at 297 K into a 0.5 mm diameter Lindemann glass capillary which was rotated around its longitudinal axes during data collection to prevent the effects of the preferred orientations. The acquisition time was 10 hours. Indexing of the X-ray powder diffraction pattern, Pawley refinement, structure solution by the rigid body direct space method using the Cu based isostructural compound as reference [10], and final Rietveld refinement were performed with the Materials Studio program [START_REF] Modeling | Materials Studio, version 5.5[END_REF]. Crystal data are summarized in Table S1 and the powder diffractogram can be found in the Supplementary Materials (Figure S1).

Computational details

Quintet, triplet and singlet states were computed for neutral molecular fragments, {Cr(pyrazine)4(OSO2CH3)2} and {Cr(pyrazine)4Cl2}, taken from the crystal structures (B3LYP/def2-TZVP) [START_REF] Weigend | [END_REF]. Exchange interactions between Cr(II) sites along the crystallographic a-and c-axes were calculated using the broken-symmetry approach [START_REF] Yamaguchi | Applied Quantum Chemistry[END_REF][START_REF] Soda | [END_REF]. Ab initio ligand field analysis and calculation of g-and D-tensors were done with the SOC-CASSCF(4,5)/def2-TZVP approach [15]. ORCA 4.0 was used for all calculations [START_REF] Neese | The ORCA program system[END_REF].

Results and discussion

Transition metal methanesulfonate salts are usually prepared through reactions between the corresponding metal oxides, carbonates and/or chlorides and methanesulfonic acid [17]. These reactions give M(OSO2CH3)2(H2O)4 complexes, where four water molecules occupy the equatorial plane and two axial methanesulfonate groups complete the octahedral coordination sphere [18]. Thermogravimetric analysis has shown that dehydration of these compounds occurs in two steps, each of them corresponding to the loss of two water molecules. The anhydrous metal methanesulfonate salts are generally stable over a wide temperature range before decomposition [19].

The Cr methanesulfonate analogue was obtained by the reaction of metallic Cr and methanesulfonic acid and isolated for the first time. Although single-crystal X-ray crystallographic analysis was not possible due to sensitivity to air and moisture, a variety of analytical techniques lead us to conclude that the formula of this methanesulfonate salt is indeed Cr(OSO2CH3)2(H2O)4. Infrared spectroscopy unambiguously reveals the presence of the sulfonate groups. The asymmetric and symmetric SO2 stretching vibrations appear in the 1270-1146 cm -1 region, and at 1068 cm -1 , respectively, while the C-S stretching vibration manifests itself at 785 cm -1 (Figure S2). Dehydration of this compound occurs in two successive steps (Figure S4), each corresponding to the loss of two water molecules, as reported in the literature for other 3d methanesulfonate salts. The eventual loss of a total of four water molecules at 140 °C confirms their initial presence in Cr(OSO2CH3)2(H2O)4.

The Cr(pyrazine)2(OSO2CH3)2 network was synthesized similarly to the Cr(pyrazine)2Cl2 parent compound [5], by means of a solvothermal reaction of the dehydrated Cr II /methanesulfonate salt and pyrazine (see experimental section). Importantly, the employment of anhydrous Cr(OSO2CH3)2 is crucial for the isolation of the final 2D compound as the reaction between the hydrated Cr II methanesulfonate and an excess of pyrazine does not yield the desired product.

Crystal structure description of Cr(pyrazine)2(OSO2CH3)2

The crystal structure of Cr(pyrazine)2(OSO2CH3)2 (Figures 1 and2) was solved from the Rietveld refinement of the X-ray powder diffraction data, which shows that it crystallizes in the orthorhombic Pnnm space group (see Supplementary Materials; Table S1 and Similar bond distances were observed in the related Cu(pyrazine)2(OSO2CH3)2 layered structure [10], where a comparable distortion is observed for the Cu II centers. Also supporting the presence of Cr II centers in Cr(pyrazine)2(OSO2CH3)2, these results contrast considerably with the structural parameters found in the Cr III (pyrazine)2Cl2 analogue, in which (i) the 2D network is closer to a square lattice with similar Cr×××Cr distances of 6.90351(4) and 6.97713(4) Å and Cr-N bond lengths of 2.003(2) and 2.059(2) Å and (ii) a significant axial distortion is also observed but perpendicularly to the Cr/pyrazine layer along the Cl-Cr-Cl axis with a Cr-Cl bond length of 2.337(1) Å [5].

Electronic structure analysis

DFT calculations capture a significant difference in the electronic structure of the molecular model fragment, {Cr(pyrazine)4(L)2}, depending on the axial ligand, L, chloride [5] versus methanesulfonate anions shown in Figure 2. In the case of chloride [5], the triplet (ST = 1) ground state is about 19 kcal mol -1 (9560 K) lower in energy than the quintet state and about 40 kcal mol -1 (20100 K) lower than the singlet one. Mulliken population analysis shows the presence of 3.18 and 2.74 unpaired electrons on the Cr site, indicating a {Cr III [(pyrazine)4] • Cl2} moiety with one unpaired electron smeared over the four pyrazine ligands for both the ST = 2 and ST = 1 states, as previously found [5]. In contrast, the axial methanesulfonate anions lead to an ST = 2 ground state, which is 22 kcal mol -1 (11100 K) lower than the triplet state and 70 kcal mol -1 (24500 K) lower than the singlet one. The Mulliken population is estimated at 4.03 unpaired electrons on the Cr site in agreement with the presence of a {Cr II (pyrazine)4(OSO2CH3)2} moiety featuring a non-reduced pyrazine scaffold. Ab initio ligand field analysis shows that the overall splitting of the d-orbitals is significantly different and quite smaller in the presence of axial methanesulfonate ligands in comparison to the chloride case (Figure 3), as expected for the lower oxidation state and the Jahn-Teller distortion in the methanesulfonate case. The {Cr II (pyrazine)4(OSO2CH3)2} moiety has the ligand field z-axis placed along the elongated Cr-N2 bonds, leading to a pronounced splitting of the dx2-y2 and dz2 orbitals (Figures 3 andS7).

Spectroscopic confirmation of the chromium oxidation state

In the recently reported coordination polymer Cr(pyrazine)2Cl2 [5], the pyrazine ligands are partially reduced by the Cr II metal ions, resulting in the oxidation of the chromium centers in the final material. To explore this possible redox phenomenon in Cr(pyrazine)2(OSO2CH3)2, X-ray absorption spectroscopy (XAS) measurements were performed at the Cr K-edge to confirm the +II-oxidation state of the chromium sites suggested by the X-ray structure analysis and DFT calculations (vide supra). XAS data were also collected for two reference compounds, trans-[Cr II Cl2(NCNH2)4] (Cr(II)) [20] and trans-[Cr III (pyridine)4Cl2](ClO4)•1/4H2O (Cr(III)) [5] in order to make a direct comparison of the XAS signature for Cr(pyrazine)2(OSO2CH3)2 and chromium-based complexes which have a well-established oxidation state. Systematically, the experimental data show intense rising edge features, corresponding to 1s → 4p transitions. The dipole-forbidden 1s → 3d transitions (Figure 4) are also observed at the pre-edge but with a much weaker intensity. As previously discussed in the literature [5,21], the first pre-edge transition is a fingerprint of the Cr oxidation state independently of the ligand field. A close examination of the XAS results reveals important similarities between Cr(pyrazine)2(OSO2CH3)2 and Cr(II), with the energy of the Cr II pre-edge transition in Cr(II) being only 0.2 eV higher in energy than in the present 2D compound.

Therefore, both the crystallographic and spectroscopic characterization demonstrate the presence of Cr II centers in Cr(pyrazine)2(OSO2CH3)2.

Magnetic properties

The temperature dependence of the magnetic susceptibility of Cr(pyrazine)2(OSO2CH3)2 was measured between 280 and 1.8 K under an applied dc field of 0.1 T (Figure 5). The molar magnetic susceptibility-temperature product, cT, is 2.9 cm 3 K mol -1 at 280 K, in good agreement with the presence of a high-spin S = 2 Cr II site (C = 3 cm 3 K mol -1 with g = 2) as already concluded from theoretical, crystallographic and spectroscopic studies (vide supra). The cT product steadily decreases with decreasing temperature and reaches a value of 0.08 cm 3 K mol -1 at 1.85 K. Such thermal behavior indicates the presence of predominant antiferromagnetic interactions between the magnetic chromium centers, which is also supported by a broad maximum of the c vs. T plot at 20 K (Figure 5). This magnetic feature is indeed commonly observed in pyrazine-based 2D quadratic-layered compounds in the presence of intra-layer antiferromagnetic interactions [2c]. The magnetic behavior of Cr(pyrazine)2(OSO2CH3)2 can also be discussed from the c -1 vs T plot (Figure S5) which shows a linear relationship in the 30-280 K temperature range consistent with a Curie-Weiss law (g = 2.11(5) and q = -40 K) and dominating antiferromagnetic interactions. Considering the 2D arrangement of the magnetic chromium centers in this system, the susceptibility data were fitted to Lines' model, which is usually applied to describe the paramagnetic phase of 2D isotropic or weakly anisotropic quadratic-layer antiferromagnets (using the spin Hamiltonian: ℋ = -2J2DSSi•Sj and the expression obtained from high-temperature series expansion techniques) [22]. This model assumes that the magnetic couplings, Ja and Jc, in the two directions of the coordinating layer are equivalent and therefore, only an average J2D value is expected in the present case. The experimental data can be nicely reproduced by this approach down to 15 K (solid red lines in Figure 5), yielding g = 2.03( 5) and an average intralayer coupling constant, J2D/kB = -1.73( 5) K. As the Cr•••Cr distances through the pyrazine ligands are indeed significantly different in the two crystallographic directions (8.086(2) Å versus 7.121(2) Å along the a and c axes, respectively; vide supra), complementary Heisenberg S = 2 spin chain models developed by Fisher [23] and Quantum Monte Carlo (QMC) [24] methods were also considered. With these approaches, the magnetic interaction, Ja, between Cr centers along the a direction is neglected, but the theory/experiment agreement (solid blue and green lines in Figure 5) is as satisfactory as the Lines' model above 15 K, yielding J1D/kB = -3.26( 5) K (g = 2.04( 5)) and J1D/kB = -3.11( 5) K (g = 2.03( 5)) for the Fisher and QMC models, respectively. Of course, these 2D and 1D models, which involve different approximations, do not give the same theoretical value of the magnetic interaction through the pyrazine ligand. While the 2D model gives J2D, an average estimation of Ja and Jc, the 1D models overestimate the main interaction Jc in order to compensate for the absence of Ja in this approach. Therefore, J2D (-1.73 K) and J1D (≈ -3.2 K) should be considered as upper limits of the weakest and strongest couplings, most likely Ja and Jc, in Cr(pyrazine)2(OSO2CH3)2. Interestingly, DFT calculations preferentially support the above 1D models, showing that the exchange interaction along the a-axis should be much smaller than along the c-axis with the following calculated values: Ja/kB = -0.01 K and Jc/kB = -5 K. It is worth mentioning that CASSCF calculations allow also an estimation of the local Cr II g-tensor [1.96, 1.99, 1.99], its magnetic anisotropy parameters D/kB = -3 K and E/D = 0.07 (using the following Hamiltonian: ℋA = DSCr,z 2 +E(SCr,x 2 -SCr,y 2 )). Due to the magnitude of the local anisotropy, its effects are not expected to influence the magnetic susceptibility above 15 K. In order to probe a possible magnetic phase transition below 15 K, heat capacity measurements were performed on Cr(pyrazine)2(OSO2CH3)2. As shown in Figure 6, a reproducible broad feature is observed in the 5-15 K temperature range, confirming the presence of an ordered magnetic ground state. The broadness of this feature could result from the thermal excitations among S = 2 microspin states (Schottky anomaly) [25] and/or from large spin correlations always present in low dimensional systems above the transition temperature [26]. In order to further analyze these calorimetric measurements, the magnetic component (Cpm, Figure 6) of the heat capacity was separated from the baseline modeled to an empirical polynomial expression (red line in Figure 6). This nonmagnetic background, mainly composed of the lattice contribution, was then subtracted from the measured heat capacity data (Cp) and the resulting magnetic component of Cp (Cpm) was plotted as shown in Figure 6 (blue dots). A relatively broad magnetic peak with a maximum centered at 9.7 K is clearly detected. The integration of Cpm/T in the 2-30 K temperature range leads an associated magnetic entropy (Sm), which reaches a saturation value of 2.7 J K -1 mol -1 (Figure S5). This estimation is lower than the calculated entropy from the simple Rln(2S + 1) relation (13.4 J K -1 mol -1 ). This discrepancy is not surprising considering the large entropy loss above the transition temperature due to the presence of short range correlations always present in low dimensional magnetic systems [26]. The combined heat capacity and magnetic measurements reveal unambiguously the occurrence of long-range antiferromagnetic order in Cr(pyrazine)2(OSO2CH3)2, with a Néel temperature around 10 K.

Conclusions

A novel Cr/pyrazine-based 2D coordination network has been synthesized by using Cr(OSO2CH3)2, a precursor salt reported here for the first time. The structure of this coordination polymer, as established from X-ray powder diffraction data, shows that this solid consists of neutral Cr(pyrazine)2(OSO2CH3)2 layers containing an array of chromium metal ions bridged by bidentate pyrazine ligands and coordinated to methanesulfonate anions in their apical positions. As also supported by quantum chemical calculations, crystallographic, spectroscopic and magnetic characterization demonstrate unequivocally the presence of high-spin Cr II centers which contrast with the Cr(pyrazine)2Cl2 analogue which incorporates Cr III metal ions and a singly reduced pyrazine scaffold [5]. The redox process involving the oxidation of Cr II in Cr III and the reduction of one pyrazine in Cr(pyrazine)2Cl2 is clearly disabled in Cr(pyrazine)2(OSO2CH3)2, likely due to the change of the redox potential at the Cr site induced by the substitution of chloride by methylsulfonate. The impact of the presence or absence of this redox process on the physical properties is colossal. While a high electrical conductivity (32 mS cm -1 ) and an ordered ferrimagnetic phase (< 55 K) are observed in Cr(pyrazine)2Cl2, the present material is an antiferromagnet below 10 K and exhibits a gigaohm resistivity. This work clearly highlights that the activation of the redox process in these 2D pyrazine coordination networks is the key parameter in the design of high TC magnets possessing high electrical conductivity.

Appendix A. Supplementary data

CCDC 1836082 contains the supplementary crystallographic data for Cr(pyrazine)2(OSO2CH3)2. These data can be obtained free of charge via http://www.ccdc.cam.ac.uk/conts/retrieving.html, or from the Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge CB2 1EZ, UK; fax: (+44)

1223-336-033; or e-mail: deposit@ccdc.cam.ac.uk.

FigureFigure 1 .

 1 Figure S1). The compound consists of stacked layers, each of them being composed of an infinite rectangular array of chromium metal ions (Cr•••Cr distances of 8.086(2) Å and 7.121(2) Å) bridged by bidentate pyrazine ligands. The coordination environment around the Cr centers is completed by two O atoms belonging to monodentate methanesulfonate anions (Figure 2). The 2D networks are spaced by 6.67 Å along the crystallographic b axis, and are arranged in such a way that the Cr centers of one layer are positioned above the middle of the rectangle created by the Cr centers in the adjacent layer. All Cr ions are six-coordinate with a distorted octahedral geometry. The Cr equatorial plane, lying in the ac crystallographic plane, contains two crystallographically independent pyrazine N atoms (N1 and N2), while the symmetry related O atoms (O1) from the methanesulfonate ligands occupy the apical coordination sites (Figure2). The Cr-N1 bond length of 2.162(1) Å is significantly shorter than the value of 2.631(2) Å between Cr-N2, while the Cr-O1 distance amounts to 1.996(6) Å. These metric parameters are consistent with a

Figure 2 .

 2 Figure 2. Local coordination environment around the Cr centre in Cr(pyrazine)2(OSO2CH3)2. Cr: dark green, N: blue, O: red, S: yellow, C: dark grey.

Figure 3 .

 3 Figure 3. Splitting of d-orbitals computed by ab initio ligand field theory for {Cr II (pyrazine)4(OSO2CH3)2} and {Cr III [(pyrazine)4] • Cl2} fragments.

Figure 4 .

 4 Figure 4. Normalized X-ray absorption spectroscopy (XAS) data at the Cr K-edge for Cr(pyrazine)2(OSO2CH3)2 at 295 K and reference compounds: Cr(II) and Cr(III) recorded both at 2.1 K. Inset: Magnification of the Cr K pre-edge region.

Figure 5 .

 5 Figure 5. Temperature dependence of the magnetic susceptibility, c (equal to M/H per mole of Cr(pyrazine)2(OSO2CH3)2; top), and the cT product (bottom) at 0.1 T (black circles are the experimental points). The solid red, blue and green lines are the best fits of the experimental data between 15 and 280 K to the Lines, Fisher and 1D-QMC models, respectively, as discussed in the main text. Insets: (top) Temperature dependence of the magnetic susceptibility at different applied field between 0.1 and 7 T for temperatures below 30 K. (bottom) Field dependence of the magnetization up to 7 T at different temperatures between 1.85 K and 30 K.

Below 15 K

 15 , the magnetic susceptibility becomes significantly field dependent, as shown in the inset of the Figure 5 top part, and the magnetization does not reach the expected 4 µB saturation value at 1.85 K and 7 T (inset of the Figure 5 bottom part). These features, as well as the non-strictly linear M vs H data, suggest a possible magnetic phase transition below 15 K toward a three-dimensional antiferromagnetic state.

Figure 6 .

 6 Figure 6. Black circles: temperature dependence of the heat capacity, Cp, per mole of Cr(pyrazine)2(OSO2CH3)2 measured on a microcrystalline sample under zero-applied field. The red solid line corresponds to the empirical polynomial (at the 6 th order) baseline used to determine the non-magnetic background of the heat capacity (Cbackground). Blue dots: Temperature dependence of the magnetic component (Cpm) of the heat capacity deduced from Cp -Cbackground.

Acknowledgements

This work was supported by the University of Bordeaux, the Région Nouvelle Aquitaine, the Centre National de la Recherche Scientifique (CNRS) and the Marie Sklodowska-Curie grant 745696 ETSMM (Postdoctoral fellowship for I.O.). The authors thank the GdR MCM-2 (Magnétisme et Commutation Moléculaires) and the MOLSPIN COST action CA15128. E.A.S. acknowledges financial support from the Russian Science Foundation (Project 16-13-10155), EPSRC (EP/N006895/1), the Supercomputer Center of Novosibirsk State University and the IRIDIS High Performance Computing Facility. X. Ma, A. Hen, M. Platunov, E. Lebraud and S. Exiga are thanked for helpful discussions and experimental assistance.