Guedes Yannick

Nicolas Bonthonneau
email: yannick.bonthonneau@univ-rennes1.fr

San Raymond

Ngoc Vu

Y Guedes Bonthonneau

AND S. V Ũ NGO . C N Raymond

Y Guedes

S Vũ

Exponential localization in 2D pure magnetic wells

come

Introduction

The question of proving the localization of a quantum state has many mathematical facets. In this article, we investigate the case of the magnetic Laplacian and prove, under a geometric confinement property on the magnetic intensity, an Agmon-type localization estimate for low-lying eigenfunctions of this operator.

The interest in the magnetic Laplacian has several origins. From a quantum mechanical viewpoint, this operator is a simplified model for describing the motion of an electron in a strong magnetic field, when the electrostatic interaction and the relativistic effects are ignored; its construction is explained for instance [START_REF] Bourbaphy | Physics in a Strong Magnetic Field[END_REF]. In the book [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF], the authors recall that the same operator also appears in the linearization of the Ginzburg-Landau functional in the domain of superconductors. In Spectral Geometry, the magnetic Laplacian is often regarded as a natural variant of the Laplace-Beltrami operator when the symplectic form of the cotangent bundle is twisted by the pull-back of a closed 2-form from the base manifold, and has proved important in the study of magnetic geodesics; see for instance [START_REF] Kordyukov | Trace formula for the magnetic laplacian[END_REF], and references therein. In the present study, we consider the magnetic Laplacian on the plane, which can be defined as follows.

When B is a real function on R 2 , a semiclassical magnetic Laplacian associated with B is a family of operators, depending on a parameter h > 0, of the form

(1.1) L h = (-ih∇ -A) 2 = (hD 1 -A 1 (x)) 2 + (hD 2 -A 2 (x)) 2 , D = -i∂ .
Here, A = (A 1 , A 2 ) is a potential vector associated with B, i.e B = ∂ 1 A 2 -∂ 2 A 1 . Notice that the semiclassical limit h → 0 is related to the limit of strong magnetic field (1/h)B.

The spectral theory of L h has received the attention of several authors; in particular, it follows from [START_REF] Helffer | Semiclassical spectral asymptotics for a two-dimensional magnetic Schrödinger operator: the case of discrete wells[END_REF] that if B is smooth and admits a global non-degenerate minimum, uniquely attained at some x ∈ R 2 , the bottom of the spectrum of L h (for h small enough) is comprised of multiplicity one eigenvalues λ 0 (h) < λ 1 (h) . . . , with

λ j (h) = b 0 h + (C 1 + C 2 j)h 2 + o(h 2 ) .
The corresponding eigenfunctions are concentrated around x, in the sense that their L 2 mass outside of a fixed neighbourhood of x is O(h ∞ ). The purpose of this article is to obtain a stronger concentration in the case when B is real analytic.

1.1. Statement of the result. From now on, we assume the following: (i). The magnetic field B has a unique minimum b 0 at x = 0. It is positive, non-degenerate, and not attained at infinity (lim inf B > b 0 ). (ii). There exists a complex strip S = R 2 + i[-a, a] 2 (a > 0) to which B can be holomorphically extended as a bounded function. (iii). The function (x 1 , x 2 ) →

x 1 0 ∂B(u,x 2 ) ∂x 2
du is bounded on the strip S.

For example, B = 2-e -|x| 2 satisfies our assumptions. We will say that a function f : R n → R goes linearly to infinity at infinity if there is a constant C > 0 such for |x| > C, f > |x|/C. Our main result is the following exponential localization estimate.

Theorem 1.1. Consider a Lipschitz function d : R 2 → R + with a unique and non-degenerate minimum at 0, d(0) = 0, and going linearly to infinity at infinity, and let K > 0. Then there exist C, h 0 , ǫ > 0 such that, for all h ∈ (0, h 0 ) and u ∈ L 2 (R 2 ) such that

L h u = hµu with µ ≤ b 0 + Kh , we have R 2 e ǫd(x)/h |u(x)| 2 dx ≤ C u 2 L 2 (R 2 ) ,
Observe that here, the third assumption (iii) seems technical, and depends on a choice of a system of coordinates, but we have not been able to remove it. Also note that since we are not trying to optimize constants in our theorem, the value of a > 0 in (ii) is not essential. As a consequence, to lighten notations, we will use a > 0 as a generic constant throughout the paper. The size of the strip on which we are working will be reduced a finite number of times. 1.2. Eigenvalues asymptotics. Strong localization of eigenfunctions, such as the one claimed by Theorem 1.1, is often a footprint of discrete spectrum. Indeed, under assumption (i), it follows from the usual theory (see [START_REF] Avron | Schrödinger operators with magnetic fields. I. General interactions[END_REF]) that below

h lim inf B, the spectrum of L h is discrete. Let λ 0 (h) ≤ λ 1 (h) ≤ . . . λ ℓ (h) ≤ • • • ≤ lim inf B
be the (possibly finite) sequence of such eigenvalues, repeated according to their multiplicity.

The following theorem has been established via a dimensional reduction in [START_REF] Helffer | Semiclassical spectral asymptotics for a magnetic Schrödinger operator with non-vanishing magnetic field[END_REF] (see also [START_REF] Helffer | Semiclassical spectral asymptotics for a two-dimensional magnetic Schrödinger operator: the case of discrete wells[END_REF] and the review paper [START_REF] Helffer | Semiclassical analysis of Schrödinger operators with magnetic wells[END_REF]) and via a Birkhoff normal form in [START_REF] Raymond | Geometry and spectrum in 2D magnetic wells[END_REF]. In fact, this theorem does not require the analyticity of B (i.e assumptions (ii) and (iii)), but rather C ∞ bounds on B. Theorem 1.2 ( [START_REF] Helffer | Semiclassical spectral asymptotics for a two-dimensional magnetic Schrödinger operator: the case of discrete wells[END_REF][START_REF] Helffer | Semiclassical spectral asymptotics for a magnetic Schrödinger operator with non-vanishing magnetic field[END_REF], [START_REF] Raymond | Geometry and spectrum in 2D magnetic wells[END_REF]).

(1.2) ∀ℓ ∈ N , λ ℓ (h) = b 0 h + 2ℓ √ det H b 0 + (Tr H 1 2 ) 2 2b 0 h 2 + o(h 2 ) ,
where b 0 = min R 2 B and H = 1 2 Hess (0,0) B.

1.3.

Complex WKB expansions. With Theorem 1.2 comes the question of describing the eigenfunctions. Inspired by the results about the semiclassical Schrödinger operator with an electric potential, we can wonder whether the complex version of the famous Wentzel-Kramers-Brillouin (WKB) Ansatz can be adapted to the magnetic case. Such constructions, solving formally the eigenvalue problem, are rather rare in the context of the pure magnetic Laplacian; see however [START_REF] Maslov | The complex WKB method for nonlinear equations. I[END_REF]VI,[START_REF] Avron | Schrödinger operators with magnetic fields. I. General interactions[END_REF]. Their existence has been established for the first time in a multi-scale framework in [START_REF] Bonnaillie-Noël | Magnetic WKB constructions[END_REF] and then in non-degenerate magnetic wells (i.e., under Assumption (i)) in [START_REF] Bonthonneau | WKB constructions in bidimensional magnetic wells[END_REF]. Let us recall the latter result (which was generalized to the Riemannian setting in [START_REF] Nguyen | Classical and Semi-classical dynamics of magnetic fields[END_REF]).

Theorem 1.3 ([15]

). Under Assumption (i), and after a rotation, we can assume

(1.3) B(x 1 , x 2 ) = b 0 + αx 2 1 + γx 2 2 + O( x 3 ) , with 0 < α ≤ γ . Let ℓ ∈ N. There exist (i). a neighborhood V of (0, 0) in R 2 , (ii). an analytic function S on V satisfying Re S(x) = b 0 2 √ α √ α + √ γ x 2 1 + √ γ √ α + √ γ x 2 2 + O( x 3 ) , (iii)
. a sequence of analytic functions (a j ) j∈N on V, (iv). a sequence of real numbers (µ j ) j∈N satisfying

µ 0 = b 0 , µ 1 = 2ℓ √ αγ b 0 + ( √ α + √ γ) 2
2b 0 , such that, for all J ∈ N, and uniformly in V,

e S/h (-ih∇ -A) 2 -h J j≥0 µ j h j e -S/h J j≥0 a j h j = O(h J+2 ) .
The WKB constructions in [START_REF] Bonnaillie-Noël | Magnetic WKB constructions[END_REF][START_REF] Bonthonneau | WKB constructions in bidimensional magnetic wells[END_REF] give a positive answer to the open problem mentioned by Helffer in [16, Section 6.1]: in generic situations with pure magnetic field, WKB constructions corresponding to the low lying spectrum exist. Once the WKB analysis is done, we want to know to which extent the Ansätze are approximations of the exact eigenfunctions u ℓ ∈ L 2 (R 2 ). It follows from Theorem 1.2 that, when h is small enough, the eigenvalues are simple and separated by a gap of order ∼ h 2 . Thanks to the Spectral Theorem, we deduce that the WKB Ansätze are approximations in the L 2 -sense, and even in a weighted L 2 -space thanks to Theorem 1.1 (up to taking a smaller ε).

Corollary 1.4. Denote by u ℓ,J = χ(x)e -S/h J j≥0 a j h j , with χ ∈ C ∞ 0 (V), and constant around the origin. Then, for fixed ℓ ∈ N and ε > 0 small enough, we have for some θ ∈ R

e εd(x)/h (e iθ u ℓ -u ℓ,J ) L 2 (R 2 ) = O(h J 2 ).
(This will be proved at the end of section 5). In contrast with Theorem 1.1, the WKB Ansätze decay like e -Re S/h away from the magnetic well; thus, the approximation should actually hold in a slight perturbation of the weighted space L 2 (e -2Re S/h ). Behind this question lies the tunneling effect problem: such exponential estimates are the heart of the analysis of the interaction between multiple magnetic wells. The present paper does not go that far1 , but establishes that the eigenfunctions decay like e -ϕ(x)/h for some non-negative function ϕ. These types of estimates are well-known and proved in the electric Schrödinger operator -h 2 ∆ + V , where they go by the name of Agmon (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] Helffer | Multiple wells in the semiclassical limit[END_REF][START_REF] Simon | Semiclassical analysis of low lying eigenvalues[END_REF]). As we will see, the purely magnetic case seems to necessitate a significantly more advanced strategy, based on the Fourier-Bros-Iagolnitzer (FBI) transform. (In [START_REF] Helffer | Multiple wells in the semiclassical limit[END_REF], the FBI transform does appear, but not for proving the exponential localization; it is used in a second step, to control the asymptotic expansion of eigenvectors and eigenvalues.) 1.4. Failure of the naive Agmon estimates. Let us explain formally why the electric strategy fails in giving the optimal Agmon estimates in the pure magnetic case (see also [START_REF] Raymond | Bound states of the magnetic Schrödinger operator[END_REF]Prop. 4.23] for a slightly different presentation). This strategy is based on the following formula:

e ϕ/h (-ih∇ -A) 2 e -ϕ/h = (-ih∇ -A + ∇ϕ) 2 ,
where ϕ is bounded and Lipschitz continuous, and on using the coercivity of the real part

Re e ϕ/h (-ih∇ -A) 2 e -ϕ/h u, u = (-ih∇ -A)u 2 -∇ϕ u 2 , where u ∈ C ∞ 0 (R 2 )
. Then, we want to use the magnetic field, and we notice that

(-ih∇ -A)u 2 ≥ h R 2 B(x)|u| 2 dx , so that, for all λ ∈ R, Re e ϕ/h (-ih∇ -A) 2 -λ e -ϕ/h u, u ≥ R 2 hB(x) -|∇ϕ| 2 -λ |u| 2 dx .
From this last inequality, we see that the only possibility to control the gradient is that ϕ actually depends on h. With the choice ϕ = h 1 2 Φ, where Φ is the Agmon distance (to 0) associated with the metric (B -b 0 -|∇Φ| 2 ) + dx 2 , we can deduce that, for eigenvalues such that λ = b 0 h + O(h 2 ), the corresponding eigenfunctions ψ(= e -ϕ/h u) satisfy, for h small enough,

(1.4) R 2 e 2Φ/h 1 2 |ψ| 2 dx ≤ C ψ 2 .
Due to the non-degeneracy of the minimum of B, Φ may be chosen with a unique and non-degenerate minimum at 0. Thus, (1.4) tells us for instance that the ground state is a priori exponentially localized at the scale h 1 4 near the minimum. This is consistent with Theorem 1.3, but much worse than expected. One should be able to prove that the eigenfunctions, just as the WKB quasi-modes suggest, are localized at the scale h 1 2 near the minimum. That it is indeed the case is the main result of this article. 1.5. Related results. Some articles have been devoted to the Agmon estimates in the presence of a magnetic field, but almost always with an additional electric potential. For instance, in [START_REF] Helffer | Effet tunnel pour l'équation de Schrödinger avec champ magnétique[END_REF], the decay estimates are inherited from the electric potential and the magnetic field is considered as a perturbation (see in particular [21, p. 629]). In the same spirit, Agmon estimates are considered in [START_REF] Nakamura | Tunneling estimates for magnetic Schrödinger operators[END_REF]Theorem 1.1] (see also the closely related articles [START_REF] Erdős | Gaussian decay of the magnetic eigenfunctions[END_REF][START_REF] Nakamura | Gaussian decay estimates for the eigenfunctions of magnetic Schrödinger operators[END_REF]) in the case of an electric well with constant magnetic field. It is proved that the magnetic field improves the decay of the eigenfunctions away from the electric well.

We will see in this paper that pure magnetic Agmon estimates at the "right" semiclassical scale can be obtained as projections of microlocal exponential estimates. Our strategy will be inspired by the ideas of Martinez [START_REF] Martinez | Microlocal exponential estimates and applications to tunneling[END_REF] (see also [START_REF] Nakamura | Agmon-type exponential decay estimates for pseudodifferential operators[END_REF], and [START_REF] Martinez | Microlocal WKB expansions[END_REF] in relation with the corresponding WKB analysis). The fact that we are able to refine this point of view, which is based on the FBI transform, and to apply it to establish our new magnetic Agmon estimates, is reminiscent of Sjöstrand's pioneer work on analytic hypo-ellipticity [START_REF] Sjöstrand | Analytic wavefront sets and operators with multiple characteristics[END_REF].

Remark 1.5. Throughout our analysis, we will meet some known close links between magnetic and Toeplitz operators. These connections are described, for instance, in [START_REF] De Verdière | Champs magnétiques classiques et quantiques[END_REF], or [START_REF] Kordyukov | Semiclassical spectral analysis of toeplitzoperators on symplectic manifolds: the caseof discrete wells[END_REF]Section 4]. In the context of Toeplitz operators, exponential decay estimates of eigenfunctions have been the subject of the recent works [START_REF] Kordyukov | Semiclassical spectral analysis of toeplitzoperators on symplectic manifolds: the caseof discrete wells[END_REF]Theorem 1.3] and [START_REF] Deleporte | Toeplitz operators with analytic symbols[END_REF]Theorem C]. In these papers, the semiclassical parameter is of the form h = p -1 , where p ∈ N is the degree of tensorization of a line bundle. 1.6. Organization and strategy. In Section 2, we perform various reductions to put the magnetic Laplacian in a "normal form". Section 3 is central in our analysis and is devoted to general properties of the Fourier-Bros-Iagolnitzer transform. Our presentation closely follows and, sometimes, completes the one exposed in the book by Martinez [26,Chapter 3]. This part of the investigation can also be considered an interpretation of the magnetic Laplacian as a Toeplitz operator. In Section 4, we prove that the FBI transform of an eigenfunction (with low energy) is exponentially localized at the scale h 1 2 near 0 ∈ R 2 × (R 2 ) * . We proceed in two steps: firstly, we prove the exponential microlocalization near the characteristic manifold (Theorem 4.4); secondly, we establish an exponential localization inside the manifold (Theorem 4.5). In Section 5, we use the microlocal exponential estimates to deduce Theorem 1.1.

Normal form

In [START_REF] Raymond | Geometry and spectrum in 2D magnetic wells[END_REF], the second and third author constructed a Fourier integral operator U h that conjugates the magnetic Laplacian L h to an operator of the form

Op w h (f (H, x 2 , ξ 2 )) + O(h ∞ ), microlocally near the characteristic set of L h , where H = h 2 D 2 x 1 + x 2 1
and Op w h denotes the Weyl quantization. If the symbol f were analytic, and the remainder O(h ∞ ) improved to O(e -C/h ), this would imply a natural (and probably optimal) exponential estimate on the bottom eigenfunctions of L h . However, the FIO U h is constructed in a relatively non-explicit fashion, including a generically divergent Birkhoff normal form, and tracking those estimates down would require quite sophisticated tools of analytic microlocal analysis. Since we "only" want to obtain decay of eigenfunctions and not the expansion of the bottom eigenvalues of L h to any power of h, we will only need a rather crude normal form.

Lemma 2.1. Under assumptions (i), (ii), (iii), there exists a > 0 and, for h > 0, a unitary operator U h acting on L 2 (R 2 , dx) such that

(2.1) U h L h U -1 h = Op w h (p L ), where p L is an h-dependent holomorphic function on R 4 + i[-a, a] 4 , such that (2.2) p L = g 11 ξ 2 1 + 2g 12 ξ 1 x 1 + g 22 x 2 1 + h 2 q
, where g 11 , g 12 , g 22 and q are holomorphic, bounded, and on R 4 they are real valued. Additionally, the g ij are critical at 0, and

(2.3) B(x, ξ) = g 11 g 22 -(g 12 ) 2 ,
when restricted to R 4 , admits a positive non-degenerate minimum at 0, uniquely attained, and not attained at infinity.

This type of operators, whose symbol is a quadratic form of some variables, with parameters, was studied by several authors in the context of hypo-ellipticity in the smooth category (see [START_REF] Boutet De Monvel | Parametrixes d'opérateurs pseudodifférentiels à caractéristiques multiples[END_REF] and references therein), and in the analytic category by Sjöstrand in [START_REF] Sjöstrand | Analytic wavefront sets and operators with multiple characteristics[END_REF]. It would be interesting to obtain a global version of Sjöstrand's results in order to give a different proof of Theorem 1.1.

Observe that the exponential decay of eigenfunctions is not preserved by general unitary operators. However, we will see that U h can be explicitly described, so this will not be an issue.

For a given magnetic field B, the choice of magnetic potential A is not unique. Any other choice A ′ differs from A by a gradient, i.e. A ′ = A + ∇f . Then, the corresponding magnetic Laplacian is obtained by conjugating L h by the multiplication operator u → e if /h u, which is unitary, both pointwise and in L 2 . Hence, it does not impact Theorem 1.1. Therefore we may, and we will, assume that (2.4)

A(x) = (0, A 2 (x)) , A 2 (x) = x 1 0 B(u, x 2 ) du ,
Notice that A 2 is real-analytic, admits a holomorphic extension to the strip S, and its derivatives are bounded on S according to assumption (iii). For d = 2 or d = 4 depending on the context and a > 0, it will be convenient to set

S a := R d + i[-a, a] d .
2.1. Normal form near the characteristic set. In this section, we prove Lemma 2.1. The operator U h will be decomposed as the composition of a change of variables and a metaplectic operator. Let us start by constructing the change of variable.

The first idea, which is quite standard, is to choose coordinates in which the magnetic field is constant as a 2 form. In that case, the natural symplectic structure becomes canonical, and all the magnetic information is transferred to a variable Riemannian metric. The guiding model is the case of constant magnetic field and constant metric, where the magnetic Laplacian takes the form

L ct h = (hD x 1 ) 2 + (hD x 2 -Bx 1 )
2 , and its bottom eigenvalue is hB. The solutions, sometimes called zero modes, to

L ct h -hB u = 0
are of the form e -Bx 2 1 /2h f , with f holomorphic, and they play an important role in the spectral analysis of the magnetic Dirac operator (see [START_REF] Barbaroux | On the semiclassical spectrum of the Dirichlet-Pauli operator[END_REF]).

Coming back to our problem, there are many diffeomorphisms κ of R 2 such that κ * B is the canonical 2 form (Darboux' Lemma), so we pick the following

(x 1 , x 2 ) = κ(x 1 , x2 ), x1 = x 1 0 B(x ′ , x 2 ) dx ′ , x2 = x 2 .
That this defines indeed a global diffeomorphism of R 2 is ensured by assumption (i).

Lemma 2.2. Under Assumption (i), (ii) and (iii), κ is a bi-Lipschitz analytic diffeomorphism of R 2 such that κ * B = dx 1 ∧ dx 2 and κ * A = x1 dx 2 . Moreover, there exists λ ≥ 1 and a > 0 such that κ and κ -1 send S a ′ to S λa ′ for all a ′ ∈ (0, a/λ).

It will be useful to let

α(x 1 , x 2 ) = x 1 0 ∂ x 2 B(u, x 2 )dx 1 .
Proof. κ is a global diffeomorphism of R 2 because B is positive, and κ -1 is well defined on S a . Next, there is a C > 0 such that |B| ≤ C and |α| ≤ C , so that κ -1 maps S a ′ into S 2Ca ′ for 0 < a ′ < a.

We can compute

d x (κ -1 ) = B(x) α(x) 0 1 .
In particular,

(d x (κ -1 )) -1 = 1 B(x) -α(x) B(x) 0 1 .
Since B ≥ b 0 > 0 on R 2 , and using Assumption (ii), there exists 0 < a 0 < a such that |B| -1 ≤ (Re B) -1 ≤ 1/(2b 0 ) on S a 0 . In particular, on S a 0 , (d x (κ -1 )) -1 is bounded.

Around each real point x, we can apply the holomorphic local inversion theorem and deduce that there are ǫ x , ǫ ′

x > 0 such that κ -1 is a biholomorphism between the ball of radius ǫ x centered at x and its image, which contains the ball of radius ǫ ′ x around κ -1 (x). One can give lower bounds to the constants ǫ x , ǫ ′

x , expressed only in terms of the C 2 norms of κ -1 , and an upper bound on (d x (κ -1 )) -1 . In particular, we can choose them independent of x.

Additionally, if κ -1 (x) = κ -1 (y) for some x, y ∈ S a ′ with 0 < a ′ < a 1 , then x 2 = y 2 , and

y 1 x 1 B = 0. Observe that 0 = Re y 1 x 1 B = Re y 1 Re x 1 Re B(t + iIm x 1 , x 2 ) dt - Im y 1 Im x 1 Im B(Re y 1 + it, x 2 ) dt. Since Re B ≥ b 0 /2 on S a 1 , we deduce that |Re (x 1 -y 1 )| ≤ C(a ′ ) 2 for some C > 0.
In particular, according to the argument above, if a ′ is small enough, this implies that x 1 = y 1 . For such an a ′ > 0, κ -1 is a biholomorphism between S a ′ and κ -1 (S a ′ ), which satisfies

S a ′′ ⊂ κ -1 (S a ′ ) ⊂ S Ca ′ ,
for some a ′′ > 0. Further, κ -1 is uniformly Lipschitz, and so is its inverse. Taking min(a ′ , a ′′ ) as the new value of a and λ the Lipschitz constant of κ, κ -1 ends the proof.

We can associate κ with a unitary operator U κ by setting

U κ f (y) = Jac(κ) 1/2 f (κ(y)).
According to Lemma A.1 and keeping the same notation, we have

(2.5) U κ L h U -1 κ = (-ih∇ y -Ã)g * (-ih∇ y -Ã) -h 2 V .
Here, V is explicit in terms of κ, and g * is the dual Riemannian metric (dκ T dκ) -1 . Note also that

(2.6) (-ih∇ y -Ã)g * (-ih∇ y -Ã) = Op w h (g * (η -A(y), η -A(y))) + O(h 2 )
, where the O(h 2 ) comes from the explicit computation of the subprincipal term with the composition formula (the operator in the right hand side is symmetric).

From explicit expressions for the remainders, we deduce that

(2.7) U κ L h U -1 κ = Op w h ( (ξ 1 , ξ 2 -x 1 ) 2 g * + O(h 2 ))
, where the remainder symbol is of the form h 2 q 1 , q 1 holomorphic and bounded on some S a with a > 0. Moreover, letting B = B • κ and α = α • κ, we get

(2.8) (ξ 1 , ξ 2 -x 1 ) 2 g * = B2 ξ 2 1 + (ξ 2 -x 1 + αξ 1 ) 2 .
We are now almost in the desired form. We consider the following symplectomorphism

κ M (x, ξ) = (x + Aξ, ξ) ,
where

A = 0 1 1 0 , A -1 = A .
It is associated with the metaplectic operator M, defined as (2.9)

Mu(x 1 , x 2 ) := 1 (2πh) 2 R 4 e i h Φ(x,y,ξ) u(y) dy dξ ,
the phase being given by

Φ(x, y, ξ) = ϕ(x, ξ) -y, ξ , ϕ(x, ξ) = x - 1 2 Aξ, ξ ,
and ϕ being the generating function of κ M . We observe that

Mu(x) = (2πh) -2 R 2 e i h x,ξ F h u(ξ)e -i 2h Aξ,ξ dξ ,
where we used the semiclassical Fourier transform

F h u(ξ) = R 2 e -i h x,ξ u(x) dx , F -1 h v(x) = (2πh) -2 R 2 e i h x,ξ v(ξ) dx .
Recalling that

F -1 h (UV ) = F -1 h (U) ⋆ F -1 h (V )
, the operator M can be written as a convolution operator (2.10)

Mu = K ⋆ u , K = F -1 h e -i 2h Aξ,ξ = 1 2πh e i 2h Ax,x ,
where we used the well-known result about the Fourier transform of a quadratic exponential.

For a symbol σ in S ′ (R 4 ) (which is surely the case of the symbols we are manipulating so far, we have the exact "Egorov" correspondence

(2.11) M -1 Op w h (σ)M = Op w h (σ • κ M ) . It follows that M -1 U κ L h U -1
κ M is in the form announced by Lemma 2.1. It remains to check the conditions on the coefficients. We find that

g 11 = ( B2 + α2 ) • κ M , g 12 = α • κ M , g 22 = 1. Then (2.12) B(x, ξ) = g 11 g 22 -(g 12 ) 2 = B(κ(κ M (x, ξ))),
is suitably non-degenerate according to assumption (i), and it remains to check that α • κ M is critical at 0. But this is true if α itself is critical at 0, and this holds since (B being critical at 0)

α = x 2 1 ∂ 2 x 2 ,x 1 B(0) + O(x 3 ).
It is important to observe that since M somehow mixes x and ξ variables, it does not preserve exponential decay of functions. However, in a sense to be precised later, we will get decay in "x and ξ", which is preserved by M.

In the sequel, it will be convenient to let (2.13)

p M := g 11 ξ 2 1 + 2g 12 x 1 ξ 1 + g 22 x 2
1 . Additionally, we will distinguish variables by setting X 1 = (x 1 , ξ 1 ), X 2 = (x 2 , ξ 2 ).

2.2.

Reduction to a bounded symbol. Our strategy strongly relies on the presentation of the Fourier-Bros-Iagolnitzer (in short, FBI) transform given in Martinez' book [START_REF] Martinez | An introduction to semiclassical and microlocal analysis[END_REF]. There, many results require that operators have symbols in the class S(1), which is the space of smooth functions on phase space that are uniformly bounded, together with all their derivatives. However, because the magnetic Laplacian is a differential operator of positive order, its symbol does not belong to that class. The statements we will use could probably be extended to the general case of symbols with more general order functions. It is to avoid this, and concentrate on the essential arguments, that we have decided to restrict ourselves to the case of a bounded magnetic field, a situation where we can reduce the problem to a problem in the S(1) class, as follows.

Initially, the symbol of magnetic Laplacian is polynomial in ξ and hence belongs to a class with gains of powers of ξ : locally in x,

(2.14) |∂ α x ∂ β ξ p κ | ≤ C α,β ξ 2-|β|
. This still holds after the change of variables κ. However, the metaplectic transform M mixes the x and ξ variables, so that we do not gain powers of ξ anymore. Recall that for a non-negative function m on R d , S R d (m) is defined as the set of functions σ that satisfy estimates

|∂ α x σ| ≤ C α m, α ∈ N d . If σ is holomorphic on a complex strip R d ⊂ S ⊂ C d , we shall say that σ ∈ S S (m) if ∀z ∈ S, |∂ α x σ(z)| ≤ C α m(Re z), α ∈ N d .
Also recall what it means for a non-vanishing smooth function m on T * R 2 = R 4 to be an admissible order function. First, one requires that m ∈ S R 4 (m). Second, there is an N > 0 such that for some C > 0 and any (x, ξ),

(x ′ , ξ ′ ) ∈ T * R 2 , (2.15) m(x, ξ) m(x ′ , ξ ′ ) ≤ C (x -x ′ , ξ -ξ ′ ) N .
Given two admissible order functions m and m ′ , then 1/m and mm ′ also are admissible and we have the following result (see for instance [12, Proposition 7.7]): if σ ∈ S(m) and σ ′ ∈ S(m ′ ), then (2.16)

Op w h (σ) Op w h (σ ′ ) = Op w h σσ ′ + h 2i {σ, σ ′ } + O S(mm ′ ) (h 2 ) ,
with the usual sign convention {f, g} = ∂ ξ f ∂ x g -∂ x f ∂ ξ g. Following the result of Boutet de Monvel-Krée [START_REF] Boutet De Monvel | Pseudo-differential operators and Gevrey classes[END_REF], a refinement of estimate (2.16) shows that if σ, σ ′ had a holomorphic extension to a strip, with uniform estimates, then the symbol of the product also does, with uniform estimates. Consider now

m M (X 1 , X 2 ) = 1 + p M (X 1 , X 2 ) .
Lemma 2.3. Assume that p M is in the form (2.13), with coefficients satisfying the conclusion of Lemma 2.1. Then m M is an admissible order function, and p M ∈ S Sa (m M ) for some a > 0.

If these assumptions are satisfied, we will introduce a bounded spectral parameter µ and work with

P = Op w h 1 1 + p M Op w h (p L -hµ) = Op w h p M -hµ 1 + p M + O S Sa (1) (h 2 ) = Op w h (p h ), (2.17)
where p L = p M + h 2 q, see (2.2), so that p h ∈ S Sa (1), uniformly with respect to h and µ.

Remark 2.4. Since the intensity of the magnetic field is given by

det Hess X 1 (p h ) |X 1 =0 + O(h)
(see Equation (2.12)) the fact that it is globally bounded is actually necessary for obtaining p h ∈ S Sa (1).

Proof. Let us check that p M ∈ S Sa (m M ). Since the coefficients g ij are in S Sa (1) for some a > 0, one finds that

|∂ α p M | ≤ C α (1 + |X 1 | 2 ) on S a .
Thus, it suffices to show that there exists λ > 0 such that (2.18)

1 + Re p M ≥ λ(1 + |X 1 | 2 ) .
Let us start by proving this on R 4 . Note that (2.18) is satisfied for example if λ ≤ 1 and everywhere

λ ≤ g 11 + g 22 -(g 11 -g 22 ) 2 + 4(g 12 ) 2 2 . Let C = sup{|g 11 | + |g 22 |} , C ′ = inf{g 11 g 22 -(g 12 ) 2 } .
The quantity in the right hand side is larger than 2(g 11 g 22 -(g 12 ) 2 )/(g 11 + g 22 ) ≥ 2C ′ /C > 0 , uniformly on T * R 2 . Now, we turn to the case that (x, ξ) = (Re x, Re ξ) + i(u, v).

Then, we can write

Re p M = Re g 11 (ξ 2 1 -v 2 1 ) + 2g 12 (ξ 1 x 1 -u 1 v 1 ) + g 22 (x 2 1 -u 2 1 ) -2Im g 11 ξ 1 v 1 + 2g 12 (ξ 1 u 1 + x 1 v 1 ) + g 22 x 1 u 1 ≥ λ ′ (Re X 1 ) 2 -Ca 2 (1 + |Re X 1 |),
where λ ′ may be smaller than the λ from before, but is still non-negative if we assume that inf{Re g 11 Re g 22 -(Re g 12 ) 2 } > 0 on R 4 + i[-a, a] 4 . Up to taking a small enough, this holds.

Finally, we consider the temperance of the symbol. We already know that for some constants C, C ′ ,

1 + p M (x, ξ) 1 + p M (x ′ , ξ ′ ) ≤ 1 + C(X ′ 1 ) 2 1 + C ′ X 2 1 , whence we find 1 + p M (x, ξ) 1 + p M (x ′ , ξ ′ ) ≤ C C ′ (1 + λ(X 1 -X ′ 1 ) 2 ) ,
for λ large enough.

About the FBI transform

Our main tool in this section will be the Fourier-Bros-Iagolnitzer (FBI) transform. Several versions exist in the literature, see [START_REF] Hitrik | Two minicourses on analytic microlocal analysis[END_REF]; in this paper we follow [START_REF] Martinez | An introduction to semiclassical and microlocal analysis[END_REF]Chapter 3], and the FBI transform we use here is defined, for u ∈ S ′ (R2 ), by

T u(x, ξ) = α h R 4 e i(x-y)ξ/h e -|x-y| 2 /2h u(y) dy , α h = 2 -1 (πh) -3 2 .
The α h is chosen so that T is isometric from L 2 (R 2 ) to L 2 (R 4 ). The knowledge of T u implies the knowledge of u via the inversion formula 2 :

(3.1)

u(y) = α h R 4
e -i(x-y)ξ/h-|x-y| 2 /2h T u(x, ξ) dx dξ = T * T u .

It will be essential later that

(3.2) (h(∂ x -i∂ ξ ) -iξ)T = 0.
In other words, T maps L 2 (R 2 ) into the closed subspace of L 2 (R 4 ) of functions of the form e -ξ 2 2h f (x -iξ), where f is holomorphic on C 2 .

3.1. Towards a Toeplitz representation. Since the naive Agmon tactic fails, it seems natural to try and use weights in phase space that depend on both x and ξ. However, it is not easy to understand the behavior of an operator of the type Op w h (e ψ(x,ξ)/h ), all the more if ψ was not bounded. (Although, in the case of a quadratic ψ, see the recent article [START_REF] Coburn | Positivity, complex FIOs, and Toeplitz operators[END_REF]). Following the strategy of [26, 3.5], [START_REF] Nakamura | On Martinez' method of phase space tunneling[END_REF], we use the FBI transform to simplify this, as e ψ(x,ξ)/h can be seen as an multiplication operator on L 2 (R 4 ). Precisely, let us consider the following quantity

(3.3) me ψ/h T Pu, e ψ/h T u L 2 (R 4 ) ,
where P is defined in (2.17), and m ∈ S(1) is multiplier (it is not an order function!). In this section, ψ ∈ S(1) and might depend on parameters uniformly with respect to the S(1)-topology, and all the bounds will depend on dψ only.

Since the FBI transform we are using has a quadratic phase, we have an exact formula T Op w h (σ) = Op w h (σ T )T , where σ T (x, ξ, x * , ξ * ) = σ(x -ξ * , x * ), valid for σ ∈ S ′ (R 4 ). From this, we get

me ψ/h T Pu, e ψ/h T u L 2 (R 4 ) = me ψ/h P T T u, e ψ/h T u L 2 (R 4 ) .
We set

P ψ T = e ψ/h P T e -ψ/h , u ψ = e ψ/h T u = T ψ u , so that (3.3) = mP ψ T u ψ , u ψ L 2 (R 4
) . Thanks to our analyticity assumption and [30, Corollary 5] or [26, Lemma 3.5.4], P ψ T is still a pseudo-differential operator with symbol in S(1). Its symbol satisfies (3.4)

p ψ h = p h (x -ξ * -i∂ ξ ψ, x * + i∂ x ψ) + O(h 2 ) .
Since we use the Weyl quantization, we have indeed O(h 2 ) and not only O(h). Now, we apply [30, Theorem 1] or [26, Theorem 3.5.1], which gives

(3.5) T Pu, me 2ψ/h T u L 2 (R 4 ) = R 4 p ψ h,m (x, ξ; h)|u ψ | 2 dx dξ + O(h 2 ) u ψ 2 , with (3.6) p ψ h,m (x, ξ; h) := m(x, ξ)p h (x + 2∂ z ψ, ξ -2i∂ z ψ) + O(h) .
Here, we have introduced the complex variable z = x + iξ, and

∂ z = 1 2 (∂ x -i∂ ξ ), ∂ z = 1 2 (∂ x + i∂ ξ ).
We stress again that the all the constants in the estimates only involve ψ via semi-norms of dψ in S(1).

Subprincipal term.

In fact, we can even describe the term estimated by O(h) and we will actually need it. For that purpose, and also for the convenience of the reader, let us revisit and refine [26, Theorem 3.5.1].

3.2.1.

General expression of the subprincipal term. Let us focus on the proof of (3.5) once we have (3.4). The following proposition shows how to explicitly write a pseudo-differential operator acting on the range of T ψ (in the sense of quadratic forms) as a multiplication operator modulo O(h 2 ).

Proposition 3.1. Consider a symbol q = q 0 (x, ξ, x * , ξ * ) ∈ S R 8 (1). We have

Op w h (q) u ψ , u ψ L 2 (R 4 ) = R 4 (q 0 (x, ξ) + hq 1 (x, ξ))|u ψ | 2 dx dξ + O(h 2 ) u ψ 2 L 2 (R 4 ) , where q0 (x, ξ) = q 0 (x, ξ, ξ -∂ ξ ψ, ∂ x ψ) , q1 (x, ξ) = 1 2 ({σ f , g} + {f, σ g }) f =g=0 , with f = x * -ξ + ∂ ξ ψ g = ξ * -∂ x ψ σ f = 1 0 ∂ x * q 0 (x, ξ, ξ -∂ ξ ψ + tf, ξ * ) dt σ g = 1 0 ∂ ξ * q 0 (x, ξ, ξ -∂ ξ ψ, ∂ x ψ + tg) dt . (3.7)
Proof. Let us follow the presentation by Martinez. The computations also appear in [START_REF] Nakamura | On Martinez' method of phase space tunneling[END_REF]. We consider

r 1 (x, ξ, x * , ξ * ) = q(x, ξ, x * , ξ * ) -q(x, ξ, ξ -∂ ξ ψ, ∂ x ψ) ,
By the Taylor formula, r 1 = f σ f + gσ g . We set F = Op w h f and G = Op w h g. Since we use the Weyl quantization, we have

Op w h (f σ f ) = 1 2 (F Op w h (σ f ) + Op w h (σ f )F ) + O(h 2 ) .
(Here, the symbol f is not in S( 1), however all its derivatives are, which is essential in the computation). Next, we observe that Equation (3.2) implies that F T ψ = iGT ψ and deduce

1 2 (F Op w h (σ) + Op w h (σ)F )u ψ , u ψ = i 2 [Op w h (σ), G]u ψ , u ψ . Thus (again, since dg ∈ S(1)) Op w h (f σ f )u ψ , u ψ = h 2 Op w h ({σ f , g})u ψ , u ψ + O(h 2 ) u ψ 2 .
In the same way, we get

Op w h (gσ g )u ψ , u ψ = h 2 Op w h ({f, σ g })u ψ , u ψ + O(h 2 ) u ψ 2 .
Therefore, iterating the argument,

Op w h r 1 u ψ , u ψ = h 2 R 4 ({σ f , g} + {f, σ g }) f =g=0 |u ψ | 2 dx dξ + O(h 2 ) u ψ 2 . Notation 3.2. When a ∈ S R 4 (1), we let a(x, ξ) = a(x + 2∂ z ψ, ξ -2i∂ z ψ) .
Corollary 3.3. We have

(3.8) T Pu, me 2ψ/h T u L 2 (R 4 ) = R 4 p ψ h,m (x, ξ; h)|u ψ | 2 dx dξ + O(h 2 ) u ψ 2 ,
where

(3.9) p ψ h,m (x, ξ; h) = m(x, ξ) p h (x, ξ) + hp ψ h,m,1 , and p ψ h,m,1 = -2∂ z m ∂ z p h + ms(x, ξ) , with s(x, ξ) = 1 2 ({σ f , g} + {f, σ g }) |f =g=0 ,
where we used the notations of Proposition 3.1 with

q 0 (x, ξ, x * , ξ * ) = p h (x -ξ * -i∂ ξ ψ, x * + i∂ x ψ) .
Proof. We apply Proposition 3.1 to the pseudo-differential operator Op w h q = mP ψ T . By the composition formula, q(x, ξ, x * , ξ * ) = mq 0 (x, ξ, x * , ξ * ) + hq 1 (x, ξ, x * , ξ * ) + O(h 2 ) , where

q 1 (x, ξ, x * , ξ * ) = (2i) -1 {m(x, ξ), p h (x -ξ * -i∂ ξ ψ, x * + i∂ x ψ)} = -(2i) -1 ∂ x m • ∂ ξ p h + (2i) -1 ∂ ξ m • ∂ x p h .
We deduce that

p ψ h,m = m(x, ξ) p h (x, ξ) + ih 2 ∂ x m • ∂ ξ p h -∂ ξ m • ∂ x p h + hs(x, ξ) , with s = 1 2 ({mσ f , g} + {f, mσ g }) |f =g=0 = m 2 ({σ f , g} + {f, σ g }) |f =g=0 + 1 2 (-∂ ξ p h • ∂ ξ m -∂ x p h • ∂ x m).

But we have

i 2 ∂ x m • ∂ ξ p h -∂ ξ m • ∂ x p h + 1 2 (-∂ ξ p h • ∂ ξ m -∂ x p h • ∂ x m) = 1 2 ∂ x m • (-∂ x p h + i∂ ξ p h ) + 1 2 ∂ ξ m • (-i∂ x p h -∂ ξ p h ) = -2∂ z m • ∂ z p h .

3.2.2.

Rough estimate of the subprincipal terms. Let us describe p ψ h,m,1 in the case when m = m(X 2 ) ∈ S(1). Recall that

p h = g 11 ξ 2 1 -2g 12 ξ 1 x 1 + g 22 x 2 1 -hµ 1 + g 11 ξ 2 1 -2g 12 ξ 1 x 1 + g 22 x 2 1 + O(h 2 ),
where the coefficients g ij are in S(1) on R 4 + i[-a, a] 4 , and µ ≥ 0. Then we notice that, since m only depends on z 2 ,

|∂ z m • ∂ z p h | = (∂ z 2 m) ∂ z 2 p h ≤ C(min(|X 1 | 2 , 1) + h 2 ) ,
and that this term is zero when m = 1. Also, we observe that a priori, s ∈ S(1), so that p ψ h,m = m p h + hmO(1) + hO(min(|X 1 | 2 , 1)) + O(h 3 ) .

3.2.3.

A more accurate description. When ψ = Ψ(X 2 ), we can give a more explicit expression for s. It will be convenient to set

(3.10) w(x, ξ, f, g) := p h (x -2∂ z ψ -g, ξ + 2i∂ z ψ + f ) ,
Then,

σ f = 1 0 ∂ f w(x, ξ, tf, g) dt , σ g = 1 0 ∂ g w(x, ξ, 0, tg) dt .
We have

{σ f , g} f =g=0 = k,j {ξ j , g k }∂ f k ∂ ξ j w(x, ξ, 0, 0) + 1 2 {f j , g k }∂ f k ∂ f j w(x, ξ, 0, 0) + {g j , g k }∂ f k ∂ g j w(x, ξ, 0, 0) , and
{f, σ g } f =g=0 = k,j {f k , x j }∂ g k ∂ x j w(x, ξ, 0, 0) + 1 2 {f k , g j }∂ g k ∂ g j w(x, ξ, 0, 0) .
From the expressions of f and g, we notice that {ξ j , g

k } = -δ jk , {f k , x j } = δ jk and {g k , g j } = -∂ 2 ξ k ,x j ψ -∂ 2 ξ j ,x k ψ , {g k , f j } = -δ k,j + ∂ 2 ξ k ,ξ j ψ + ∂ 2 x k ,x j ψ . Since ψ = Ψ(X 2
), the only non-zero terms involving ψ are obtained for j = k = 2. Thus,

(3.11) 2s(x, ξ) = k -∂ f k ∂ ξ k + 1 2 ∂ f k ∂ f k + ∂ g k ∂ x k + 1 2 ∂ g k ∂ g k w(x, ξ, 0, 0)+R 1 ,
where

R 1 = O(|d 2 ψ||d 2 X 2 p h |).
Let us look at the first term in the right-hand side of (3.11) and recall (3.10). Then, we can write it as

k -∂ f k ∂ ξ k + 1 2 ∂ f k ∂ f k + ∂ g k ∂ x k + 1 2 ∂ g k ∂ g k w(x, ξ, 0, 0) = - 1 2 ∆p h + R 2 ,
where again,

R 2 = O(|d 2 ψ||d 2 X 2 p h |), so that finally, s = - 1 4 ∆p h + O(min(|X 1 | 2 , 1)) + O(h 2 ).
We can summarize the discussion above in the following.

(i). When m = 1, r = 0 (ii). When ψ = Ψ(X 2 ), s = - 1 4 ∆p h (x, ξ) + Ř , where Ř ∈ S(1) and Ř = O(|d 2 Ψ| min(|X 1 | 2 + h 2 , 1)
). Moreover, all estimates are uniform for h small and dψ varying in a bounded subset of S(1).

Noticing that Ř is zero when ψ = 0, we get the following.

Proposition 3.5. When σ ∈ S(1), Op w h (σ)u, u L 2 (R 2 ) = T Op w h (σ)u, T u L 2 (R 4 ) = R 4 σ(x, ξ) - h 4 ∆σ(x, ξ) + O(h 2 ) |u ψ | 2 dx dξ . (3.12)
Remark 3.6. This classical proposition (see [26, Corollary 3.5.7 & Section 3.6, Ex. 7] and consider also [START_REF] Zworski | Semiclassical analysis[END_REF]Theorem 13.10]) is also true when σ is a quadratic form, and in this case the remainder O(h 2 ) is zero.

Microlocal Agmon estimates

In this section, we establish Agmon estimates with respect to X 1 in an exponentially weighted space with respect to X 2 . These estimates are stated in Theorem 4.3 and 4.4. They imply Theorem 4.1. In this whole section we will consider u ∈ L 2 (R 2 ) solving the equation

(4.1) Pu = Op w h 1 1 + p M Op w h (p M -hµ + h 2 q)u = 0.
with p L = p M +h 2 q satisfying the conclusion of Lemma 2.1, so that the conclusions of Scholium 3.4 applies.

Theorem 4.1. Let Ψ 1 , Ψ 2 be non-negative Lipschitz functions with a unique and non-degenerate minimum at 0 with minimum value 0. We also assume that they go linearly to infinity at infinity. We set ψ(x, ξ) = Ψ 1 (X 1 ) + Ψ 2 (X 2 ). Given K > 0, there exist ε, h 0 , C > 0, such that, for all h ∈ (0, h 0 ), µ ≤ b 0 + Kh and u solving (4.1), we have

R 4 e 2εψ(x,ξ)/h |T u| 2 dx dξ ≤ C R 4
|T u| 2 dx dξ (= C u 2 ).

4.1.

Decay away from the characteristic manifold. In this section, we establish the exponential decay of T u with respect to X 1 .

4.1.1. First estimate. One will need the following elementary lemma.

Lemma 4.2. Recall that p M (x, ξ) = g 11 ξ 2 1 -2g 12 ξ 1 x 1 + g 22 x 2 1 . Then, there exist non-negative numbers γ, c 1 , c 2 , c 3 such that (i). for all

|X 1 | ≥ γ, p M 1+p M ≥ c 1 , (ii). for all |X 1 | ≤ γ, p M 1+p M ≥ c 2 |X 1 | 2 . If, moreover, ε is small enough, (i). for all |X 1 | ≥ γ, Re p M 1+ p M ≥ c 1 , (ii). for all |X 1 | ≤ γ, Re p M 1+ p M ≥ Re p M -c 3 |X 1 | 4 , and Re p M ≥ c 2 |X 1 | 2
, where we used Notation 3.2. Theorem 4.3. Given K > 0, there exist ε, h 0 , C > 0 such that, for all h ∈ (0, h 0 ), µ ≤ K and u solving (4.1), we have

R 4 e 2ε(Ψ 1 (X 1 )+Ψ 2 (X 2 ))/h |T u| 2 dx dξ ≤ C R 4 e 2εΨ 2 (X 2 )/h |T u| 2 dx dξ .
Proof. Assume temporarily that Ψ 1 is bounded. Let us use Scholium 3.4 with m = 1. Then, taking the real part, we get

R 4 (Re p h -Ch)|u ψ | 2 dx dξ ≤ 0 . Recall p h (x, ξ) = p M (x, ξ) -hµ 1 + p M (x, ξ) + O(h 2 ) . Since p M ≥ 0, R 4 Re p M 1 + p M -C(1 + K)h |u ψ | 2 dx dξ ≤ 0 .
Consider R > 0 and the set

J R = {X ∈ R 4 : |X 1 | ≥ Rh 1 2 } . We write J R Re p M 1 + p M -C(1 + K)h |u ψ | 2 dx dξ ≤ - ∁J R Re p M 1 + p M -C(1 + K)h |u ψ | 2 dx dξ ,
and notice

∁J R Re p M 1 + p M -Ch |u ψ | 2 dx dξ ≤ C R h(1 + K) R 4 e 2Ψ 2 (X 2 )/h |T u| 2 dx dξ .
From Lemma 4.2, we get c2 > 0 such that on J R ,

Re p M 1 + p M -C(1 + K)h ≥ c2 R 2 h -C(1 + K)h .
Choosing R large enough, we get

J R |u ψ | 2 dx dξ ≤ C R R 4 e 2Ψ 2 (X 2 )/h |T u| 2 dx dξ ,
and then

R 4 |u ψ | 2 dx dξ ≤ C R 4 e 2Ψ 2 (X 2 )/h |T u| 2 dx dξ .
If Ψ 1 is not bounded, we introduce an appropriate cutoff function. For example, we apply the previous estimates to Ψ 1,k := χ(k -1 εΨ 1 (X 1 ))εΨ 1 (X 1 ) and send k to +∞. The estimates are independent of k because dΨ 1,k is uniformly bounded in S(1). Then, we conclude with the Fatou lemma. Then, for M > 0, there exist ǫ, h 0 , C > 0 such that, for all h ∈ (0, h 0 ), µ ≤ M and u solving (4.1), we have (mRe

R 4 me 2ε(Ψ 1 (X 1 )+Ψ 2 (X 2 ))/h |T u| 2 dx dξ ≤ C R 4 (m + h)e 2εΨ
p h (x, ξ; h) -Chm) |u ψ | 2 dx dξ ≤ Ch 2 R 4 e 2εΨ 2 (X 2 )/h |T u| 2 dx dξ .
Then, the analysis the same lines as in the proof of Theorem 4.4. The same splitting of the integral in the left-hand-side gives the conclusion.

4.2. Subprincipal decay estimates. Let us now prove an exponential estimate with respect to all the phase space variables. In the previous section, we essentially used the ellipticity of the operator outside of the characteristic set. The results, while new in this precision as far as we know, are not surprising. However, in this section, we have to understand what is happening directly on the characteristic set, i.e understand in detail the subprincipal terms. This is a much finer analysis. At the microlocal level, the computations are similar to the ones in [START_REF] Sjöstrand | Analytic wavefront sets and operators with multiple characteristics[END_REF]; however, instead of using the Boutet de Monvel calculus for polynomial operators, we directly use the invertibility of an effective harmonic oscillator.

Theorem 4.5. For M > 0, there exist ε, h 0 , C > 0 such that, for all h ∈ (0, h 0 ), µ ≤ b 0 + Mh and u solving (4.1), we have

R 4 e 2εΨ 2 (X 2 )/h |T u| 2 dx dξ ≤ C R 4
|T u| 2 dx dξ .

Proof. This time, ψ = Ψ(X 2 ). Let us use Scholium 3.4 again with m = 1. We get

R 4 Re p h - h 4 ∆p h -Ř -Ch 2 |u ψ | 2 dx dξ ≤ 0 .
With Theorem 4.3, we can estimate Ř and get

R 4 Re p h - h 4 ∆p h -Ch 2 |u ψ | 2 dx dξ ≤ 0 .
Observe that (4.2)

p h + O(h 2 ) = p M -hµ 1 + p M = p M -hµ + hµ p M 1 + p M - p 2 M 1 + p M .
The fourth term in the right-hand side is O(min(|X 1 | 4 , 1)), and can be absorbed using Theorem 4.3, and replaced by a O(h 2 ). The third term can also be absorbed in the same fashion, and replaced by O(h 2 µ). We deduce that

R 4 Re p M -b 0 h - h 4 Re ∆p h -C(1 + K)h 2 |u ψ | 2 dx dξ ≤ 0 .
Using Equation (4.2) to estimate the contribution from ∆p h , and using the same arguments,

R 4 Re p M -b 0 h - h 4 Re ∆ X 1 p M -C(1 + K)h 2 |u ψ | 2 dx dξ ≤ 0 ,
Now, we will approximate p M by a quadratic form in X 1 , with coefficients depending only on X 2 . To this end, let

Q X 2 (X 1 ) := g 11 |X 1 =0 ξ 2 1 -2 g 12 |X 1 =0 ξ 1 x 1 + g 22 |X 1 =0 x 2 1 .
(Observe that since ψ does not depend on X 1 , and differentiation in X 1 commute). Since the coefficients g ij are assumed to be critical at 0, and dψ(0) = 0, we find

Re p M - h 4 Re ∆ X 1 p M = Re Q X 2 (X 1 ) - h 2 Tr Re Q X 2 +O(|X 1 | 2 (min(|X 1 | 2 + |X 2 | 2 , 1) + h|X 1 | 2 + h|X 1 | min(|X 2 |, 1))) Using Theorem 4.4, we can absorb O(h k |X 1 | 2ℓ min(|X 2 | 2 , 1
)) and replace it by

O(h k+ℓ (min(|X 2 | 2 , 1) + h)).
Therefore, using also

|X 1 ||X 2 | ≤ ε -1 |X 1 | 2 + ε|X 2 | 2 , we get Re Q X 2 (X 1 ) -b 0 h - h 2 Tr Re Q X 2 -Cεh min(|X 2 | 2 , 1) |u ψ | 2 dX 1 dX 2 ≤ (1 + ε -1 + K)h 2 |u ψ | 2 dX 1 dX 2 .
For fixed X 2 , we recognize the Bargmann symbol of the "harmonic oscillator" in X 1 (see Remark 3.6) and thus

R 2 Re Q X 2 (X 1 ) - h 2 Tr Re Q X 2 |u ψ | 2 dX 1 ≥ h det Re Q X 2 |u ψ | 2 dX 1 . So that R 4 det Re Q X 2 -b 0 -Cε min(|X 2 | 2 , 1) -C(1 + ε -1 + K)h |u ψ | 2 dx dξ ≤ 0 . Recall now that B = √ det Q, so that √ det Re Q = B(1 + O(Tr Re Q -1 Im Q)) = B(1 + εO(min(|X 2 | 2 , 1))).
Under the conclusion on Lemma 2.1, we get the estimate

R 4 min(|X 2 | 2 , 1)(1 -Cǫ) -C(1 + ε -1 + K)h |u ψ | 2 dx dξ ≤ 0 .
The conclusion follows from the usual Agmon arguments, and again the fact that the constant only depend on derivatives of ψ.

Space exponential decay

We are now in position to prove Theorem 1.1. Let û ∈ L 2 (R 2 ) such that L h û = hµû and let u = M -1 U κ û = M * U κ û. We have Pu = 0, see Equation (4.1).

Remark 5.1. Following [START_REF] Martinez | An introduction to semiclassical and microlocal analysis[END_REF]Theorem 4.1.2], with Theorem 4.1, one could deduce (up to technicalities) that, if K is a compact set away from 0, we have

û L 2 (K) = O(e -c/h )
for some c > 0. Below, one will get a more explicit result. Already observe that we can drop the factor U κ . Indeed, since κ is uniformly bi-Lipschitz, it preserves spatial exponential decay. So we can concentrate on ũ := Mu = U κ û.

With the notation of Theorem 4.1, we have, for ǫ small enough,

T u ∈ L 2 ǫψ (R 4 ), where L 2 ǫψ (R 4 ) := L 2 (R 4 ; e ǫψ dx dξ) , with a uniform bound: T u L 2 ǫψ (R 4 ) ≤ C u L 2 (R 2 )
, where C does not depend on h. From this exponential decay in phase space, we wish to obtain exponential decay in the position variable x for ũ. We start with the inversion formula (3.1):

û = MT * (T u).
Let ϕ be a non-negative Lipschitz function, going linearly to infinity at infinity, having a unique and non-degenerate minimum at the origin, with minimal value 0 (let us call these functions admissible weights). We would like to obtain a uniform bound e ǫ ′ ϕ û L 2 (R 2 ) ≤ C û L 2 (R 2 ) , for some ǫ ′ > 0 small enough. Thus, it is enough to prove that the operator (5.1)

MT * : L 2 ǫψ (R 4 ) → L 2 ǫ ′ ϕ (R 2
) is uniformly bounded with respect to h ∈ ]0, h 0 ]. Lemma 5.2. Let ϕ 1 , ϕ 2 be admissible weights on R d . Then there exists C > 0 such that

(5.2) C -1 ϕ 1 ≤ ϕ 2 ≤ Cϕ 1 .
Proof. By the Taylor formula at the origin, the estimate (5.2) is valid in a neighborhood of 0. By the linear behavior at infinity, it is valid outside of a compact set. On the remaining compact subset of R d \ {0}, it is enough to use that the range of ϕ j is a compact interval of ]0, +∞].

A consequence of the lemma is that the choice of ϕ and ψ in (5.1) is not relevant, as long as we don't seek the optimal constants and are allowed to play with ǫ, ǫ ′ . Proposition 5.3. Given an admissible weight ϕ on R 2 , there exists an admissible weight ψ on R 4 of the form required by Theorem 4.1, and a constant C > 0 independent of h, such that for all ǫ ′ ≤ ǫ/C, ǫ ≤ 1, the operator MT * defined in (5.1) is bounded by O(1).

As a consequence, there exists ǫ ′ 0 such that if ǫ ′ ≤ ǫ ′ 0 , then there exists C > 0 such that

R 2 e ǫ ′ ϕ(x)/h |û(x)| 2 dx ≤ C û 2 L 2 (R 2 ) .
Proof. By Lemma 5.2, we can always change the function ϕ, so we will pick a convenient one. First, consider the C 1 Lipschitz function f defined by f (ρ) = ρ 2 if ρ ∈ [0, 1] and f (ρ) = 2ρ -1 if ρ ≥ 1. Notice that (5.3) ∀ρ ≥ 0 , f (2ρ) ≤ 4f (ρ) .

We define now the admissible weight ϕ(x) := f (|x|), x ∈ R 2 .

A formula for MT * can be obtained from the action of the FBI transform T on arbitrary metaplectic operators, see [26, 3.4]; here we derive it explicitly. We have We wish to conclude on the L 2 continuity of MT * by applying the Schur lemma.

For given (x, ξ), we make a change of variables to get Using again the convexity of ϕ, On the other hand, for a given y, we make an analogous change of variables: With this Proposition 5.3, the proof of Theorem 1.1 is complete. It would be very interesting to investigate the optimality of (ϕ, ǫ ′ ) for which Proposition 5.3 holds, in particular by relating ϕ ′′ (0) to the behaviour of the magnetic field at the origin.

(

Scholium 3 . 4 .

 34 Under the conclusion of Lemma 2.1, consider ψ bounded with dψ ∈ S(1) and m = m(X 2 ) ∈ S(1). Then me -ψ/h T Pu, e -ψ/h T u = R 4 |u ψ | 2 m p h + hms + hr + O(h 2 ) dX 1 dX 2 , where r, s ∈ S(1) and |r| ≤ C(min(|X 1 | 2 , 1)). Moreover, we have the following properties.

Theorem 4 . 4 .

 44 Consider m = m(X 2 ) non-negative with m ∈ S(1).

α h √ 2 , 4 e-.-

 24 T u(x, ξ) = α h e -ξ 2 2h (L ⋆ u)(z), with L(x) = e -x 2 2h , z := x -iξ ∈ C 2 .From (2.10) we obtainT M * u(x, ξ) = α h e -ξ 2 2h ((L ⋆ K) ⋆ u)(z), where L ⋆ K is a complex Gaussian that can be computed explicitly, using in particular that(I + iA) -1 = 1 2 (I -iA): L ⋆ K(y) = √ 2πhe )(z-y),z-y u(y) dy , αh =and therefore, taking the adjoint, we have forv ∈ L 2 ǫψ (R 4 ) (MT * )v(y) = αh R iA)(z-y),z-y v(x, ξ) dx dξ , z = x + iξ .Let K MT * (y, x, ξ) be the Schwartz kernel of e ǫ ′ ϕ h MT * e -ǫψ h , viewed as an operatorL 2 (R 4 ) → L 2 (R 2 ), i.e. K MT * (y, x, ξ) = αh e ǫ ′ ϕ(y) h -|ξ| 2 2h -1 4h (I-iA)(z-y),z-y -ǫψ(x,ξ) h .We haveRe (I -iA)(z -y), z -y = |x -y| 2 -|ξ| 2 -2 Aξ, x -y = |Aξ -(x -y)| 2 -2 |ξ| 2 ,where in the second line we used |Aξ| 2 = |ξ| 2 . Therefore,|K MT * (y, x, ξ)| ≤ αh e ǫ ′ ϕ(y) hLet us choose now, as we may, ψ(x, ξ) := ϕ(x) + |ξ| 2 / ξ . Indeed, ψ is not of the form Ψ 1 + Ψ 2 , but is bounded from below by a function of this form (this can be written explicitly, or by invoking Lemma 5.2). By convexity of ϕ, ϕ(y) ≤ 1 2 ϕ(2x) + 1 2 ϕ(2(y -x)) , and hence|K MT * (y, x, ξ)| ≤ αh e ǫ ′ ϕ(2x) 2h -ǫϕ(x) h + ǫ ′ ϕ(2(y-x)) 2h -|x-y-Aξ| 2 2h -ǫ|ξ| 2 h ξ .If ǫ ′ ≤ ǫ/2 then, by (5.3), ǫ ′ ϕ(2x)/2 ≤ ǫϕ(x) for all x ∈ R 2 so that (5.4) |K MT * (y, x, ξ)| ≤ αh e ǫ ′ ϕ(2(y-x)) 2h

R 2 |K 2 |K

 22 MT * (y, x, ξ)| dy = R MT * (y + x + Aξ, x, ξ)| dy.From (5.4) we have|K MT * (y + x + Aξ, x, ξ)| ≤ αh e ǫ ′ ϕ(2(y+Aξ)) 2h-

R 4 |K 4 |K 2 2h -ǫξ 2 h ξ Applying Equation ( 5 . 5 ) 4 |K 2 e -ǫξ 2 2h ξ dξ R 2 eǫ 4 |K

 442554224 MT * (y, x, ξ)| dx dξ = R MT * (y, x + y + Aξ, ξ)| dx dξ and write (5.4) as |K MT * (y, x + y + Aξ, ξ)| ≤ αh e ǫ ′ ϕ(2(x+Aξ)) 2h -|x| with y replaced by x, and choosing ǫ′ ≤ 2C ϕ ǫ, gives R MT * (y, x + y + Aξ, ξ)| dx dξ ≤ αh R ′ ϕ(4x) 4h -|x| 2 2h dx .Using that both integrals are O(h), we haveR MT * (y, x + y + Aξ, ξ)| dx dξ ≤ Ch 2 αh .Hence, the Schur lemma givesMT * = O( αh h 3/2 ) = O(1) : L 2 ǫψ (R 4 ) → L 2 ǫ ′ ϕ (R 2 ).

  4.1.2. Agmon estimate with multiplier. Let us now add a multiplier in the previous estimate. This can be done modulo O(h).

  2 (X 2 )/h |T u| 2 dx dξ . Re p ψ h,m (x, ξ; h)|u ψ | 2 dx dξ ≤ Ch 2 R 4 e 2εΨ 2 (X 2 )/h |T u| 2 dx dξ . (mRe p h (x, ξ; h) -Chm) |u ψ | 2 dx dξ ≤ Ch (X 2 )/h |T u| 2 dx dξ .Using again Theorem 4.3 with a smaller ε to absorb the |X 1 | 2 term,

	Then, by Scholium 3.4,
	R 4		R 4	|X 1 | 2 |u ψ | 2 dx dξ
	+ Ch 2 e 2εΨ 2 R 4 R 4
	Proof. We use again Scholium 3.4, this time without assuming that m = 1. We
	have
	Re	R 4	p ψ h,m (x, ξ; h)|u ψ | 2 dx dξ = O(h 2 ) u ψ	2 ,
	so that, with Theorem 4.3,
	R 4

  From Lemma 5.2, there existsC ϕ > 0 such that, ϕ(4Aξ) ≤ C ϕ |ξ| 2 / ξ . Hence, if ǫ ′ ≤ 4C ϕ ǫ, we get ǫ ′ ϕ(4Aξ)

	5.5)	ǫ ′ ϕ(2(y + Aξ)) 2h	-	|y| 2 2h	-	ǫξ 2 h ξ	≤	ǫ ′ ϕ(4y) 4h	+	ǫ ′ ϕ(4Aξ) 4h	-	|y| 2 2h	-	ǫξ 2 h ξ	.
			4h	-ǫξ 2 h ξ ≤ 0, and
		R 2	|K MT R 2	e	ǫ ′ ϕ(4y) 2h	-	|y| 2 2h dy .

* (y + x + Aξ, x, ξ)| dy ≤ αh Using Laplace's method, the integral on the right-hand side is O(h) provided ǫ ′ < 1/16. Hence, R 2 |K MT * (y, x, ξ)| dy ≤ Chα h .

The only known (and optimal) result of pure magnetic tunnelling has recently been proved in a two-dimensional setting in[START_REF] Bonnaillie-Noël | On a purely magnetic tunnel effect in two dimensions[END_REF] by means of microlocal dimensional reductions.

sometimes called coherent state decomposition; in relation with the magnetic Laplacian, it has been used in[START_REF] Bonnaillie-Noël | Magnetic WKB constructions[END_REF] Section 2.3]

Acknowledgments

N.R. is deeply grateful the Mittag-Leffler Institute where this work was started. N.R. also thanks Martin Vogel for many stimulating discussions in the Mittag-Leffler library.

Corollary 1.4. In L 2 (R 2 ), we can decompose u ℓ,J = αe iθ u ℓ + w with w orthogonal to u ℓ , θ ∈ R and α ≥ 0. Since the eigenvalues of L h are h 2 separated, the Spectral Theorem implies that (L h -λ ℓ (h))u ℓ,J ≥ Ch 2 w , so w ≤ Ch J . In particular, 1 -α ≤ (Ch J ) 2 /2. We then get 2 . Now, we turn to exponentially weighted spaces. We consider ǫ > 0 small enough so that Theorem 1.1 applies to 2ǫ. Then we observe that

Appendix A. Change of variables

Since L h is invariantly defined by the 2-form B and the Riemannian metric on M, its principal and subprincipal Weyl symbols are well defined, which implies that a change of variables like the one defined in Lemma 2.2 and used in Lemma 2.1 acts naturally on the Weyl symbol modulo terms of order O(h 2 ) (see also [START_REF] Morin | A semiclassical Birkhoff normal form for symplectic magnetic wells[END_REF]).

Here we give a direct proof of this and compute explicitly the O(h 2 ) remainder.

We have

)) + g * ∇(|g|

and

where

In terms of forms, this means that

We let P = -ih∇ y -Ã(y) and notice that

and then

, P ] ψ g * dy , so that

[|g| 1 4 , P ] ψ, |g| 4 , P ] ψ 2 g * dy .

Since [P, |g| 4 ) and à is real-valued, we deduce that 2Re ). Therefore,

V (y)|Uψ| 2 dy , and the conclusion follows.


Alexandre Leclercq

Alexandra Moura

Guillaume Vales

Nathalie Tessaud-Rita

Christine Aguilhon

Marc Lecuit
email: marc.lecuit@pasteur.fr

Nathalie Tessaud- Rita

Listeria thailandensis sp. nov

Keywords: Listeria, Thailand, chicken, new taxa, whole genome sequencing, ANI, AAI, POCP

Main text

The genus Listeria currently comprises 19 species which has been proposed to be subdivided in two major groups [START_REF] Chiara | Comparative genomics of Listeria Sensu Lato: Genus-wide differences in evolutionary dynamics and the progressive gain of complex, potentially pathogenicity-related traits through lateral gene transfer[END_REF][START_REF] Orsi | Characteristics and distribution of Listeria spp., including Listeria species newly described since 2009[END_REF][START_REF] Mclauchlin | Genus Listeria[END_REF]: (i) Listeria sensu stricto, constituted by the species Listeria monocytogenes [START_REF] Pirie | The genus Listerella Pirie[END_REF], L. innocua [START_REF] Seeliger | Nonpathogenic listeriae: L. innocua sp. n. (Seeliger et Schoofs, 1977) (author's transl)[END_REF], L. welshimeri [START_REF] Rocourt | Listeria welshimeri sp. nov. and Listeria seeligeri sp. nov[END_REF], L. seeligeri [START_REF] Rocourt | Listeria welshimeri sp. nov. and Listeria seeligeri sp. nov[END_REF], L. ivanovii [START_REF] Seeliger | Listeria ivanovii sp. nov[END_REF] and L. marthii [START_REF] Graves | Listeria marthii sp. nov., isolated from the natural environment, Finger Lakes National Forest[END_REF], and (ii) Listeria sensu lato, constituted by the species L. grayi [START_REF] Rocourt | Assignment of Listeria grayi and Listeria murrayi to a single species, Listeria grayi, with a revised description of Listeria grayi[END_REF][START_REF] Larsen | A mannitol fermenting Listeria: Listeria grayi sp[END_REF], L. rocourtiae [START_REF] Leclercq | Listeria rocourtiae sp. nov[END_REF], L. fleischmannii [START_REF] Den Bakker | Genome sequencing identifies Listeria fleischmannii subsp. coloradonensis subsp. nov., isolated from a ranch[END_REF][START_REF] Bertsch | Listeria fleischmannii sp. nov., isolated from cheese[END_REF], L. weihenstephanensis [START_REF] Halter | Listeria weihenstephanensis sp. nov., isolated from the water plant Lemna trisulca taken from a freshwater pond[END_REF],

L. floridensis [START_REF] Den Bakker | Listeria floridensis sp. nov., Listeria aquatica sp. nov., Listeria cornellensis sp. nov., Listeria riparia sp. nov. and Listeria grandensis sp. nov., from agricultural and natural environments[END_REF], L. aquatica [START_REF] Den Bakker | Listeria floridensis sp. nov., Listeria aquatica sp. nov., Listeria cornellensis sp. nov., Listeria riparia sp. nov. and Listeria grandensis sp. nov., from agricultural and natural environments[END_REF], L. cornellensis [START_REF] Den Bakker | Listeria floridensis sp. nov., Listeria aquatica sp. nov., Listeria cornellensis sp. nov., Listeria riparia sp. nov. and Listeria grandensis sp. nov., from agricultural and natural environments[END_REF], L. riparia [START_REF] Den Bakker | Listeria floridensis sp. nov., Listeria aquatica sp. nov., Listeria cornellensis sp. nov., Listeria riparia sp. nov. and Listeria grandensis sp. nov., from agricultural and natural environments[END_REF], L. grandensis [START_REF] Den Bakker | Listeria floridensis sp. nov., Listeria aquatica sp. nov., Listeria cornellensis sp. nov., Listeria riparia sp. nov. and Listeria grandensis sp. nov., from agricultural and natural environments[END_REF], L. booriae [START_REF] Weller | Listeria booriae sp. nov. and Listeria newyorkensis sp. nov., from food processing environments in the USA[END_REF], L. newyorkensis [START_REF] Weller | Listeria booriae sp. nov. and Listeria newyorkensis sp. nov., from food processing environments in the USA[END_REF], L. costaricensis [START_REF] Núñez-Montero | Listeria costaricensis sp. nov[END_REF] and L. goaensis [START_REF] Doijad | Listeria goaensis sp. nov[END_REF].

Of those, L. monocytogenes and L. ivanovii are reported to be pathogenic to human and other animals [START_REF] Allerberger | Listeriosis: a resurgent foodborne infection[END_REF] and constitute a major concern for the food industry and public health authorities.

In 2015, a Listeria-like bacterium was recovered from a food retailer in Thailand. The isolate was obtained from a fried chicken sample during a screening of Listeria spp. in food, using the enzyme linked fluorescent immunoassay VIDAS LIS and LPT methods (bioMérieux, Marcy l'Etoile, France). The isolate CLIP 2015/00305 T (originally designated as bioMérieux S.A. R&D Industrie) was received at the World Health Organization Collaborating Centre for Listeria, Paris, France, for characterization. Species identification by MALDI-TOF mass spectrometry using either the MicroFlex LT system with MBT library DB-5989 (Bruker Daltonics, Bremen, Germany) [START_REF] Thouvenot | MALDI-TOF mass spectrometry-based identification of Listeria species in surveillance: A prospective study[END_REF] or the VITEK 2 MS system (bioMérieux, Marcy l'Etoile, France) were inconclusive. Similarly, the API Listeria (bioMérieux API web database, version 2.0), VITEK 2 (bioMérieux, Marcy l'Etoile, France) biochemical systems also failed to provide any reliable identification (profiles 2731 and 510200021533601, respectively).

The genome of isolate CLIP 2015/00305 T was obtained after DNA extraction (DNeasy Blood & Tissue kit, Qiagen, København, Denmark), library preparation (Nextera XT DNA Sample kit, Illumina, California, USA), and sequencing using the NextSeq 500 (2 x 150 bp) platform (Illumina), according to the manufacturer's protocol, as previously described [START_REF] Moura | Real-time wholegenome sequencing for surveillance of Listeria monocytogenes, France[END_REF].

Phylogenetic analyses were performed based on the 16S rRNA gene sequence comparisons and on the concatenated deduced amino acid sequences of 243 core genes present in all Listeria species [START_REF] Núñez-Montero | Listeria costaricensis sp. nov[END_REF], defined using Roary v.3.11 [START_REF] Page | Roary: Rapid largescale prokaryote pan genome analysis[END_REF] and a BLASTP identity cut-off of 80%. Sequences were aligned using MUSCLE v.3.8 [START_REF] Edgar | MUSCLE: Multiple sequence alignment with high accuracy and high throughput[END_REF].

Maximum likelihood phylogenetic trees were inferred by using IQ-Tree v.1.5 [START_REF] Nguyen | IQ-TREE: A fast and effective stochastic algorithm for estimating maximum-likelihood phylogenies[END_REF] and visualized in MEGA v.7.0 [START_REF] Kumar | MEGA7: Molecular Evolutionary Genetics Analysis version 7.0 for bigger datasets[END_REF].

Whole-genome pairwise average nucleotide and amino acid identity comparisons (ANI and AAI, respectively) against the Listeria species reference genomes were determined using the enveomics package [START_REF] Rodriguez-R Lm | The enveomics collection : a toolbox for specialized analyses of microbial genomes and metagenomes[END_REF]. Dendrograms based on the unweighted pair group method with arithmetic mean (UPGMA) method were obtained from the distance matrixes using Bionumerics v.7.6 (Applied Maths, Sint-Martens-Latem, Belgium).

The percentage of conserved proteins (POCP) was calculated as described [START_REF] Qin | A proposed genus boundary for the prokaryotes based on genomic insights[END_REF].

Multiplex and in silico PCR-serogrouping were determined as described previously [START_REF] Doumith | Differentiation of the major Listeria monocytogenes serovars by multiplex PCR[END_REF][START_REF] Moura | Whole genomebased population biology and epidemiological surveillance of Listeria monocytogenes[END_REF]. The draft genome assembly was obtained using CLC Assembly Cell 4.3.0 (Qiagen, Hilden, Germany) as previously described [START_REF] Moura | Real-time wholegenome sequencing for surveillance of Listeria monocytogenes, France[END_REF], and annotated with Prokka v.1.12 [START_REF] Seemann | Prokka: Rapid prokaryotic genome annotation[END_REF].

The draft assembly of CLIP 2015/00305 T obtained from high-quality reads (final coverage of 108 X) contained 15 contigs, with a total length of 2.70 Mb and a N50 length of 554,335 bp. Data obtained followed the quality standards for its use for taxonomic purposes [START_REF] Chun | Proposed minimal standards for the use of genome data for the taxonomy of prokaryotes[END_REF]. The genome of CLIP 2015/00305 T contained 2,689 predicted protein-coding sequences, 1,107 (41%) of which with no known function.

16S rRNA gene phylogenetic analysis showed highest similarity with L. floridensis (98.9%) and L. costaricensis (98.8%; Fig. 1), in the borderline of the proposed species cut-off based on 16S sequence similarity (98.7-99.0%; [START_REF] Kim | Towards a taxonomic coherence between average nucleotide identity and 16S rRNA gene sequence similarity for species demarcation of prokaryotes[END_REF][START_REF] Erko | Taxonomic parameters revisited: tarnished gold standards[END_REF]). Maximum likelihood phylogeny based on the amino acid sequences of Listeria core genes (Fig. 2) placed the novel taxon within the L. aquatica, L. floridensis, L. fleishmannii and L. costaricensis clade (Listeria sensu lato), with highest similarity to L. aquatica. ANI analysis revealed that CLIP 2015/00305 T shared less than 90% genome sequence identity with all known Listeria species (Fig. 3), thus lower than the proposed genomic species cut-off of 95% [START_REF] Goris | DNA-DNA hybridization values and their relationship to whole-genome sequence similarities[END_REF]. Highest ANI values were obtained L. aquatica (two-way ANI of 85.3±4.4%, based on 8082 genome fragments). Consistently, analyses based on the deduced proteomes also showed highest identities L. aquatica (two-way AAI of 91.4±9.4%, based on 2180 orthologous proteins; POCP of 83.1%). Both AAI and POCP pairwise analyses were higher than proposed cut-offs for genus delineation of 60% and 50%, respectively [START_REF] Goris | DNA-DNA hybridization values and their relationship to whole-genome sequence similarities[END_REF][START_REF] Rodriguez-R Lm | Bypassing cultivation to identify bacterial species[END_REF], confirming that the CLIP 2015/00305 T represents a novel Listeria taxon.

In silico and multiplex PCR-serogrouping were positive for the prs gene (serogroup L, typical of Listeria species with the exception for L. monocytogenes [START_REF] Doumith | Differentiation of the major Listeria monocytogenes serovars by multiplex PCR[END_REF]).

Phenotypic analyses of isolate CLIP 2015/00305 T were carried out on BHI agar plates, Agar Listeria according to Ottaviani and Agosti (ALOA; bioMérieux, Craponne, France) after incubation for 24h at 30 and 37 °C. The presence of a capsule was tested using the India ink test [START_REF] Hussey | Endospore stain protocol[END_REF]. Gram staining, catalase and oxidase activities, respiratory characteristics and endospore formation were tested as described in the Bacteriological Analytical Manual (BAM) [START_REF] Hitchins | Detection of Listeria monocytogenes in foods and environmental samples, and enumeration of Listeria monocytogenes in foods[END_REF]. Growth characteristics were determined on BHI agar and broth at 4 °C for 10 days and at 22, 30, 37 and 42 °C for 7 days.

Growth was considered positive if there was an increase in cell number of at least 1.0 log (cfu.ml -1 ) over 14 days. Motility was tested by stab-inoculation in mannitol-mobility semi-solid agar (Bio-Rad, Marnes-La-Coquette, France) and incubation at different growth temperatures (4 °C, 22 °C and 37 °C) for 10 days in aerobic conditions. L. monocytogenes ATCC 35152 T and L. booriae CIP 111022 T were used as positive and negative controls, respectively. Nitrate reduction and Voges-Proskauer tests were performed as described in the BAM [START_REF] Hitchins | Detection of Listeria monocytogenes in foods and environmental samples, and enumeration of Listeria monocytogenes in foods[END_REF]. Haemolysis was tested using either Columbia agar plates containing 5 % defibrinated horse blood (bioMérieux) or the Christie, Atkins, Munch-Petersen (CAMP) test on Columbia agar containing 5 % defibrinated sheep blood (bioMérieux) as described in the BAM [START_REF] Hitchins | Detection of Listeria monocytogenes in foods and environmental samples, and enumeration of Listeria monocytogenes in foods[END_REF]. Biochemical tests performed with API Listeria strips (bioMérieux), the API50CH system (bioMérieux), RAPIDEC Lmono (bioMérieux) and VITEK 2 version 07.01 (bioMérieux) with Gram positive (GP) card as recommended by the manufacturer [START_REF] Bille | API Listeria, a new and promising one-day system to identify Listeria isolates[END_REF]. API Listeria tests were recorded after incubation at 37 °C for 24h. API50CH tests were recorded after 2, 5, 10 and 15 days of incubation at 37 °C. Susceptibility to a wide range of antibacterial agents was determined with the disk diffusion method on Mueller-Hinton agar plates (Bio-Rad), using the interpretative criteria and recommendations from the French Microbiology Society and the European Committee on Antibiotic Susceptibility Testing [40,[START_REF]Testing breakpoint tables for interpretation of MICs and zone diameters[END_REF].

According to its phenotypic and biochemical profile (Table 1), CLIP 2015/00305 T clustered together with L. aquatica (Fig. 4).

Cells of CLIP 2015/00305 T were Gram-stain-positive facultative anaerobic rods, oxidase negative and catalase positive. Colonies on BHI were opaque, flat, with a diameter of 0.5 to 1.0 mm. On Palcam agar, colonies were approximately 2mm in diameter, grey-green in colour with a black sunken centre and a black halo against a cherry-red medium background. On ALOA agar, colonies were blue, due to βglucosidase activity, and not surrounded by a white halo, denoting the absence of phosphatidylinositol-specific phospholipase C (PI-PLC) activity (Table 1).

CLIP 2015/00305 T showed growth between 22 °C and 42 °C but not at 4 °C. No capsule or spores were detected. Motility was also absent.

Voges-Proskauer and nitrate reductase tests were positive (Table 1).

The CAMP test was negative and no haemolysis was detected in either horse or sheep bloods. The lack of haemolysis was consistent with the absence of Listeria pathogenicity islands (LIPI-1 to -4) [START_REF] Vázquez-Boland | Listeria pathogenesis and molecular virulence determinants[END_REF][START_REF] Maury | Uncovering Listeria monocytogenes hypervirulence by harnessing its biodiversity[END_REF] within the draft genome of CLIP 2015/00305 T , combined with absence of internalins inlABCDEFGHJK genes and genes coding bile resistance, suggesting that this novel taxon is not pathogenic.

CLIP 2015/00305 T was sensitive to penicillin G, ampicillin, amoxicillin, imipenem, kanamycin, streptomycin, gentamicin, rifampicin, erythromycin, levofloxacin, moxifloxacin, tetracycline, chloramphenicol, fusidic acid, trimethoprim and vancomycin. However, CLIP 2015/00305 T showed resistance to nalidixic acid, fosfomycin, sulphonamides, clindamycin, ciprofloxacin and cefotaxime, as other Listeria species [START_REF] Troxler | Natural antibiotic susceptibility of Listeria species: L. grayi, L. innocua, L. ivanovii, L. monocytogenes, L. seeligeri and L. welshimeri strains[END_REF]. Within the draft genome of CLIP 2015/00305 T , genes conferring resistance to nalidixic acid (norB), fosfomycin (fosX), sulphonamides (sul), lincomycin (lmrB) and quaternary ammonium compounds (qacA) used in food industry for cleaning and disinfection were present. Thus, on the basis of the molecular findings described above as well as the phenotypic distinctiveness of strain CLIP 2015/00305 T , we propose that this strain should be classified as a member of a novel species of the genus Listeria for which the name Listeria thailandensis sp. nov. is proposed.

DESCRIPTION OF LISTERIA THAILANDENSIS SP. NOV.

Listeria thailandensis (thai.land.en'sis N.L. fem. adj. thailandensis, "from Thailand", the country where the type strain was originally isolated).

Cells are straight, Gram-stain-positive, non-motile and non-spore-forming rods.

Facultative aerobic, catalase-positive and oxidase-negative. Capsule is not formed.

Colonies are opaque, not pigmented, with a flat shape and entire margin on BHI. On ALOA, colonies are blue centred without white halo, typical of Listeria species. Growth occurs at 22-42 °C, with optimal growth at 30-37 °C. Negative for haemolysis and nitrite

Conflicts of interest

The authors declare no conflict of interests.

Abbreviations:

AAI

  reduction. Positive for Voges-Proskauer and nitrate reduction tests. Acid is produced from aesculin, N-acetylglucosamine, amygdalin, arbutin, D-cellobiose, D-fructose, gentiobiose, D-glucose, glycerol (weakly positive), D-mannose, L-rhamnose, D-ribose, salicin, dulcitol, D-trehalose, D-tagatose, methyl α-D-glucoside, D-arabitol, inositol and D-xylose. Acid is not produced from D-adonitol, D-arabinose, L-arabitol, D-galactose, L-fucose, D-lactose, D-maltose, D-saccharose, methyl α-D-mannopyranoside, methyl α-D-glucopyranoside, methyl -D-xylopyranoside, potassium gluconate, potassium 5ketogluconate, L-arabinose, glucose 1-phosphate, xylitol, starch, erythritol, D-fucose, glycogen, inulin, D-mannitol, D-melezitose, D-melibiose, potassium 2-ketogluconate, D-raffinose, D-sorbitol, L-sorbose, D-turanose, D-lyxose, L-xylose. It can be differentiated from other species of the genus Listeria by the production of acid from D-tagatose and inositol. The type strain CLIP 2015/00305 T (=CIP 111635 T = DSM 107638 T ) was isolated in March 2015 from a fried chicken sample in Thailand. The genomic DNA G+C content of the type strain is 40.3 mol%. Cell Cultures GmbH (Braunschweig, Germany) and the Collection of Institut Pasteur (Paris, France) for the deposit of the type strain in their collections.

FIGURE 1 .

 1 FIGURE 1. Phylogenetic analysis of the 16S rRNA gene based on the maximum likelihood method. Distance estimation was obtained by the model of Kimura 2parameter [45]. Selected members of Brochothrix genus were used as outgroup. Positions containing gaps and missing data were eliminated, resulting in a total of 1072 positions. Branch lengths represent the number of nucleotide substitutions per site and bootstrap percentages of 1,000 replicates are shown. GenBank accession numbers are provided in brackets. The newly Listeria thailandensis isolate is highlighted in bold.

FIGURE 2 .

 2 FIGURE 2. Maximum likelihood phylogenetic analysis based on the concatenated amino acid sequences of 243 core genes present in all Listeria species. Distance estimation was obtained by the Whelan and Goldman (WAG) model [46]. Branch lengths represent the number of amino acid substitutions per site and bootstrap percentages of 1,000 replicates are shown. GenBank accession numbers are provided in brackets.

FIGURE 3 .

 3 FIGURE 3. UPGMA clustering based on the genomic average nucleotide difference (ANI).The vertical dashed bar represents the proposed 95% ANIb species cut-off that correlates with the 70% DNA-DNA hybridization threshold[START_REF] Goris | DNA-DNA hybridization values and their relationship to whole-genome sequence similarities[END_REF]. Scale bar represents the percentage of similarity.

FIGURE 4 . UPGMA analysis based on the 33 biochemical characteristics of Listeria species shown in Table 1 .

 41 FIGURE 4. UPGMA analysis based on the 33 biochemical characteristics ofListeria species shown in Table1. Unknown data and traits with variations between different isolates and/or studies were ignored in the pairwise analyses. Scale bar represents the percentage of similarity of phenotypical profiles.

  FIGURE 2

  FIGURE 3

  FIGURE 4

Acknowledgments

We thank the P2M platform (Institut Pasteur, Paris, France) for genome sequencing.

We also thank the Leibniz Institute DSMZ-German Collection for Microorganisms and

Funding information

This work was supported by Institut Pasteur and Inserm.

All species/strains are positive for aesculine and acid production from N-acetylglucosamine, amygdalin, arbutin, D-cellobiose, D-fructose, Dmannose and salicin.


Kattia Núñez-Montero

Alexandre Leclercq

Alexandra Moura

Guillaume Vales

Johnny Peraza

Javier Pizarro-Cerdá
email: javier.pizarro-cerda@pasteur.fr

Marc Lecuit
email: marc.lecuit@pasteur.fr

Listeria costaricensis sp. nov

Keywords: Listeria, Costa Rica, new taxa, whole genome sequencing, ANI, AAI, POCP

the average nucleotide BLAST identity (ANI<80%), indicated that this isolate belonged to a new species. Results on the pairwise amino acid identity (AAI>70%) and on the percentage of conserved proteins (POCP>68%) with currently known Listeria species, as well as on biochemical characterization, confirmed that the strain constituted a new species within the Listeria genus. The name Listeria costaricensis sp. nov. is proposed for the novel species, and is represented by the type strain CLIP 2016/00682 T (=CIP 111400 T =DSM 105474 T ).

The GenBank/EMBL/DDBJ accession numbers for the draft genome sequence of strain CLIP 2016/00682 T are FXUT01000001-FXUT01000044.

Main text

With the rapid development of whole genome sequencing technologies, the genus Listeria has expanded since 2009 to reach 17 species with validly published names [START_REF] Orsi | Characteristics and distribution of Listeria spp., including Listeria species newly described since 2009[END_REF]. Recently, the subdivision of Listeria genus in two major groups has been proposed [START_REF] Orsi | Characteristics and distribution of Listeria spp., including Listeria species newly described since 2009[END_REF][START_REF] Chiara | Comparative Genomics of Listeria sensu lato: Genus-Wide Differences in Evolutionary Dynamics and the Progressive Gain of Complex, Potentially Pathogenicity-Related Traits through Lateral Gene Transfer[END_REF][START_REF] Den Bakker | Genome sequencing identifies Listeria fleischmannii subsp. coloradonensis subsp. nov., isolated from a ranch[END_REF][START_REF] Den Bakker | Listeria floridensis sp. nov., Listeria aquatica sp. nov., Listeria cornellensis sp. nov., Listeria riparia sp. nov. and Listeria grandensis sp. nov., from agricultural and natural environments[END_REF][START_REF] Mclauchlin | Genus I. Listeria Piere 1940a 383 AL[END_REF]: (i) Listeria sensu stricto, constituted by the species Listeria monocytogenes [START_REF] Pirie | The Genus Listerella Pirie[END_REF],

L. innocua [START_REF] Seeliger | Nonpathogenic Listeriae: L. innocua sp. n[END_REF], L. welshimeri [START_REF] Rocourt | Listeria welshimeri sp. nov. and Listeria seeligeri sp. nov[END_REF], L. seeligeri [START_REF] Rocourt | Listeria welshimeri sp. nov. and Listeria seeligeri sp. nov[END_REF] and L. ivanovii [START_REF] Seeliger | Listeria ivanovii sp. nov[END_REF], and (ii) Listeria sensu lato, presumably non-pathogenic and constituted by the species L. grayi [START_REF] Larsen | A mannitol fermenting Listeria: Listeria grayi sp[END_REF][START_REF] Rocourt | Assignment of Listeria grayi and Listeria murrayi to a single species, Listeria grayi, with a revised description of Listeria grayi[END_REF], L. marthii [START_REF] Graves | Listeria marthii sp. nov., isolated from the natural environment, Finger Lakes National Forest[END_REF], L. rocourtiae [START_REF] Leclercq | Listeria rocourtiae sp. nov[END_REF], L. fleischmannii [START_REF] Den Bakker | Genome sequencing identifies Listeria fleischmannii subsp. coloradonensis subsp. nov., isolated from a ranch[END_REF][START_REF] Bertsch | Listeria fleischmannii sp. nov., isolated from cheese[END_REF], L. weihenstephanensis [START_REF] Halter | Listeria weihenstephanensis sp. nov., isolated from the water plant Lemna trisulca taken from a freshwater pond[END_REF], L. floridensis, L. aquatica, L. cornellensis, L. riparia, L. grandensis [START_REF] Den Bakker | Listeria floridensis sp. nov., Listeria aquatica sp. nov., Listeria cornellensis sp. nov., Listeria riparia sp. nov. and Listeria grandensis sp. nov., from agricultural and natural environments[END_REF], L. booriae and L.

newyorkensis [START_REF] Weller | Listeria booriae sp. nov. and Listeria newyorkensis sp. nov., from food processing environments in the USA[END_REF]. The two species L. monocytogenes and L. ivanovii are pathogenic for human and animals [START_REF] Allerberger | Listeriosis: a resurgent foodborne infection[END_REF]. Due to the rarity of L. ivanovii infection cases, only L. monocytogenes represents a worldwide public health concern [START_REF] Guillet | Human listeriosis caused by Listeria ivanovii[END_REF][START_REF] Leclercq | Global burden of listeriosis: the tip of the iceberg[END_REF].

In 2016, the World Health Organization Collaborating Centre for Listeria (WHOCCL), Paris, France, received the isolate CLIP 2016/00682 (originally designated ES106) from the Institute of Technology of Costa Rica. The isolate was collected from a food processing drainage system in the province of Alajuela, Costa Rica, in August 2015, and characterized using the method described in the Bacteriological Analytical Manual (BAM) [START_REF] Hitchins | Detection of Listeria monocytogenes in Foods and Environmental Samples, and Enumeration of Listeria monocytogenes in Foods[END_REF]. Briefly, selective enrichment was performed using buffered Listeria enrichment broth (BLEB, Oxoid, Basingstoke, United Kingdom) with subsequent selective plating in Modified Oxford Listeria selective agar (MOX, Oxoid). Typical Listeria species colonies growing on MOX agar were characterized by PCR using the primers List-univ 1 and List-univ 2, as described [START_REF] Cocolin | Direct identification in food samples of Listeria spp. and Listeria monocytogenes by molecular methods[END_REF]. Positive PCR amplification was assumed as Listeria spp. and the isolate was sent to WHOCCL for further identification. Species identification of the isolate was attempted using MALDI-TOF mass spectrometry, based on the mass spectra obtained from full protein extraction protocol analyzed on a MicroFlex LT system (Bruker Daltonics, Bremen, Germany), according to the manufacturer's instructions [START_REF] Thouvenot | MALDI-TOF mass spectrometry-based identification of Listeria species in surveillance: A prospective study[END_REF]. Spectra analyses against the MBT library (DB-5989 MS; Bruker Daltonics), which is limited to 8 Listeria species [START_REF] Thouvenot | MALDI-TOF mass spectrometry-based identification of Listeria species in surveillance: A prospective study[END_REF], were inconclusive. The genome of isolate CLIP 2016/00682 T was sequenced after DNA extraction (DNeasy Blood & Tissue kit, Qiagen, København, Denmark), library preparation (Nextera XT DNA Sample kit, Illumina, California, USA), using NextSeq 500 (2 x 150 bp) platform (Illumina), according to the manufacturer's protocol, as described [START_REF] Moura | Real-time whole-genome sequencing for surveillance of Listeria monocytogenes, France[END_REF]. The data obtained followed the quality standards for its use for taxonomic purposes [START_REF] Chun | Proposed minimal standards for the use of genome data for the taxonomy of prokaryotes[END_REF]. The draft genome assembly was obtained from high-quality reads (final coverage of 108 X) using CLC Assembly Cell 4.3.0 (Qiagen, Hilden, Germany) as described [START_REF] Moura | Real-time whole-genome sequencing for surveillance of Listeria monocytogenes, France[END_REF], and annotated with Prokka v.1.9 [START_REF] Seemann | Prokka: rapid prokaryotic genome annotation[END_REF]. The draft assembly contained 44 contigs, with a total length of 3.18 Mb and a N50 length of 224,949 bp.

Phylogenetic analyses were performed based on the 16S rRNA gene sequence comparisons and on the concatenated deduced amino acid sequences of 243 core genes present in all Listeria species, defined in this study using Roary v.3.11 [START_REF] Page | Roary: rapid largescale prokaryote pan genome analysis[END_REF] and a BLASTP identity cut-off of 80%. Sequences were aligned using MUSCLE v.3.8 [START_REF] Edgar | MUSCLE: multiple sequence alignment with high accuracy and high throughput[END_REF] and maximum likelihood phylogenetic trees were inferred by using IQ-Tree v.1.5 [START_REF] Nguyen | IQ-TREE: a fast and effective stochastic algorithm for estimating maximum-likelihood phylogenies[END_REF].

16S rRNA gene phylogenetic analysis showed highest similarity with L. floridensis (98.71%; Fig. 1), i.e. at the lower cut-off value previously proposed for species delineation based on 16S sequence similarity (98.7-99.0%; [START_REF] Kim | Towards a taxonomic coherence between average nucleotide identity and 16S rRNA gene sequence similarity for species demarcation of prokaryotes[END_REF][START_REF] Stackebrandt | Taxonomic parameters revisited: tarnished gold standards[END_REF]). Maximum likelihood phylogeny based on the amino acid sequences of Listeria core genes (Fig. 2) placed the novel taxon within the L. fleishmannii, L. floridensis and L. aquatica clade (Listeria sensu lato).

Species identification was determined by whole-genome pairwise average nucleotide and amino acid BLAST identity comparisons (ANI and AAI, respectively) against

Listeria species reference genomes deposited at NCBI database, using the enveomics package [START_REF] Rodriguez-R L | The enveomics collection : a toolbox for specialized analyses of microbial genomes and metagenomes[END_REF]. The percentage of conserved proteins (POCP) was calculated as described by Qin et al. [START_REF] Qin | A proposed genus boundary for the prokaryotes based on genomic insights[END_REF]. ANI analysis revealed that CLIP 2016/00682 T shared less than 80% genome sequence identity with all known Listeria species (Fig. 3), thus lower than the proposed genomic species cut-off of 95% [START_REF] Goris | DNA-DNA hybridization values and their relationship to whole-genome sequence similarities[END_REF]. Further analysis based on the deduced proteomes showed the highest AAI with L. fleischmannii subsp. fleischmannii (two-way AAI of 70.40%, based on 2041 orthologous proteins) and L. fleischmannii subsp. cornellensis (two-way AAI of 70.09%, based on 2056 orthologous proteins), above the 60% cut-off proposed for genus delineation [START_REF] Rodriguez-R Lm | Bypassing cultivation to identify Bacterial species[END_REF]. The pairwise POCP, which takes also into account the size of deduced proteomes [START_REF] Qin | A proposed genus boundary for the prokaryotes based on genomic insights[END_REF], between CLIP 2016/00682 T and L. fleischmannii subspecies (POCP=68%), was also higher than the genus boundary cut-off of 50%. These results confirmed that the unusual isolate represented a novel Listeria taxon.

In silico PCR-serogrouping [START_REF] Doumith | Differentiation of the major Listeria monocytogenes serovars by multiplex PCR[END_REF] was positive for the prs gene (serogroup L, typical of Listeria species with the exception for L. monocytogenes), although negative by multiplex PCR due to primers mismatches.

The isolate CLIP 2016/00682 T was also characterized at phenotypic and biochemical levels Table 1 (adapted from [START_REF] Weller | Listeria booriae sp. nov. and Listeria newyorkensis sp. nov., from food processing environments in the USA[END_REF]).

CLIP 2016/00682 T was grown on BHI agar plates, Agar Listeria according to Ottaviani and Agosti (ALOA; bioMérieux, Craponne, France) and on MOX (Oxoid) agar plates after incubation for 24h at 30 or 37°C. Colonies on BHI were opaque, flat and yellow pigmented, with a diameter of 0.5 to 1.0 mm. The production of a yellow pigment in CLIP 2016/00682 T (Supplementary Fig. S1) is atypical in Listeria species. On MOX agar, colonies were dimpled, round and pewter coloured with black haloes. On ALOA Agar, colonies were blue, due to β-glucosidase activity, and not surrounded by a white halo, denoting the absence of phosphatidylinositol-specific phospholipase C (PI-PLC) activity (Table 1).

CLIP 2016/00682 T did not show the presence of capsule after performing the India ink test as described by Hughes and Smith (2013) [START_REF] Hussey | Endospore Stain Protocol. Laboratory protocols[END_REF]. Gram staining, catalase and oxidase activities, respiratory characteristics and endospore formation were tested as described in the BAM [START_REF] Guillet | Human listeriosis caused by Listeria ivanovii[END_REF]. The isolate was a Gram-stain-positive, rod-shaped bacterium, that was oxidase negative, non-spore-forming and facultative anaerobic.

CLIP 2016/00682 T was catalase negative. The organism was catalase negative, in contrast to all other 17 Listeria species. Expression of catalase is actually a major phenotypic characteristic of the Listeria genus, although there are reports of catalase negative Listeria monocytogenes strains [START_REF] Cepeda | Listeriosis due to infection with a catalase-negative strain of Listeria monocytogenes[END_REF][START_REF] Swartz | Catalase-negative Listeria monocytogenes causing meningitis in an adult. Clinical and laboratory features[END_REF]. The lack of catalase activity in CLIP 2016/00682 T was consistent with the absence of catalase gene in the draft genome.

To determine growth characteristics, isolates were grown on BHI agar and broth at 4 °C for 10 days and at 22, 30, 37 and 42 °C for 7 days. Growth was considered positive if there was an increase in cell number of at least 1.0 log (cfu.ml -1 ) over 14 days. CLIP 2016/00682 T showed growth between 22 °C and 42 °C but not at 4°C. Motility was tested by stab-inoculation in mannitol-mobility semi-solid agar (Bio-Rad, Marnes-La-Coquette, France) and incubation at different growth temperatures (4 °C, 22 °C and 37 °C) for 10 days in aerobic conditions. Strains L. monocytogenes ATCC 35152 T and L. booriae CIP 111022 T were used as positive and negative controls, respectively. Contrarily to other Listeria sensu lato species [START_REF] Den Bakker | Listeria floridensis sp. nov., Listeria aquatica sp. nov., Listeria cornellensis sp. nov., Listeria riparia sp. nov. and Listeria grandensis sp. nov., from agricultural and natural environments[END_REF][START_REF] Weller | Listeria booriae sp. nov. and Listeria newyorkensis sp. nov., from food processing environments in the USA[END_REF], CLIP 2016/00682 T was motile (only at 37 °C), giving a typical umbrella-like growth pattern, due to the presence of the flagella biosynthesis regulon typical of Listeria sensu stricto. Nitrate reduction and Voges-Proskauer tests were performed as described in the BAM [START_REF] Guillet | Human listeriosis caused by Listeria ivanovii[END_REF].

Nitrate reductase tests were positive (Table 1), whereas nitrite reduction was negative.

Voges-Proskauer were positive (Table 1).

No haemolysis was detected using either Columbia agar plates containing 5 % defibrinated horse blood (bioMérieux) or the Christie, Atkins, Munch-Petersen (CAMP) test on Columbia agar containing 5 % defibrinated sheep blood (bioMérieux) as described in the BAM [START_REF] Guillet | Human listeriosis caused by Listeria ivanovii[END_REF]. The lack of haemolysis was consistent with the absence of Listeria pathogenicity islands (LIPI-1 to -4) [START_REF] Maury | Uncovering Listeria monocytogenes hypervirulence by harnessing its biodiversity[END_REF][START_REF] Vazquez-Boland | Listeria pathogenesis and molecular virulence determinants[END_REF] within the draft genome of CLIP 2016/00682 T , suggesting that this novel taxon is not pathogenic.

Biochemical tests performed with API Listeria strips (bioMérieux) and the API50CH system (bioMérieux) as recommended by the manufacturer [START_REF] Bille | API Listeria, a new and promising one-day system to identify Listeria isolates[END_REF]. API Listeria tests were recorded after incubation at 37 °C for 24h. API50CH tests were recorded after 2, 5, 10 and 15 days of incubation at 30 °C. Results are summarized in Table 1. CLIP 2016/00682 T fit the API Listeria numerical profile 2730 of Listeria ivanovii (identification at 99.7%; bioMérieux API web database, version 04/2017) with divergent tests for Darylamidase and glucose-1-phosphate.

Interestingly, according to its phenotypic profile (Table 1), isolate CLIP 2016/00682 T clustered together with Listeria grayi and the subgroup of Listeria sensu stricto (Fig. 4).

Susceptibility to a wide range of antibacterial agents was determined with the disk diffusion method on Mueller-Hinton agar plates (Bio-Rad), using the interpretative criteria and recommendations from the French Microbiology Society and the European Committee on Antibiotic Susceptibility Testing [START_REF] Eucast | Breakpoint tables for interpretation of MICs and zone diameters, Version 7.1[END_REF]43]. CLIP 2016/00682 T was sensitive to penicillin G, ampicillin, amoxicillin, imipenem, kanamycin, streptomycin, gentamicin, fosfomycin, rifampicin, erythromycin, levofloxacin, moxifloxacin, clindamycin, tetracycline, chloramphenicol, sulphonamides, trimethoprim and vancomycin. However, CLIP 2016/00682 T showed resistance to nalidixic acid and cefotaxime, and intermediate resistance to ciprofloxacin and fusidic acid, as other Listeria species [START_REF] Troxler | Natural antibiotic susceptibility of Listeria species: L. grayi, L. innocua, L. ivanovii, L. monocytogenes, L. seeligeri and L. welshimeri strains[END_REF].

Thus, on the basis of the molecular findings described above as well as the phenotypic distinctiveness of strain CLIP 2016/00682 T , we propose that this strain should be classified as a member of a novel species of the genus Listeria for which the name Listeria costaricensis sp. nov. is proposed.

DESCRIPTION OF LISTERIA COSTARICENSIS SP. NOV.

Listeria costaricensis (cos.ta.ri.cen'sis N.L. fem. adj. costaricensis, "from Costa Rica", the country where the type strain was isolated).

Cells are straight, Gram-stain-positive, straight rods. Capsule is not formed. Spores are not produced. Colonies are opaque, yellow pigmented (atypical in Listeria species), with a flat shape and entire margin on BHI. On ALOA, colonies are blue centred without white halo, typical of Listeria species. On MOX, colonies are convex and pewter.

Growth occurs at 22-42 °C, with optimal growth at 30-37 °C. Motile at 37°C. Negative

FIGURE 1 .

 1 FIGURE 1. Phylogenetic analysis of the 16S rRNA gene based on the maximum likelihood method. Distance estimation was obtained by the model of Kimura 2-Parameter [45]. Selected members of Brochothrix genus were used as outgroup. Positions containing gaps and missing data were eliminated, resulting in a total of 1071 positions. Branch lengths represent the number of nucleotide substitutions per site and bootstrap percentages of 1,000 replicates are shown. GenBank accession numbers are provided in brackets. The newly Listeria costaricensis isolate is highlighted in bold.

FIGURE 2 .

 2 FIGURE 2. Maximum likelihood phylogenetic analysis based on the concatenated amino acid sequences of 243 core genes present in all Listeria species. Distance estimation was obtained by the model of WAG [46]. Branch lengths represent the number of amino acid substitutions per site and bootstrap percentages of 1,000 replicates are shown. GenBank accession numbers are provided in brackets. The newly Listeria costaricensis isolate is highlighted in bold.

FIGURE 3 .

 3 FIGURE 3. Unweighted pair group method with arithmetic mean (UPGMA)clustering based on the genomic average nucleotide difference (ANI). The vertical dashed bar represents the proposed 95% ANI species cut-off that correlates with the 70% DNA-DNA hybridization threshold[START_REF] Goris | DNA-DNA hybridization values and their relationship to whole-genome sequence similarities[END_REF]. Scale bar represents the percentage of similarity. The newly Listeria costaricensis isolate is highlighted in bold.

FIGURE 4 . 1 .

 41 FIGURE 4. Unweighted pair group method with arithmetic mean (UPGMA) analysis based on the 33 biochemical characteristics of Listeria species shown in Table1. Unknown data and traits with variations between different isolates and/or studies were ignored in the clustering analysis. Scale bar represents the percentage of similarity of biochemical profiles.

FIGURE 1 .

 1 FIGURE 1.

  FIGURE 3.

FIGURE 4 .

 4 FIGURE 4.

This work was supported by Institut Pasteur, INSERM and the Vicerrectoría de Investigación y Extensión-Instituto Tecnológico de Costa Rica (Research Project VIE-1510068). Acknowledgments We thank the P2M platform (Institut Pasteur, Paris, France) for genome sequencing and the Leibniz Institute DSMZ-German Collection for Microorganisms and Cell Cultures GmbH (Braunschweig, Germany) and the Collection of Institut Pasteur (Paris, France) for the deposit of the type strain in their collections.

Conflicts of interest

The authors declare no conflict of interests. A B

Abbreviations