Frédéric Hérau

David Krejčiřík

Nicolas Raymond

SEMICLASSICAL ASYMPTOTICS OF THE BLOCH-TORREY OPERATOR IN TWO DIMENSIONS

The Bloch-Torrey operator -h 2 ∆ + e iα x 1 on a bounded smooth planar domain, subject to Dirichlet boundary conditions, is analyzed. Assuming α ∈ 0, 3π 5 and a non-degeneracy assumption on the left-hand side of the domain, asymptotics of the eigenvalues with the smallest real part in the limit h → 0 are derived. The strategy is a backward complex scaling and the reduction to a tensorized operator involving a real Airy operator and a complex harmonic oscillator.

Introduction

Let Ω be a smooth bounded open connected set in R 2 . Given a small positive parameter h and a fixed real constant α ∈ [0, π], we consider the operator (1.1) L h,α = -h 2 ∆ + e iα x 1 in L 2 (Ω), subject to Dirichlet boundary conditions. On its natural domain Dom(L h,α ) = H 2 (Ω)∩H 1 0 (Ω), the operator is closed, has non-empty resolvent set and compact resolvent. Consequently, the spectrum is purely discrete and can be written as an infinite sequence of complex numbers tending to +∞ in modulus or as a (possibly empty) finite sequence. The latter cannot be a priori excluded because L h,α is non-selfadjoint unless α ∈ {0, π}. Our goal is to show the existence of "low-lying" eigenvalues and derive their asymptotics in the semiclassical limit h → 0.

1.1.

Motivations. There are two sources of motivation for this work. First, the selfadjoint situation α = 0 has been recently analysed in [START_REF] Cornean | On the two-dimensional quantum confined Stark effect in strong electric fields[END_REF] in the context of semiconductor devices exposed to a strong uniform electric field. Indeed, h -2 L h,0 is the Hamiltonian of an electron confined to a nanostructure of shape Ω, subject to singularly scaled electric potential h -2 x 1 . The following geometric hypothesis is adopted in [START_REF] Cornean | On the two-dimensional quantum confined Stark effect in strong electric fields[END_REF]: Assumption 1. The minimum min{x 1 : x ∈ Ω} is uniquely attained at a point A 0 , assumed to be (0, 0) (without loss of generality). Moreover, the (signed) curvature κ 0 of ∂Ω (computed with respect to the inner normal of Ω) at A 0 = (0, 0) is positive.

Let (λ n (h)) n⩾1 denote the non-decreasing sequence of the eigenvalues of L h,0 , where each eigenvalue is repeated according to its multiplicity. The following asymptotic estimate of each individual eigenvalue was established in [START_REF] Cornean | On the two-dimensional quantum confined Stark effect in strong electric fields[END_REF]: Theorem 1.1 ( [START_REF] Cornean | On the two-dimensional quantum confined Stark effect in strong electric fields[END_REF]). Assume α = 0 and Assumption 1. Then, for all n ⩾ 1,

(1.2) λ n (h) = z 1 h 2 3 + (2n -1)h κ 0 2 + o(h)
as h → 0, where z 1 is the absolute value of the smallest zero of the Airy function Ai.

The eigenvalue splitting given by the second term containing the curvature is experimentally spectacular, for it enables one to determine the shape of a convex nanostructure by imposing uniform electric fields in various directions [START_REF] Pedersen | Stark-localization as a probe of nanostructure geometry[END_REF].

Second, there have been an intensive study of the operator (1.1) for the purely imaginary choice α = π 2 in various geometric settings (and even for more general electric potentials) [START_REF] Almog | The stability of the normal state of superconductors in the presence of electric currents[END_REF][START_REF] Almog | Superconductivity near the normal state in a half-plane under the action of a perpendicular electric currents and an induced magnetic field[END_REF][START_REF] Almog | Superconductivity near the normal state in a half-plane under the action of a perpendicular electric current and an induced magnetic field, part II: The large conductivity limit[END_REF][START_REF] Henry | Spectre et pseudospectre d'opérateurs non-autoadjoints[END_REF][START_REF] Almog | Spectral analysis of a complex Schrödinger operator in the semiclassical limit[END_REF][START_REF] Grebenkov | On spectral properties of the Bloch-Torrey operator in two dimensions[END_REF][START_REF] Almog | On a Schrödinger operator with a purely imaginary potential in the semiclassical limit[END_REF][START_REF] Almog | Spectral semi-classical analysis of a complex Schrödinger operator in exterior domains[END_REF][START_REF] Almog | The spectrum of a Schrödinger operator in a wire-like domain with a purely imaginary degenerate potential in the semiclassical limit[END_REF][START_REF] Grebenkov | On the spectral properties of the Bloch-Torrey equation in infinite periodically perforated domains[END_REF]17]. Among the variety of physical motivations mentioned in these references, let us point out the Bloch-Torrey equation describing the diffusion-precession of spin-bearing particles in nuclear magnetic resonance experiments.

In particular, in [START_REF] Grebenkov | On spectral properties of the Bloch-Torrey operator in two dimensions[END_REF]Theorem 1.1], quasimodes are constructed and allow to conjecture the behavior of the eigenvalues with the smallest real part. Motivated by these constructions, the behavior of the real part of the left-most spectrum has then been analyzed, see [START_REF] Almog | On a Schrödinger operator with a purely imaginary potential in the semiclassical limit[END_REF]Theorem 1.6]. For our linear electric potential, we can apply, for instance, [START_REF] Henry | Spectre et pseudospectre d'opérateurs non-autoadjoints[END_REF]Theorem 4.1.1] and [START_REF] Almog | Spectral analysis of a complex Schrödinger operator in the semiclassical limit[END_REF]Theorem 1.1], and we get the following typical result.

Theorem 1.2 ( [START_REF] Almog | Spectral analysis of a complex Schrödinger operator in the semiclassical limit[END_REF]). Assume α = π 2 and Assumption 1. Then

(1.3) inf Re sp(L h,α ) = z 1 h 2 3 2 + o(h 2 3
)

as h → 0.

As observed in [START_REF] Almog | Spectral analysis of a complex Schrödinger operator in the semiclassical limit[END_REF]Introduction], the lower bound in Theorem 1.2 can be proved without Assumption 1 (see also [START_REF] Almog | The stability of the normal state of superconductors in the presence of electric currents[END_REF][START_REF] Henry | Spectre et pseudospectre d'opérateurs non-autoadjoints[END_REF]).

In this article we explain the transition between α = 0 (Theorem 1.1) and α = π 2 (Theorem 1.2). First of all, we show how α enters the constant coefficient in the first term of the asymptotic expansions (1.2) and (1.3). We also aim at providing the reader with an accurate description of the spectrum by exhibiting spectral gaps in the left-most part of the spectrum (similarly to Theorem 1.1 in the case when α = 0). This question is all the more interesting that, when α ∈ (0, π), the operator L h,α is not selfadjoint and therefore, classical tools and strategies such as the min-max and spectral theorems (used, for instance, in [START_REF] Cornean | On the two-dimensional quantum confined Stark effect in strong electric fields[END_REF]) have to be replaced by unconventional arguments. Throughout this paper, we use the nickname Bloch-Torrey operator for (1.1) even if α ̸ = π 2 .

1.2. Heuristics. Before stating our main results, let us explain the intuitive origin of Theorems 1.1 and 1.2. This is also the opportunity to discuss the heuristics of our main theorem, which is stated in Section 1.3. When α = 0 and under Assumptions 1, due to the Agmon estimates, we can check that the eigenfunctions associated with the lowest eigenvalues are localized near A 0 . We will see that such a localization behavior persists in some sense for certain eigenfunctions when α ∈ [0, π], especially for those associated with the left-most eigenvalues when α ∈ 0, π 2 . Anyway, this naively suggests to use the classical tubular coordinates near the (outer) boundary defined through the map

(1.4) Γ(s, t) = γ(s) -tn(s) = (Γ 1 (s, t), Γ 2 (s, t)),
where γ is the arc-length parametrization of the outer boundary of Ω, denoted by ∂Ω 0 , and n is the outward pointing normal of Ω. Let L > 0 be the half-length of ∂Ω 0 and consider the torus T 2L = R/(2LZ). The map Γ induces a smooth diffeomorphism from B δ 0 = T 2L × (0, δ 0 ) to the tubular neighborhood T δ 0 of width δ 0 > 0 of ∂Ω 0 lying inside Ω.

In the coordinates (s, t), the operator (1.1) becomes

-h 2 (1 -tκ(s)) -1 ∂ t (1 -tκ(s))∂ t -h 2 (1 -tκ(s)) -1 ∂ s (1 -tκ(s)) -1 ∂ s + e iα Γ 1 (s, t) ,
acting in the Hilbert space L 2 (B δ 0 , (1 -tκ(s))dsdt). Here the curvature function κ is defined via the Frenet formula n ′ = κγ ′ .

According to Assumption 1 (which involves Γ(0, 0) = A 0 ), we have Γ 1 (s, t) = t + κ 0 2 s 2 + O(ts 2 + |s| 3 ). Since 1 -tκ(s) ≃ 1 when t is small, this suggests to consider the operator (1.5)

P h,α = -h 2 ∂ 2 s + -h 2 ∂ 2 t + e iα κ 0 2 s 2 + t ,
acting on L 2 (R 2 + , dsdt), subject to Dirichlet boundary condition at t = 0. Taking profit of the analyticity (since it is linear) in the variable t, we make the formal dilation t = ue -iα/3 . The model operator P h,α then becomes

(1.6) N h,α = e 2iα 3 (h 2 D 2 u + u) + h 2 D 2 s + e iα κ 0 s 2 2 ,
which is, up to multiplications by complex constants, the sum of a real Airy operator and a complex harmonic oscillator, whose resolvent and spectra are rather well-known.

Heuristically, this allows us to describe the spectrum of L h,α accurately in appropriate regions of the complex plane.

1.3. The main result. The main result of this article is the following theorem, which can be guessed from the heuristics of the previous section.

Theorem 1.3. Consider α ∈ 0, 3π 5
and R > 0 with R ̸ ∈ (2N -1) κ 0 2 . Under Assumption 1, there exist h 0 > 0 and N ∈ N such that for all h ∈ (0, h 0 ) the following holds. The spectrum of L h,α lying in the disk D(h 2 3 e 2iα/3 z 1 , Rh) is made of exactly N eigenvalues of algebraic multiplicity 1 and they satisfy, for all n ∈ {1, . . . , N },

(1.7) λ n (α, h) = h 2 3 e 2iα/3 z 1 + (2n -1)he iα/2 κ 0 2 + o(h)
as h → 0. Moreover, for all α ∈ [0, π 2 ), there exist C, h 0 > 0 such that, for all h ∈ (0, h 0 ), we have

(1.8) inf Re sp(L h,α ) ⩾ z 1 h 2 3 cos 2α 3 -Ch 4 3 .
In particular,

inf Re sp(L h,α ) = z 1 h 2 3 cos 2α 3 + o(h 2 3
) as h → 0.

Theorem 1.3 is illustrated on Figure 1: there is exactly one eigenvalue (with algebraic multiplicity) in each small circle (which has radius o(h)) and there is no spectrum in the gray region when α ∈ 0, π 2 . Remark 1.4.

(i) Theorem 1.3 gives an accurate description of the spectrum in large balls of size h when α ∈ 0, 3π 5 , but it only states the one-term asymptotics of the eigenvalue with the smallest real part when α ∈ 0, π 2 . When α ∈ 0, π 2 , we will see that elliptic estimates using the real part of an operator (which is isospectral to L h,α ) are enough to establish the semiclassical localization near (0, 0) (in the Agmon sense) of the eigenfunctions associated with eigenvalues having a real part less than M h 2 3 . This localization is the key to get the lower bound (1.8). When α ∈ π 2 , 3π 5 , these considerations must be slightly adapted by introducing a parameter β and by multiplying the operator by e -iβ . This rotation is the reason why the control of infimum of the real part is lost with our method. This aspect is discussed in more detail in Section 1.4.

(ii) Our assumptions allow us to deal with the case α = π 2 and to get the asymptotic estimate (1.7). For more general potentials, see [START_REF] Almog | Spectral analysis of a complex Schrödinger operator in the semiclassical limit[END_REF]Theorem 1.1], only the existence of one eigenvalue in the disk is ensured (the one corresponding to n = 1). Not only our theorem gives the existence of more eigenvalues, it also states that they are algebraically simple and that they are the only ones in the disk. The proof of this simplicity involves rather subtle and tedious elliptic estimates, especially to exclude the existence of Jordan blocks. (iii) As we explain in Section 1.5, the analysis used to establish Theorem 1.3 strongly relies on the analyticity of V (x) = x 1 . However, it seems that arguments such as analytic dilations have not yet been used to investigate the spectrum of such Bloch-Torrey operators on domains. We believe that our method is of independent interest. It can easily be extended to more general analytic potentials V (still satisfying the generic assumptions in [START_REF] Almog | Spectral analysis of a complex Schrödinger operator in the semiclassical limit[END_REF][START_REF] Almog | On a Schrödinger operator with a purely imaginary potential in the semiclassical limit[END_REF]) and we may even think that it could be used to deal with smooth V by means of almost analytic extensions. (iv) Unfortunately, our strategy does not allow us to recover Theorem 1.2, even though, at a formal level, (1.8) would give the appropriate lower bound when α = π 2 . In this case, the real parts of two networks of eigenvalues cross, see Section 1.4. (v) Our theorem does not say anything about the eigenfunctions (even if one could prove that they are localized near A 0 when α ∈ 0, π 2 ). Their accurate localization properties (in the Agmon sense) would be quite natural to investigate. 1.4. Consequences and extensions. The analysis in this article can be used to get an a priori location of the spectrum in the case when α ∈ π 2 , 3π 5 .

Proposition 1.5. Consider α ∈ π 2 , 3π 5 . There exist β ∈ 0, π 10 with 2α 3 ∈ β -π 2 , β + π 2 , C > 0, and h ∈ (0, h 0 ) such that, for all h ∈ (0, h 0 ),

inf Re e -iβ sp(L h,α ) ⩾ z 1 h 2 3 cos 2α 3 -β -Ch 4 3 .
In others terms, the eigenvalues λ of L h,α belong to the half-plane given by

cos β Re λ + sin β Im λ ⩾ z 1 h 2 3 cos 2α 3 -β -Ch 4 3 .
Proposition 1.5 is illustrated by Figure 2: there is no spectrum on the left of the dashed oblique line. In fact, in this case, there exist eigenvalues with a smaller real part (as one can see on the same figure). They are related to the right-most part of the domain.

z 1 e 2iα 3 h 2 3 c h c h = e iα x 1,max + z 1 e -2i α 3 h 2 3 e iα x 1,max 2 α 3 β Re c h -Ch 4 3 α 2 α 2 α = π -α Figure 2. Case when α ∈ π 2 , 3π 5 .
Assumption 2. The maximum max{x 1 : x ∈ Ω} = x 1,max is uniquely attained at a point A 1 . Moreover, the curvature κ 1 of ∂Ω at A 1 is positive.

We let α = π -α and we consider α ∈ 2π 5 , π so that α ∈ 0, 3π 5 . Then, the affine change of variable y = F (x) = (-x 1 + x 1,max , x 2 ) transforms L * h,α into the unitarily equivalent operator

U * L * h,α U = L h, α + e -iα x 1,max , Dom(L h, α) = H 2 ( Ω) ∩ H 1 0 ( Ω) , Ω = F (Ω)
. Therefore, under Assumption 2, we can apply Theorem 1.3 to L h, α and we get the following.

Corollary 1.6. Consider α ∈ 2π 5 , π and R > 0 with R ̸ ∈ (2N -1) κ 1 2 . Then, the spectrum of L h,α lying in the disk D(e iα x 1,max +h 2 3 e -2i α/3 z 1 , Rh) is made of N eigenvalues of algebraic multiplicity 1 and satisfying, for all n ∈ {1, . . . , N },

λn (α, h) = e iα x 1,max + h 2 3 e -2i α/3 z 1 + (2n -1)he -i α/2 κ 1 2 + o(h) as h → 0. Moreover, when α ∈ π 2 , π , inf Re sp(L h,α ) = cos(α)x 1,max + z 1 h 2 3 cos 2α 3 + o(h 2 3
)

as h → 0.

When α ∈ 2π 5 , 3π 5 , under Assumptions 1 and 2, Theorem 1.3 and Corollary 1.6 apply. Therefore, we have coexistence of (at least) two networks of eigenvalues. This phenomenon is illustrated on Figure 2 when α ∈ π 2 , 3π 5 : we see that the right-most part of Ω determines the left-most part of the spectrum. 1.5. Organization and strategy. The article is organized as follows. In Section 2, we make the analytic dilation argument rigorous. We introduce an analytic family of operators (L h,α,θ ) θ∈Θ in the sense of Kato. To do so, we use a real dilation with respect to the distance to the outer boundary t (acting only near the boundary). This is where we take advantage of the fact that Γ (see Lemma 2.5). By the Kato theory, it is also isospectral to L h,α,-i α 3 (see Corollary 2.8). In Section 3, we see that this special choice of complex parameter is particularly convenient since we can prove that the eigenfunctions of M h,α associated with eigenvalues located in a half-plane of the form Re (e -iβ λ) ⩽ M h 2 3 are exponentially localized near A 0 , see Proposition 3.5. The introduction of the parameter β and the constraint on α ∈ 0, 3π 5 originate from these localization arguments, which are based on ellipticity/coercivity estimates, induced by the complex electric potential (after the change of coordinates and the complex dilation), see Lemma 3.4.

(1.4)) is always analytic in t. When θ ∈ R, L h,α,θ is isospectral to M h,α = L h,α (see
Section 4 is devoted to the spectral analysis of M h,α . When α ∈ 0, π 2 , the asymptotic estimate of the infimum part of the spectrum is obtained, see Proposition 4.1 and its proof given in Section 4.1. This proves (1.8) (see also Remark 4.4, which proves Proposition 1.5). When α ∈ 0, 3π 5 , we first prove that the spectrum in the disk mentioned in Theorem 1.3 is necessarily close (essentially at a distance of order h 3 2 ) to the eigenvalue of a model operator

µ n (h, α) = h 2 3 e 2iα/3 z 1 + (2n -1)he iα/2 κ 0 2 , see Proposition 4.2.
Then, we prove that there is exactly one eigenvalue (with algebraic muliplicity 1) in these small discs (see Proposition 4.3 and Figure 1), and we deduce Theorem 1.3.

The proof of Proposition 4.2 relies on three important ingredients. First, it requires resolvent estimates of N h,α (we recall that N h,α is given in (1.6)), see Proposition 4.5. The fact that we performed an analytic dilation in t is a crucial help to get the control of the resolvent (by estimating the real part of the operator). The second ingredient is to show that the eigenfunctions associated with the eigenvalues in our disc are good quasimodes for N h,α , see Proposition 4.6. To do so, we need to prove optimal localization estimates with respect to the curvilinear abscissa s (see Proposition 4.12) in order to estimate the remainders of order s 3 when Taylor expanding the electric potential. We stress that estimating the real part of M h,α is a key to get such estimates (and that this argument succeeds thanks to the analytic dilation). Proposition 4.6 and the resolvent estimate are then enough to locate the spectrum in the small discs.

The fact that the rank of the Riesz projector is at most one requires more work. This is where the third ingredient comes into play. We assume that this rank is at least two and even that we have a Jordan block (in the worst scenario) and we prove that a generalized eigenfunction also satisfies accurate localization estimates, see Section 4.4 and especially Proposition 4.13. This part of the proof is technically more involved and it is somewhat reminiscent of Caccioppoli estimates, see Proposition 4.14. There remains to estimate the Riesz projectors to get a contradiction, see Section 4.5.1. To prove that the projectors are non-zero, we consider a quasimode built from the Airy and Hermite functions, see Section 4.5.2.

The analytic dilation

2.1. The sesquilinear form. Before introducing the main idea of this paper, let us stress that the operator L h,α from (1.1) is rigorously introduced via its sesquilinear form defined on H 1 0 (Ω) by

L h,α (φ, ψ) = Ω ∇φ ∇ψ dx + e iα Ω x 1 φ ψ dx . Notice that Re L h,α (ψ, ψ) ⩾ ∥∇ψ∥ 2 -sup x∈Ω |x 1 | ∥ψ∥ 2 ,
which enables to apply the standard Lax-Milgram theorem. An elementary argument shows that

sp(L h,α ) ⊂ {λ ∈ C : 0 ⩽ Im λ ⩽ (sin α) sup Ω x 1 } . 2.
2. An isospectral operator. Following the intuition described in Section 1.2, we would like to perform a complex scaling in the normal variable to the outer boundary. By doing that, we will preserve the spectrum as soon as we have a family of type (B) in the sense of Kato [15, Chap. VII]. This will reveal some hidden elliptic properties of the new operator. Let δ ∈ (0, δ 0 ) where we recall that δ 0 is defined just after (1.4). The heuristic considerations of Section 1.2 lead to introduce the following unitary transform U θ , depending on the real parameter θ. For all φ ∈ L 2 (Ω), we let

U θ φ = (φ |Ω\T δ , φ |T δ • Γ(s, J θ (u))) , with J θ given by t = J θ (u) = ue θχ(u)
, where χ is non-increasing smooth function from [0, δ] to [0, 1] such that χ = 1 near 0 and χ = 0 near δ. For all ϵ > 0, we can choose χ so that

(2.1) ∥χ ′ ∥ ∞ ⩽ 1 + ϵ δ .
Note that t = J θ (u) = ue θ near 0 and that t = J θ (u) = u at a distance larger than δ of the outer boundary and that the change of variable is smooth in between. There exists θ 0 > 0 such that, for all θ ∈ (-∞, θ 0 ), the map J θ : (0, δ) → (0, δ) is smooth diffeomorphism and, for all u ∈ (0, δ),

(2.2) J ′ θ (u) = (1 + θuχ ′ (u))e θχ(u) > 0 . We let (2.3) m θ (s, u) = 1 -J θ (u)κ(s) ,
where κ(s) is the curvature at the point of curvilinear coordinate s. Thanks to a change of variables, we have the following.

Lemma 2.1. For all θ ∈ (-∞, θ 0 ), U θ is an isometry from L 2 (Ω) to the product

E θ := L 2 (Ω \ T δ ) × L 2 (B δ , m θ (s, u)J ′ θ (u)dsdu).
As vector spaces, we have

E θ = E 0 = L 2 (Ω \ T δ ) × L 2 (B δ ).
Then, let us describe the effect of U θ on the form domain of the operator L h,α , which is H 1 0 (Ω).

Lemma 2.2. We have

U θ (H 1 0 (Ω)) = {(φ 1 , φ 2 ) ∈ H 1 (Ω \ T δ ) × H 1 (B δ ) : φ 2 (s, 0) = 0 & φ 1 (Γ(s, δ)) = φ 2 (s, δ)} = U 0 (H 1 0 (Ω)) .
Proof. Let us first notice that, by standard trace theorems, the functions ϕ 2 is well defined a.e. on {t = δ} (where coordinates t and u coincide), as well as ϕ 1 • Γ on ∂B δ . For a smooth function ϕ, we have φ 1 (Γ(s, δ)) = φ 2 (s, δ) and the result follows by density and the fact that U θ is an isometry. □

Let us now consider the quadratic form induced by U θ from L h,α .

Proposition 2.3. Letting, for all φ ∈ U 0 (H 1 0 (Ω)), ℓ h,α,θ (φ, φ) = Ω\T δ |h∇φ 1 | 2 + e iα x 1 |φ 1 | 2 dx + B δ (m -2 θ |h∂ s φ 2 | 2 + [J ′ θ ] -2 |h∂ u φ 2 | 2 + e iα Γ 1 (s, J θ (u))|φ 2 | 2 )m θ J ′ θ (u)dsdu ,
we have

L h,α (U -1 θ φ, U -1 θ φ) = ℓ h,α,θ (φ, φ) .
Proof. For all φ ∈ U 0 (H 1 0 (Ω)), we let ψ = U -1 θ φ ∈ H 1 0 (Ω). Let us first describe the kinetic part:

Ω |h∇ψ| 2 dx = Ω\T δ |h∇ψ| 2 dx + T δ |h∇ψ| 2 dx = Ω\T δ |h∇φ 1 | 2 dx + B δ (m -2 θ |h∂ s φ 2 | 2 + [J ′ θ ] -2 |h∂ u φ 2 | 2 )m θ (s, u)J ′ θ (u)dsdu .
Using the changes of variable x → (s, u) on B δ for the non-kinetic part completes the proof. □ From Proposition 2.3, we see that

U θ L h,α U -1 θ
is the operator associated with ℓ h,α,θ in the ambient space E θ (with the weighted scalar product, which depends on θ). To avoid the θ-dependence of the ambient space through its scalar product, we can consider the isometry

V θ : φ → (φ 1 , m 1 2 θ (J ′ θ ) 1 2 φ 2 =ϕ 2 ) , from L 2 (Ω \ T δ ) × L 2 (B δ , m θ (s, u)J ′ θ (u)dsdu) to L 2 (Ω \ T δ ) × L 2 (B δ , dsdu). Lemma 2.4. Let φ ∈ U 0 (H 1 0 (Ω)) and ϕ = V θ φ = (φ 1 , ϕ 2 ). We have B δ (m -2 θ |h∂ s φ 2 | 2 + [J ′ θ ] -2 |h∂ u φ 2 | 2 )m θ (s, u)J ′ θ (u)dsdu = B δ (m -2 θ |h∂ s ϕ 2 | 2 + [J ′ θ ] -2 |h∂ u ϕ 2 | 2 + h 2 V θ (s, u)|ϕ 2 | 2 )dsdu + h 2 L -L W θ (s)|ϕ 2 (s, δ)| 2 ds ,
where, letting

X = m -1/2 θ (J ′ θ ) -1/2 we have V θ = m -2 θ (∂ s X) 2 + (J ′ θ ) -2 (∂ u X) 2 -∂ s (m -2 θ )X∂ s X -∂ u (J ′ θ ) -2 X∂ u X and W θ (s) = (J ′ θ )(s, δ)X(s, δ)(∂ u X)(s, δ) .
Proof. This follows from two integrations by parts and from the fact that ϕ 2 (s, 0) = 0 a.e. and that J ′ θ is constant near ∂B δ . As we shall see later, the exact values of V θ and W θ are unimportant; we only note that they are smooth. □

These considerations lead to define the following quadratic form, in the ambient Hilbert space

L 2 (Ω \ T δ ) × L 2 (B δ , dsdu), for all φ ∈ U 0 (H 1 0 (Ω)), L h,α,θ (φ, φ) = Ω\T δ |∇φ 1 | 2 + e iα x 1 |φ 1 | 2 dx + B δ (m -2 θ |h∂ s ϕ 2 | 2 + [J ′ θ ] -2 |h∂ u ϕ 2 | 2 + (e iα Γ 1 (s, J θ (u)) + h 2 V θ (s, u))|ϕ 2 | 2 )dsdu + h 2 L -L W θ (s)|ϕ 2 (s, δ)| 2 ds , (2.4)
where we recall that ϕ = V θ φ = (φ 1 , ϕ 2 ).

We get the following lemma.

Lemma 2.5. The operator associated with

L h,α,θ is L h,α,θ = V θ U θ L h,α U -1 θ V -1 θ .
In particular, the spectrum of the operator L h,α,θ is the same as that of L h,α .

2.3.

Complex deformation parameters. According to our heuristic discussion, we would like to consider complex θ. More precisely, we would like the family (L h,α,θ ) θ∈Θ to be analytic of type (B) in the sense of Kato, where Θ is a connected open set containing θ = 0 and θ = -i α 3 . First, we notice that the form domain U 0 (H 1 0 (Ω)) is independent of θ and that, for all φ ∈ U 0 (H

1 0 (Ω)), Θ ∋ θ → L h,α,θ (φ, φ) ∈ C
is analytic. Then, it is sufficient to check that the form L h,α,θ is sectorial and closed on U 0 (H 1 0 (Ω)) for θ ∈ Θ. Lemma 2.6. Let θ 0 > 0 and β 0 ∈ (0, π 4 ). For η > 0 let us consider the rectangle Θ η = (-θ 0 , η) + i(-β 0 , η). Then, if η and δ are small enough, there exists c > 0 such that, for all θ ∈ Θ η , and all u ∈ (0, δ),

Re J ′-2 θ (u) ⩾ c > 0 . Proof. Writing θ = θ 1 + iθ 2 with θ 1 , θ 2 ∈ R and taking u ∈ (0, δ), we notice that J ′-2 θ (u) = |J ′ θ | -4 J ′2 θ = |J ′ θ | -4 (1 + θ 1 uχ ′ -iθ 2 uχ ′ ) 2 e 2θ
1 χ e -2iθ 2 χ , so that, by using that δ is small, we can write

(2.5) J ′-2 θ (u) = |J ′ θ | -4 J ′2 θ = e 2θ 1 χ |J ′ θ | -4 e -2i arctan θ 2 uχ ′ 1+θ 1 uχ ′ -2iθ 2 χ = e 2θ 1 χ |J ′ θ | -4 e -iA(u,θ) ,
where the argument A(u, θ) is given by

(2.6) A(u, θ) = 2 arctan θ 2 uχ ′ 1 + θ 1 uχ ′ + 2θ 2 χ.
When θ 2 is positive, we notice that, for all a > 0, by choosing η small enough, and by using (2.1), we have, for all θ 2 ∈ (0, η), |A(u, θ)| ⩽ a.

When θ 2 is non-positive, namely θ 2 ∈ (-β 0 , 0], we have, by using again (2.1), that

-2β 0 ⩽ 2θ 2 ⩽ 2θ 2 χ ⩽ A(u, θ) ⩽ 2θ 2 uχ ′ 1 + θ 1 uχ ′ ⩽ 2(1 + 2η)|θ 2 ||uχ ′ | . Thus, - π 2 < -2β 0 ⩽ A(u, θ) ⩽ (1 + 4η)2β 0 < π 2 .
We therefore get the result by (2.5). □ Proposition 2.7. There exist c, C > 0 such that, for all φ ∈ U 0 (H

1 0 (Ω)), Re L h,α,θ (φ, φ) ⩾ c∥h∇φ∥ 2 E 0 -C∥φ∥ 2 E 0 .
Proof. Thanks to (2.4), there exists C > 0 such that the following holds: for all δ, ϵ > 0, there exists C δ,ϵ > 0 such that, for all φ ∈ U 0 (H 1 0 (Ω)),

Re L h,α,θ (φ, φ) ⩾ B δ |h∂ s ϕ 2 | 2 dsdu + B δ Re J ′-2 θ (u)|h∂ u ϕ 2 | 2 dsdu + ∥h∇φ 1 ∥ 2 L 2 (Ω\T δ ) -Cδ∥h∇ϕ 2 ∥ 2 L 2 (B δ ) -C∥φ∥ 2 E 0 -C(ϵ∥h∇ϕ 2 ∥ 2 + C δ,ϵ h 2 ∥ϕ 2 ∥ 2 )
, where we used the classical estimate

L -L |ϕ 2 (s, δ)| 2 ds ⩽ ϵ∥∇ϕ 2 ∥ 2 + C δ,ϵ ∥ϕ 2 ∥ 2 .
This concludes the proof. □ Corollary 2.8. The operators L h,α and L h,α,-iα/3 are isospectral.

Proof. It is a consequence of the analytic pertubation theory, upon observing that -i α 3 ∈ Θ η since -β < -α 3 is equivalent to α < 3β which is satisfied for all α ∈ [0, 3π 4 ) as soon as β is close enough to π 4 . □

Localization estimates

In virtue of Corollary 2.8, we now focus on the spectral analysis of L h,α,-iα/3 , for which ellipticity properties are established in the present section. Definition 3.1. We denote M h,α = L h,α,-iα/3 and by M h,α the associate quadratic form on E 0 , see (2.4) and Lemma 2.1.

In the following series of lemmas, we show bounds from below for the potential part of e -iβ M h,α , where β is introduced to correct a lack of coercivity of the real part when α is larger than π 2 . These lemmas lead to Proposition 3.5, which provides us with a precise semiclassical localization of the eigenfunctions of M h,α .

The first lemma shows that the result of Lemma 2.6 remains true if we insert e -iβ .

Lemma 3.2. Consider α ⩾ 0 and β ∈ R such that

β - 2α 3 > - π 2 , β + 2α 3 < π 2 .
Then, there exists C(α, β) >0 such that, for all u ∈ [0, δ],

Re (e -iβ J ′-2 θ (u)) ⩾ C(α, β) . Proof. We have

Re (e -iβ J ′-2 θ ) = Re e -iβ+ 2iα 3 χ (1 -i α 3 uχ ′ ) -2 = 1 + α 2 9 (uχ ′ ) 2 -1 Re e -iβ+ 2iα 3 χ-2i arctan( α 3 |uχ ′ |)
= Re e -iβ-iA(u,-iα/3) , where we recall A(u, θ) is defined in (2.6). Note that, by using (2.1), we have, for ϵ > 0 small enough,

- π 2 < β - 2α 3 ⩽ β - 2α 3 χ + 2 arctan( α 3 |uχ ′ |) ⩽ β + 2α 3 (1 + ϵ) < π 2 ,
so that Re (e -iβ-iA(u,-iα/3) ) is uniformly bounded from below by a positive constant. This gives the result. □

The following lemma is a preparation lemma in order to get the ellipticity of the electric potential in M h,α . Lemma 3.3. We let

T = (α, β) ∈ R + × R : β - 2α 3 > - π 2 , β + 2α 3 < π 2 , - π 2 < α -β < π 2 .
Then sup{α : (α, β) ∈ T } = 3π 5 . Moreover, for all α ∈ 0, 3π 5 , we have (α, π 10 ) ∈ T . For all α ∈ [0, π

2 ), we have (α, 0) ∈ T . We also have that π 2 , β ∈ T for β positive and small enough.

Proof. These estimates follow from straightforward computations, which are conveniently supported by drawing a picture. We leave the details to the reader. □ Lemma 3.4. Assume that α ∈ 0, 3π 5 and consider β such that (α, β) ∈ T . There exist s 0 , δ 0 , c > 0 such that the following holds. For all δ ∈ (0, δ 0 ), for all (s, u) ∈

[-L, L) × (0, δ), if |s| ⩾ s 0 , then Re (e i(α-β) Γ 1 (s, J θ (u))) ⩾ c , and, if |s| ⩽ s 0 , Re (e i(α-β) Γ 1 (s, J θ (u))) ⩾ c(u + s 2 ) .
Proof. Consider (α, β) as in the statement. We have

Re (e i(α-β) Γ 1 (s, J θ (u))) = Re (e i(α-β) Γ 1 (s, ue -i α 3 χ )) . From (1.4), we have Γ 1 (s, e -i α 3 χ u) = γ 1 (s) -ue -i α 3 χ n 1 (s) , and thus, by using the Taylor expansion with respect to s near 0 and Assumption 1, we get

Γ 1 (s, e -i α 3 χ u) = ue -i α 3 χ + κ 0 2 s 2 + O(|s| 3 + us 2 ) . Therefore, Re (e i(α-β) Γ 1 (s, J θ (u))) = u cos α -β - α 3 χ + κ 0 2 s 2 cos(α -β) + O(|s| 3 + us 2 ) . Since (α, β) ∈ T , - π 2 < 2α 3 -β ⩽ α -β - α 3 χ < π 2 and - π 2 < α -β < π 2 .
Therefore, there exist s 0 , δ 0 > 0 such that, for all δ ∈ (0, δ 0 ), there exists c > 0 such that, for all s such that |s| ⩽ s 0 and all u ∈ (0, δ), we have

Re (e i(α-β) Γ 1 (s, J θ (u))) ⩾ c(u + s 2 ) .

Let us now study the case when |s| ⩾ s 0 . We first introduce u) . We notice that when |s| ⩾ s 0 , and choosing δ small enough, we have γ 1 (s)-u cos( α 3 χ)n 1 (s) ⩾ c 0 > 0 uniformly. This implies that

B(s, u) = arctan u sin( α 3 χ(u))n 1 (s) γ 1 (s) -u cos( α 3 χ(u))n 1 (s) , so that Γ 1 (s, e -i α 3 χ u) = |γ 1 (s) -ue -i α 3 χ n 1 (s)|e iB(s,
Re (e i(α-β) Γ 1 (s, J θ (u))) ⩾ |γ 1 (s) -ue -i α 3 χ n 1 (s)| cos(α -β + B(s, u)) ⩾ c 0 cos(α -β + B(s, u)) .
From the expression of B(s, u), we deduce that, with a possibly smaller c, we have, for all |s| ⩾ s 0 ,

Re (e i(α-β) Γ 1 (s, J θ (u))) ⩾ c > 0 . This completes the proof. □

The following proposition gives Agmon type localization estimates for some eigenfunctions of M h,α . Proposition 3.5. Let α ∈ [0, 3π 5 ) and consider suitable parameters β, δ 0 introduced in Lemma 3.4. Then for any M > 0 and 0 < δ < δ 0 , there exists h 0 , C > 0 such that for all h ∈ (0, h 0 ), all eigenvalue λ (of M h,α ) such that Re (e -iβ λ) ⩽ M h 2 3 and all associated eigenfunction φ = (φ 1 , ϕ 2 ), we have

(3.1) Ω\T δ e 2|x|/h 2 3 |φ 1 | 2 dx + B δ e 2|Γ(s,u)|/h 2 3 |ϕ 2 | 2 dsdu ⩽ C∥φ∥ 2 E 0 ,
and

Ω\T δ e 2|x|/h 2 3 |h∇φ 1 | 2 dx + B δ e 2|Γ(s,u)|/h 2 3 |h∇ s,u ϕ 2 | 2 dsdu ⩽ Ch 2 3 ∥φ∥ 2 E 0 .
Proof. The proof essentially follows from the classical Agmon estimates. Considering φ = (e 2|x|/h 2/3 φ 1 , e 2|Γ(s,u)|/h 2/3 ϕ 2 ), we see that φ ∈ U 0 (H 1 0 (Ω)). We have then

(3.2) ⟨M h,α φ, φ⟩ = λ ∥e |x|/h 2/3 φ 1 ∥ 2 Ω\T δ + ∥e |Γ(s,u)|/h 2/3 ϕ 2 ∥ 2 T δ
, and, recalling Definition 3.1, we can write that

(3.3) e -iβ M h,α (φ, φ) = I + II + III , with I = e -iβ h 2 ⟨∇φ 1 , ∇(e 2|x|/h 2/3 φ 1 )⟩ Ω\T δ + e i(α-β) Ω\T δ x 1 |e |x|/h 2/3 φ 1 | 2 dx , II = B δ e -iβ m -2 θ h 2 ∂ s ϕ 2 ∂ s (e 2|Γ|/h 2/3 ϕ 2 ) + e -iβ [J ′ θ ] -2 h 2 ∂ u ϕ 2 ∂ u (e 2|Γ|/h 2/3 ϕ 2 ) + e i(α-β) Γ 1 (s, J θ (u))|e |Γ|/h 2/3 ϕ 2 | 2 dsdu
and

III = h 2 B δ V θ (s, u)|e |Γ|/h 2/3 ϕ 2 | 2 dsdu + h 2 L -L W θ (s)|e |Γ|/h 2/3 ϕ 2 (s, δ)| 2 ds .
Let us now bound the real part of I and II from below. We have

Re I = Re e -iβ h 2 ⟨∇φ 1 , ∇(e 2|x|/h 2/3 φ 1 )⟩ Ω\T δ + cos(α -β)∥ √ x 1 e |x|/h 2/3 φ 1 ∥ 2 .
Then, with the chain rule, we get

h 2 ⟨∇φ 1 , ∇(e 2|x|/h 2/3 φ 1 )⟩ Ω\T δ = h 2 ⟨e |x|/h 2/3 ∇φ 1 , ∇(e |x|/h 2/3 φ 1 )⟩ Ω\T δ + h 4 3 ⟨∇φ 1 , (∇|x|)(e 2|x|/h 2/3 φ 1 )⟩ Ω\T δ = h 2 ∥∇(e |x|/h 2/3 φ 1 )∥ 2 Ω\T δ + O(h 4
3 )∥e |x|/h 2/3 φ 1 ∥∥∇(e |x|/h 2/3 φ 1 )∥

+ h 4 3 ⟨∇φ 1 , ∇|x|(e 2|x|/h 2/3 φ 1 )⟩ Ω\T δ = h 2 ∥∇(e |x|/h 2/3 φ 1 )∥ 2 Ω\T δ + O(h 4 3 )∥e |x|/h 2/3 φ 1 ∥∥∇(e |x|/h 2/3 φ 1 )∥ + O(h 2 3 )∥e |x|/h 2/3 φ 1 ∥ 2 .
Thus, with the Young inequality, we deduce that

(3.4) Re I ⩾ c(α, β) h 2 ∥∇(e |x|/h 2/3 φ 1 )∥ 2 Ω\T δ + ∥ √ x 1 e |x|/h 2/3 φ 1 ∥ 2 Ω\T δ -Ch 2 3 ∥e |x|/h 2/3 φ 1 ∥ 2 Ω\T δ .
Note that we did not use any integration by parts in the last computation, so that no boundary term appears. We proceed rather similarly to see that

B δ m -2 θ h 2 ∂ s ϕ 2 ∂ s (e 2|Γ|/h 2/3 ϕ 2 )dsdu = B δ m -2 θ h 2 |∂ s (e |Γ|/h 2/3 ϕ 2 )| 2 dsdu + O(h 1/3 )∥e |Γ|/h 2/3 ϕ 2 ∥∥h∂ s (e |Γ|/h 2/3 ϕ 2 )∥ + O(h 2 3 )∥e |Γ|/h 2/3 ϕ 2 ∥ 2 = B δ h 2 |∂ s (e |Γ|/h 2/3 ϕ 2 )| 2 dsdu + O(h 1/3 )∥e |Γ|/h 2/3 ϕ 2 ∥∥h∂ s (e |Γ|/h 2/3 ϕ 2 )∥ + O(δ)∥h∂ s (e |Γ|/h 2/3 ϕ 2 )∥ 2 B δ + O(h 2 3 )∥e |Γ|/h 2/3 ϕ 2 ∥ 2 ,
where we used m θ = 1 + O(δ) from (2.3). In the same way, we get

B δ [J ′ θ ] -2 h 2 ∂ u ϕ 2 ∂ u (e 2|Γ|/h 2/3 ϕ 2 ) = B δ [J ′ θ (u)] -2 h 2 |∂ u (e |Γ|/h 2/3 ϕ 2 )| 2 dsdu + O(h 4/3 )∥e |Γ|/h 2/3 ϕ 2 ∥∥∂ u (e |Γ|/h 2/3 ϕ 2 )∥ + O(δ)h 2 ∥∂ u (e |Γ|/h 2/3 ϕ 2 )∥ 2 B δ + O(h 2 3 )∥e |Γ|/h 2/3 ϕ 2 ∥ 2 .
where we used that the exponential is bounded on {(s, u) ∈ B δ : u + s 2 ⩽ Rh

Spectral analysis

The aim of this section is to prove the following three propositions, which imply Theorem 1.1 (since M h,α is isospectral to L h,α , see Lemma 2.8). 2 ) and consider M > 0. There exist C, h 0 > 0 such that, for all h ∈ (0, h 0 ), 2 . Then there exist h 0 > 0 and N ∈ N such that, for all h ∈ (0, h 0 ),

(4.1) sp(L h,α ) ∩ {z ∈ C : Re (z) < M h 2 3 } ⊂ z ∈ C : Re z ⩾ z 1 h 2 3 cos (2α/3) -Ch 4 3 ), 0 ⩽ Im z ⩽ (sin α) max{x 1 : x ∈ Ω .
(4.2) sp(M h,α ) ∩ D(z 1 e 2iα 3 h 2 3 , Rh) ⊂ N n=1 D(µ n (h, α), h 3 2 -2η ) , with (4.3) µ n (h, α) = h 2 3 e 2iα/3 z 1 + (2n -1)he iα/2 κ 0 2 .
Moreover, for all n ∈ {1, . . . , N }, the Riesz projector

Π n,h := 1 2iπ C n,h (z -M h,α ) -1 dz , where C n,h = ∂D(µ n (h, α), h 3 2 -2η ) ,
is of rank at most one.

Proposition 4.3 (Existence of the spectrum). Consider α ∈ 0, 3π 5 . There exists h 0 > 0 such that, for all h ∈ (0, h 0 ), the rank of Π n,h is exactly one. Let us first note that, in a first naive approach, when α ∈ [0, π 2 ),

Re ⟨L h,α ψ, ψ⟩ ⩾ 0 , Im ⟨L h,α ψ, ψ⟩ = sin α Ω x 1 |ψ| 2 dx .
This gives the estimate on the imaginary part of the spectrum, and it remains to refine the estimate on the real part. For this, we consider a smooth cutoff function χ h in the form χ h (s, u) = χ(h -1 3 +η s, h -2 3 +η u) , where η > 0 and we let (4.4)

ϕ cut 2 = χ h ϕ 2 .
Consider now an eigenfunction φ of M h,α associated with an eigenvalue λ such that Re λ ⩽ M h 2 3 . From (2.4), we get

Re λ∥φ∥ 2 = Re L h,α,θ (φ, φ) ⩾ B δ Re (m -2 θ )|h∂ s ϕ 2 | 2 + Re ([J ′ θ ] -2 )|h∂ u ϕ 2 | 2 + Re e iα Γ 1 (s, J θ (u))|ϕ 2 | 2 dsdu -Ch 2 ∥ϕ 2 ∥ 2 ,
where we used the trace theorem and Proposition 3.5 to control the boundary term.

Then, with a Taylor expansion in the expressions (2.3) of m θ and (2.2) of J ′ θ and with Proposition 3.5, we get

Re λ∥φ∥ 2 = Re L h,α,θ (φ, φ) ⩾ B δ |h∂ s ϕ cut 2 | 2 + cos 2α 3 |h∂ u ϕ cut 2 | 2 + u|ϕ cut 2 | 2 dsdu -Ch 4 3 ∥ϕ 2 ∥ 2 .
From the min-max theorem applied to the real Airy operator, we deduce that

Re λ∥φ∥ 2 ⩾ z 1 h 2 3 cos 2α 3 ∥ϕ cut 2 ∥ 2 -Ch 4 3 ∥ϕ 2 ∥ 2 .
By using again Proposition 3.5, we infer that

Re λ ⩾ z 1 h 2 3 cos 2α 3 -Ch 4 3 .
This proves Proposition 4.1. ). In that case we do not even a priori have Re ⟨M h,α ψ, ψ⟩ ⩾ 0 because of the electric potential. Choosing β such that (α, β) ∈ T where T is defined in Lemma 3.3, we can still use the localization properties proved in Section 3. In particular, working with e -iβ M h,α instead of M h,α and using Proposition 3.5, as well as Taylor expansions of m θ and of J ′ θ , we get that, for any eigenvalue λ of L h,α , we have

Re e -iβ λ ⩾ z 1 h 2 3 cos 2α 3 -β -Ch 4 3 .

Resolvent estimates.

To prepare the proof of Proposition 4.2, let us describe the spectrum and resolvent of our model operator

(4.5) N h,α = e 2iα 3 (h 2 D 2 u + u) + h 2 D 2 s + e iα κ 0 s 2 2 ,
where D = -i∂.

Proposition 4.5. Let R > 0 with R ̸ ∈ (2N -1) κ 0 2 . There exist C, h 0 > 0 and N ∈ N such that the following holds. The spectrum of N h,α in D(z 1 e 2iα 3 h 2 3 , Rh) is made of N eigenvalues of algebraic multiplicity one, which are the (µ n (h, α)) 1⩽n⩽N as given in (4.3). Moreover, for all z ∈ D(z 1 e 2iα 3 h 2 3 , Rh) such that z / ∈ {µ n (h, α) , n ∈ {1, . . . , N }}, we have

∥(z -N h,α ) -1 ∥ ⩽ Ch -2 3 + C dist(z, sp(N h,α ))
.

Proof. Note that N h,α is the sum of two decoupled operators, an Airy operator and a harmonic oscillator. It has compact resolvent and its spectrum and eigenfunctions are completely known. Consider

(4.6) Ψ m,n,h (s, u) = h -1 3 -1 4 Ai(h -2 3 u -z m )f n (h -1 2 e iα/2 κ 0 2 s) ,
where Ai is the usual Airy function (and z m its m-th positive zero) and f n the n-th normalized Hermite function. We have

N h,α Ψ m,n,h = z m h 2 3 e 2iα 3 + (2n -1)h κ 0 2 Ψ m,n,h .
Moreover, there are no other eigenvalues and they are all of multiplicity one. Indeed, by analytic dilation, we see that N h,α is isospectral to the normal operator

e 2iα 3 (h 2 D 2 u + u) + e iα 2 h 2 D 2 s + k 0 s 2 2 .
Let us now turn to the estimate of the resolvent in the disk D(z

1 e 2iα 3 h 2 3 , Rh). Consider z = z 1 e 2iα 3 h 2 3 + ζh , ζ ∈ D(0, R) ,
with ζ avoiding the numbers e iα 2 (2n -1) k 0 2 . We have

N h,α -z = e 2iα 3 (h 2 D 2 u + u -z 1 h 2 3 ) + h 2 D 2 s + e iα k 0 s 2 2 -ζh ,
and also

e -iα 2 (N h,α -z) = e iα 6 (h 2 D 2 u + u -z 1 h 2 3 ) + e -iα 2 h 2 D 2 s + e iα k 0 s 2 2 -ζh .
Let us denote g h the (explicit) positive normalized groundstate of h 2 D 2 u + u and consider the orthogonal projection

P h ψ(s, u) = ⟨g h , ψ(s, •)⟩ L 2 (R + ) g h (u) .
We have

Re ⟨e -iα 2 (N h,α -z)(Id -P h )ψ, ψ⟩ ⩾ h 2 3 cos(α/6)(z 2 -z 1 ) -Rh ∥(Id -P h )ψ∥ 2 ⩾ ch 2 3 ∥(Id -P h )ψ∥ 2 , (4.7)
which implies that the restriction of N h,α -z to Ker(P h ) is injective and therefore bijective since N h,α is Fredholm of index 0. Moreover, we have the orthogonal decomposition

e -iα 2 (N h,α -z) = e -iα 2 (N h,α -z)(Id -P h ) + e -iα 2 h 2 D 2 s + e iα κ 0 s 2 2 -ζh P h .
From (4.7), the fact that ζ avoids the numbers e iα 2 (2n-1) k 0 2 , we get that e -iα 2 (N h,α -z) is bijective and that its inverse is given by

e -iα 2 (N h,α -z) -1 =e iα 2 (N h,α -z) Ker(P h ) -1 (Id -P h ) + e iα 2 h 2 D 2 s + e iα k 0 s 2 2 -ζh -1 P h .
Let us notice that there exists C > 0 such that for all ζ ∈ D(0, R) avoiding the numbers e iα 2 (2n -1) k 0 2 , we have

h 2 D 2 s + e iα k 0 s 2 2 -ζh -1 ⩽ C dist(sp(h 2 D 2 s + e iα k 0 s 2 2 ), ζh)
.

Thus, from the above orthogonal decomposition, we deduce that

∥[e -iα 2 (N h,α -z)] -1 ∥ ⩽ (ch 2
3 ) -1 + C dist(sp(h 2 D 2 s + e iα k 0 s 2 2 ), ζh)
.

This concludes the proof. □ with M h,α denoting the formal operator (4.9)

M h,α = hD s (m -2 θ )hD s + hD u (J ′ θ ) -2 hD u + e iα Γ 1 (s, J θ (u)
) . This section is devoted to the proof of the following proposition. We recall that the model operator N h,α is defined in (4.5) and that ϕ cut 2 is defined as a truncation of ϕ 2 in (4.4). Proposition 4.6. We have

(N h,α -λ)ϕ cut 2 = O(h 3 2 -3η )∥ϕ cut 2 ∥
. This proposition essentially says that ϕ cut 2 is a good quasimode for our model operator N h,α . The proof will be done in several steps including elliptic estimates and a refined localization in s.

Preliminary estimates.

Recall that χ h (s, u) = χ(h -1 3 +η s, h -2 3 +η u) with χ a smooth cutoff function and η fixed and small, and that ϕ cut 2 = χ h ϕ 2 . Lemma 4.7. We have

M h,α ϕ cut 2 = λϕ cut 2 + r h , with r h = [M h,α , χ h ]ϕ 2
, where the commutator is given by

(4.10) [M h,α , χ h ]ϕ = hD s (m -2 θ (hD s χ h )) ϕ + 2m -2 θ (hD s χ h )(hD s ϕ) + hD u ((J ′ θ ) -2 (hD u χ h )) ϕ + 2(J ′ θ ) -2 (hD u χ h )(hD u ϕ) . Moreover, ∥r h ∥ = O(h ∞ )∥ϕ cut 2 ∥ .
Proof. The expression (4.10) follows from a straightforward computation. The estimate

∥r h ∥ = O(h ∞ )
ϕ cut 2 = χ 0 (h -1 2 +η s)ϕ cut 2 + O(h ∞ )
, where the remainder is estimated in H 1 -norm.

Moreover, for all (α 1 , α 2 , α 3 ) ∈ N 3 , (4.12) )Ψ, Ψ⟩ ⩾ 0 . This, combined with the fact that γ 1 (s) is bounded from below by h 1-2η on the support of χ far,h , implies that, for some c > 0, Using a commutator in the last term of the previous expression, we also have

∥s α 1 (hD s ) α 2 (hD u ) α 3 ϕ cut 2 ∥ ⩽ Ch α 3 3 + α 1 +α
Re ⟨(e -iα/2 (N h,α -λ)(χ far,h ϕ cut 2 ), χ far,h ϕ cut 2 ⟩ ⩾ ch 1-2η ∥χ far,h ϕ cut 2 ∥ 2 + c∥hD s (χ far,h ϕ cut 2 )∥ 2 .
|⟨(1 -m -2 θ )χ far,h (hD s ) 2 ϕ cut 2 , χ far,h ϕ cut 2 ⟩| ⩽ Ch 2 3 -η ∥(hD s )(χ far,h ϕ cut 2 )∥ 2 + Ch 4 3 -3η ∥χ far,h ϕ cut 2 ∥ 2 + Ch∥hD s (χ far,h ϕ cut 2 )∥∥χ far,h ϕ cut 2 ∥ + O(h ∞ )∥ϕ cut 2 ∥ 2
, where χ far has the same properties as χ far and is such that χ far χ far = χ far .

In a similar way, we get that , Rh) associated with φ = (φ 1 , ϕ 2 ) ∈ ker(M h,α -λ) 2 with ∥φ∥ = 1 and such that φ / ∈ ker(M h,αλ) (if it exists). We still denote ϕ cut 2 = χ h ϕ 2 . The following proposition states that ϕ cut 2 is a generalized quasimode of N h,α . Its proof is the object of the following two sections. Proposition 4.13. We have satisfies the same localization estimates as in the previous section. Let us explain this. We have

⟨[N h,α , χ far,h ]ϕ cut 2 , χ far,h ϕ cut 2 ⟩| ⩽ Ch 4 3 -2η ∥χ far,h ϕ cut 2 ∥ 2 + Ch∥hD s (χ far,h ϕ cut 2 )∥∥χ far,h ϕ cut 2 ∥ + O(h ∞ )∥ϕ cut 2 ∥ 2 . It follows that c 2 ∥hD s (χ far,h ϕ cut 2 )∥ 2 + ch 1-2η ∥χ far,h ϕ cut 2 ∥ 2 ⩽ Ch 4 3 -3η ∥χ far,h ϕ cut 2 ∥ 2 + O(h ∞ )
(N h,α -λ) 2 ϕ cut 2 = O(h 13
(M h,α -λ) 2 φ = 0 ,
for φ normalized and (M h,α -λ)φ = f ̸ = 0. We have (M h,α -λ)f = 0 and thus f satisfies the estimates of the previous section. For instance, we have

f cut 2 = χ(h -1 2 +η s)f cut 2 + O(h ∞ )∥f ∥ ,
where f cut 2 is defined without ambiguity and satifies, from Proposition 4.12, (4.17)

∥s α 1 (hD s ) α 2 (hD u ) α 3 f cut 2 ∥ ⩽ Ch α 3 3 + α 1 +α 2 2 ∥f cut 2 ∥ for all (α 1 , α 2 , α 3 ) ∈ N 3 . Coming back to the eigenvalue equation, this implies that (4.18) (h 2 D 2 u + u -z 1 h 2 3 )f cut 2 = O(h)∥f cut 2 ∥
. We can easily adapt the proof of the Agmon estimates given in Proposition 3.5 with the right-hand side f to get

(4.19) Ω\T δ e 2|x|/h 2 3 |φ 1 | 2 dx + B δ e 2|Γ(s,u)|/h 2 3 |ϕ 2 | 2 dsdu ⩽ C∥φ∥ 2 E 0 + Ch -2 3 ∥f ∥ 2 E 0 , and
(4.20) Ω\T δ e 2|x|/h 2 3 |h∇φ 1 | 2 dx + B δ e 2|Γ(s,u)|/h 2 3 |h∇ s,u ϕ 2 | 2 dsdu ⩽ Ch 2 3 ∥φ∥ 2 E 0 + C∥f ∥ 2 E 0 .
These estimates imply that

(4.21) (M h,α -λ)ϕ cut 2 = f cut 2 + r h ,
where r h has the same expression as in Lemma 4.7 and satisfies

r h = O(h ∞ )(∥f ∥ + ∥φ∥) .
In the following proposition we prove that, in fact, f is small compared to ϕ. This estimate is reminiscent of the famous Caccioppoli estimates (see the original article [START_REF] Caccioppoli | Limitazioni integrali per le soluzioni di un'equazione lineare ellittica a derivate parziali[END_REF], and, for instance, the article [START_REF] Iwaniec | Caccioppoli estimates and very weak solutions of elliptic equations[END_REF] or the book [8, Section 5.

∥(N h,α -λ)ϕ cut 2 ∥ ⩽ Ch 3 2 -η ∥ϕ cut 2 ∥
. Then, we use the resolvent estimate of Proposition 4.5. We write λ = z 1 e 2iα 3 h 2 3 + ζh, with ζ ∈ D(0, R). If ζ does not belong to the spectrum of our complex harmonic oscillator, then, we have

1 ⩽ Ch 3 2 -η Ch -2 3 + C dist(sp(h 2 D 2 s + e iα k 0 s 2 2 ), ζh) . We deduce that dist ζ, {(2n -1)e iα 2 k 0 2 , 1 ⩽ n ⩽ N } ⩽ Ch 1 2 -η ,
which implies (4.2). Let us now discuss the rank of the Riesz projector Π n,h . From the estimate (4.2), we can draw the circle C n,h with center µ n (h, α) and radius h 3 2 -3η in the resolvent set of M h,α . Let us assume that the rank of the projector is at least two. There are two possibilities. Either there are two distinct (possibly not simple) eigenvalues (which coincide with µ n (h, α) modulo O(h 3 2 -η )), or there is an eigenvalue with algebraic multiplicity at least two. The strategy is to evaluate the Riesz projector on the corresponding (possibly generalized) eigenfunctions

Πn,h = 1 2iπ C n,h (z -N h,α ) -1 dz ,
whose rank is one by Proposition 4.5. Consider the first case and ψ and ψ corresponding normalized eigenfunctions. Let us denote F h = span(ψ, ψ), which is of dimension two. Then, the map Q h : F h ∋ f → χ h f 2 is injective. Indeed, from the Agmon estimates satisfied by the eigenfunctions, we see that ∥χ h f 2 ∥ = ∥f ∥ + O(h ∞ )∥f ∥ , and, in particular, for h small enough, ∥f ∥ ⩽ 2∥Q h f ∥ . Moreover, we have, for all f ∈ F h ,

∥(N α,h -λ)Q h f ∥ ⩽ Ch 3 2 -η ∥Q h f ∥ , λ = µ n (h, α) . We notice that Πn,h Q h f = Q h f + 1 2iπ C n,h (z -λ) -1 (z -N h,α ) -1 (N h,α -λ) Q h f dz . Thus, ∥ Πn,h Q h f -Q h f ∥ ⩽ Ch η ∥Q h f ∥ ⩽ 1 2 ∥Q h f ∥ .
This shows that rank Πn,h = 2, which is a contradiction.

Let us now consider the second case of an eigenvalue with algebraic multiplicity at least two. This implies the existence of φ = (φ 1 , ϕ 2 ) ∈ ker(M h,α -λ) 2 such that φ / ∈ ker(M h,α -λ).

Then, we write

Πn,h ϕ cut 2 = ϕ cut 2 + 1 2iπ C n,h (z -λ) -2 (z -N h,α ) -1 (N h,α -λ) 2 ϕ cut 2 dz .

Combining Propositions 4.13 and 4.5, we get

∥ Πn,h ϕ cut 2 -ϕ cut 2 ∥ = o(1)∥ϕ cut 2 ∥ ,
where we used the fact that 13/6 > 2. We conclude that the range of Πn,h has dimension at least two. This is a contradiction with Proposition 4.5. where Ψ 1,n,h is defined in (4.6) and χ is a smooth function with compact support equal to 1 near 0 and equal to 0 outside a small neighborhood of (0, 0). Then, we consider

Π n,h ψ h = 1 2iπ C n,h (z -M h,α ) -1 ψ h dz = 1 2iπ C n,h (z -M h,α ) -1 ψ h dz ,
where C n,h is a circle with the same center as C n,h , but with radius of order ϵh for ϵ small enough. Given z ∈ C n,h , we consider φ h,z = (φ h,z,1 , ϕ h,z,2 ) the unique solution of (z -M h,α )φ h,z = ψ h .

Then, φ h,z satisfies the Agmon estimates with a right-hand side ( Then, we also deduce that

C n,h (N h,α -z) -1 R h,z dz = o(1)∥ψ h ∥ .
This shows that

Π n,h ψ h = 1 2iπ C n,h (z -N h,α ) -1 ψ cut h,2 dz + o(1)∥ψ h ∥ .
Recalling the resolvent formula

(z -N h,α ) -1 -(z -µ n (h, α)) -1 = (z -N h,α ) -1 (z -µ n (h, α)) -1 (N h,α -µ n (h, α)) ,
and that

(N h,α -µ n (h, α))ψ h,2 = O(h ∞ ) ,
we get Π n,h ψ h = ψ h + o(1)∥ψ h ∥ .

Therefore, Π n,h is not zero for h small enough. Recalling the discussion at the beginning of this section, this completes the proof of Proposition 4.3.

Figure 1 .e 2iα 3 h 2 3

 12 Figure 1. The spectrum in the disk of center z 1 e 2iα 3 h 2 3 and radius Rh, when α ∈ 0, 3π 5 .

2 3

 2 }. The estimate (3.1) follows by choosing R large enough. Then, the estimate of the gradient follows by gathering (3.2), (3.3), (3.4), (3.5), (3.6) and by using (3.1).□

Proposition 4 . 1 (

 41 Rough localization of the spectrum). Let α ∈ [0, π

Proposition 4 . 2 (

 42 Refined localization of the spectrum). Consider α ∈ 0, 3π 5 and R > 0 with R ̸ ∈ (2N -1) κ 0

Proposition 4 . 4 . 1 .

 441 1 is proved in Section 4.1. Propositions 4.2 and 4.3 are proved in Section 4.5. Proposition 4.2 is a consequence of the analysis in Sections 4.2 and 4.3 where, by inserting an appropriate quasimode in the Riesz projector, we can prove that Π n,h is not zero. Rough localization of the spectrum. In this section, we prove Proposition 4.1.

Remark 4 . 4 .

 44 Let us mention what happens in the case α ∈ ( π 2 , 3π 5

4. 3 .e 2iα 3 h 2 3

 32 Quasimodes and localization estimates.If φ = (φ 1 , ϕ 2 ) is a normalized eigenfunction of M h,α associated with λ ∈ D(z 1 , Rh), we have in particular (4.8) M h,α ϕ 2 = λϕ 2 ,

6 )∥ϕ cut 2 ∥ . 4 . 4 . 1 .

 441 Localization estimates. The function ϕ cut 2

4. 5 . 2 .

 52 Proof of Proposition 4.3. Considering the result of Proposition 4.2, it is sufficient to show that for each fixed n ∈ {1, • • • N }, the Riesz projector Π n,h is not zero. For this consider the function ψ h (x) = (0, χ(s, u)Ψ 1,n,h (s, u)) ,

  is a consequence of Proposition 3.5 and support considerations. □ Refined localization in s. Thanks to the Agmon estimates in Proposition 3.5, we have proved so far a localization of order h 1 3 in the variable s. We improve this in the following proposition. Proposition 4.12. Consider χ 0 ∈ C ∞ 0 (R) equal to 1 in a neighborhood of 0 and η > 0.

	4.3.2. Then,
	(4.11)
	Remark 4.8. With a straightforward computation, we can check that, for all k, ℓ ∈ N,
	∥D k s D ℓ u r h ∥ = O(h ∞ ).
	Lemma 4.9. We have
	∥hD s ϕ cut 2 ∥ 2 + ∥hD u ϕ cut 2 ∥ 2 ⩽ Ch	2 3 ∥ϕ cut 2 ∥ 2 .

  Let us consider χ far,h (s) = χ far (h -1 2 +η s), where χ far is supported away from 0 and equal to 1 away form a compact set. Then, we use Proposition 4.11 to get that(4.13) ⟨(N h,α -λ)(χ far,h ϕ cut 2 ), χ far,h ϕ cut 2 ⟩ = ⟨χ far,h R h , χ far,h ϕ cut 2 ⟩ + ⟨[N h,α , χ far,h ]ϕ cut 2 , χ far,h ϕ cut 2 ⟩ .

			2	2	∥ϕ cut 2 ∥ .
	Proof. Notice that -α/2 + 2α/3 = α/6 and that, for all Ψ ∈ H 1 0 (R + ),
	(4.14)	cos(α/6)⟨(h 2 D 2 u + u -z 1 h	2 3

  ∥ϕ cut 2 ∥ 2 . By choosing η small enough, and by using an induction argument, we get that, for allN ∈ N, χ far,h ϕ cut 2 = O(h N )∥ϕ cut 2 ∥ , 4.4. Quasimodes for (N h,α -λ) 2 .Since we are in a non-selfadjoint context, the algebraic and geometric dimension associated with a given eigenvalue λ may differ. For further use, we now deal with localization estimates similar to the ones in Proposition 4.6, but in the case of generalized eigenfunctions. For this, let us consider such a λ ∈ D(z 1 e

	2iα 3 h	2 3

  4.1]), since it allows us to control the derivatives of ϕ with ϕ. Proof of Proposition 4.2. Let us first consider φ = (φ 1 , ϕ 2 ) an eigenfunction of M h,α . From Proposition 4.6, we have

	4.5. Proof of Propositons 4.2 & 4.3.
	4.5.1.
	Proposition 4.14. We have
	(4.22)	∥f ∥ ⩽ Ch∥φ∥ ⩽ Ch∥ϕ cut 2 ∥ ,
	and, for all (α 1 , α 2 ) ∈ N 2 ,
	(4.23)	∥(hD α 1 3 h	α 2 2 ∥φ∥ .

u ) α 1 (hD s ) α 2 ϕ cut 2 ∥ ⩽ Ch

  4.19) and (4.20). In particular, we have, in H 1 -norm,(4.31) φ h,z,1 = O(h ∞ )(∥φ h,z ∥ + ∥ψ h ∥)and, with similar notations as in (4.4),(4.32) ϕ h,z,2 = ϕ cut h,z,2 + O(h ∞ )(∥φ h,z ∥ + ∥ψ h ∥). One needs to estimate ∥φ h,z ∥. To do so, let us consider ϕ cut h,z,2 , which satisfies(M h,α -z)ϕ cut h,z,2 = -ψ cut h,2 + O(h ∞ )(∥φ h,z ∥ + ∥ψ h ∥) .As in (the beginning of) the proof of Proposition 4.14, we get the following.∥ 2 + ∥ψ h ∥∥ϕ cut h,z,2 ∥ + O(h ∞ )(∥φ h,z ∥ 2 + ∥ψ h ∥ 2 ) ,and∥hD s ϕ cut h,z,2 ∥ 2 + ∥sϕ cut h,z,2 ∥ 2 ⩽ Ch∥ϕ cut h,z,2 ∥ 2 + C∥ψ h ∥∥ϕ cut h,z,2 ∥ + O(h ∞ )(∥φ h,z ∥ 2 + ∥ψ h ∥ 2 ). Similarly, we get the control of the second order derivative with respect to s. Lemma 4.17. We have∥(hD s ) 2 ϕ cut h,z,2 ∥ ⩽ Ch∥ϕ cut h,z,2 ∥ + ChProof. Adapting (4.26) with our notations gives∥(hD s ) 2 ϕ cut h,z,2 ∥ 2 ⩽ Ch∥hD s ϕ cut h,z,2 ∥ 2 + Ch∥ψ cut h,2 ∥∥ϕ cut h,z,2 ∥ + Ch∥(hD s ) 2 ϕ cut h,z,2 ∥∥hD s ϕ cut h,z,2 ∥ + O(h ∞ )∥(hD s ) 2 ϕ cut h,z,2 ∥(∥ψ h ∥ + ∥φ h,z ∥) .With the Young inequality, this gives∥(hD s ) 2 ϕ cut h,z,2 ∥ 2 ⩽ Ch 2 ∥ϕ cut h,z,2 ∥ 2 + Ch∥ψ cut h,2 ∥∥ϕ cut h,z,2 ∥ + O(h ∞ )(∥ψ h ∥ 2 + ∥φ h,z ∥ 2 ) .We first write(N h,α -z)ϕ cut h,z,2 = -ψ cut h,2 + R h,z , with ∥R h,z ∥ ⩽ Ch2 3 -η ∥(hD s ) 2 ϕ cut h,z,2 ∥ + Ch∥(hD s )ϕ cut h,z,2 ∥ + C∥s 3 ϕ cut h,z,2 ∥ + O(h ∞ )(∥φ h,z ∥ + ∥ψ h ∥) .

	The proof is complete.			□
	From Lemmas 4.16 and 4.17, we get
	∥R h,z ∥ ⩽ Ch	2 3 -2η+ 1 2 ∥ϕ cut h,z,2 ∥ + Ch	2 3 -2η ∥ψ h ∥	1 2 ∥ϕ cut h,z,2 ∥
	Lemma 4.16. We have
	∥hD u ϕ cut h,z,2 ∥ 2 ⩽ Ch	2 3 ∥ϕ cut h,z,2 1 2 ∥ψ cut h,2 ∥	1 2 ∥ϕ cut h,z,2 ∥

1 2 + O(h ∞ )(∥φ h,z ∥ + ∥ψ h ∥) . 1 2 + O(h ∞ )(∥φ h,z ∥ + ∥ψ h ∥) .

By using Proposition 4.5 and the fact that z ∈ C n,h , we infer that

∥ϕ cut h,z,2 ∥ ⩽ Ch -1 ∥ψ h ∥ + O(h ∞ )(∥φ h,z ∥ + ∥ψ h ∥) .

With (4.31) and (4.32), this gives ∥φ h,z ∥ ⩽ Ch -1 ∥ψ h ∥ .

Acknowledgments

This work was conducted within the France 2030 framework programme, the Centre Henri Lebesgue ANR-11-LABX-0020-01. The authors thank the Fédération de recherche Mathématiques des Pays de Loire and its regional project Ambition Lebesgue Loire, which funded the stay of D. K. at the Laboratoire Angevin de Recherche Mathématique in May 2022, when this work started. D.K. was also partially supported by the EXPRO grant No. 20-17749X of the Czech Science Foundation. N.R. is grateful to Anne-Sophie Bonnet -Ben Dhia and Marc Lenoir for stimulating discussions at the Oberwolfach Research Institute for Mathematics in September 2022. He also wishes to thank Martin Averseng for enlightening discussions about the Caccioppoli estimates and the presentation in Section 1.4. The authors also thank Laura Baldelli for communicating them references about Caccioppoli estimates. F. H. and N.R. are also grateful to Bernard Helffer for useful discussions.

We deduce that

Re (e i(α-β) Γ 1 (s, J θ (u)))|e |Γ|/h 2/3 ϕ 2 | 2 dsdu + O(h 4/3 )∥e |Γ|/h 2/3 ϕ 2 ∥∥∇(e |Γ|/h 2/3 ϕ 2 )∥ + O(δ)h 2 ∥∇(e |Γ|/h 2/3 ϕ 2 )∥ 2

3 )∥e |Γ|/h 2/3 ϕ 2 ∥ 2 .

With Lemma 3.2, we get, for some c(α, β) > 0,

Re II ⩾c(α, β)

Re (e i(α-β) Γ 1 (s, J θ (u)))|e |Γ|/h 2/3 ϕ 2 | 2 dsdu + O(h 4/3 )∥e |Γ|/h 2/3 ϕ 2 ∥∥∇(e |Γ|/h 2/3 ϕ 2 )∥ + O(δ)h 2 ∥∇(e |Γ|/h 2/3 ϕ 2 )∥ 2

3 )∥e |Γ|/h 2/3 ϕ 2 ∥ 2 , and thus, perhaps after changing the value of c(α, β) > 0,

Re II ⩾c(α, β)

Re (e i(α-β) Γ 1 (s, J θ (u)))|e |Γ|/h 2/3 ϕ 2 | 2 dsdu .

(

Moreover, by using a H 1 2 -trace theorem, we get that, for all ϵ ∈ (0, 1), there exists C ϵ such that

To conclude, we now split the integral into two parts according to the decomposition

} . The rest of the proof follows from the usual manipulations à la Agmon. Indeed, thanks to Lemma 3.4, there exists C M such that

Proof. Due to the Dirichlet boundary condition of ϕ 2 on the external part of ∂B δ and the cutoff χ h , the function ϕ cut 2 satisfies the Dirichlet condition on ∂B δ . Let us now choose β such that (α, β) ∈ T as defined in Lemma 3.3. With an integration by parts using the expression of e -iβ M h,α , we get that, for some c > 0, Re e -iβ ⟨M h,α ϕ cut 2 , ϕ cut 2 ⟩ ⩾ c(∥hD s ϕ cut 2 ∥ 2 + ∥hD u ϕ cut 2 ∥ 2 ) . Then, by Lemma 4.7, we have that

The conclusion follows. □

For further use, we check now that we even have a control of higher order derivatives.

Lemma 4.10. We have

. Thus, after computing a commutator, we get

) . Then, we use Lemma 4.9 and we proceed as in its the proof to get

We proceed in the same way to get the control of (hD u ) 2 ϕ cut 2 . The conclusion follows. □

Let us consider the following intermediate operator,

s + e iα γ 1 (s) , which differs from N h,α in (4.5) only through its potential part. Proposition 4.11. We have

It is sufficient to use Lemma 4.7 and the explicit expression of M h,α . The estimate of R h follows from support considerations and Lemma 4.10 (note also that n

□ in H 1 -norm with respect to s. Coming back to (4.13), we also get the control of hD u . This gives (4.11).

Let us now turn to (4.12), which are better estimates than those in Lemmas 4.9 and 4.10. We again use Proposition 4.11 and see that

By using (4.14), we get that

and the estimate for

Thus,

By using the Young inequality and Proposition 4.11 to deal with the last term, we get

Coming back to (4.15), we also get

To get the control of s 2 , it is sufficient to notice that

and to estimate again a commutator. This concludes the case |(α 1 , α 2 , α 3 )| = 2. The proof of (4.12) for general (α 1 , α 2 , α 3 ) follows then by induction using the same method. □ 4.3.3. Proof of Proposition 4.6. We are now in position to complete the proof of Proposition 4.6, namely that ϕ cut 2 is indeed a good quasimode for N h,α . For this, we consider an operator P h defined through:

We check that P h can be written in the following way:

where the remainders r h,j all belong to S R 2 (1) and r h,5 with support avoiding a fixed neighborhood of (0, 0). We recall that S

For the other terms, we use the support property in the variable u and Proposition 4.12 to get

The conclusion follows.

Proof. Let us start by noticing that, from (4.21), we have

, and thus, with (4.16) and localization estimates, we get

where, for all Ψ ∈ B 1 0 (R + ) := {Ψ ∈ H 1 0 (R + ) :

3 )|Ψ| 2 duds ⩾ 0 .

We will denote the corresponding operator by A h and we observe that ϕ cut 2 and f cut 2 belong to its domain. At this stage, we still have to control f . By using (4.21), an integration by parts and (4.17), we have

Multiplying by e -iα/2 , taking the real part and estimating commutators give

, where we note that hD s ϕ cut 2 is also in B 1 0 (R + ). Proceeding in the same way, we find

, and thus, with (4.18),

, so that, using the definition of q Ai,h , we get

Thus, with (4.26), (4.27) and the Young inequality, we get, for some constant c > 0,

Recalling (4.21) to bound the right-hand side, we deduce that (4.28)

∥ϕ cut 2 ∥ . Proof. The proof is similar to that of Proposition 4.12 since we can write

, since we can apply Proposition 4.12 to the eigenfunction f and use (4.22). Following the same lines as in the proof of Proposition 4.12, we get (4.29).

Let us explain (4.30). Let us consider the case |(α 1 , α 2 , α 3 )| = 1. When α 3 = 1 or α 2 = 1, the estimate comes from Proposition 4.14. Then, we recall (4.24) and (4.22) and we get (4.30) with α = (1, 0, 0). For |(α 1 , α 2 , α 3 )| ⩾ 2, the result follows by induction. □ 4.4.2. Proof of Proposition 4.13. We have

Then, from Proposition 4.14 and (4.21), we get that

. By means of Proposition 4.15, we deduce Proposition 4.13. Let us explain this. Among the terms on the right-hand side (coming from the definition of P h ), we have to estimate the following

2 ∥ + C∥s 3 uϕ cut 2 ∥ + C∥s 5 ϕ cut 2 ∥ ⩽Ch 13 6 . All the other terms can be analyzed in the same way. It appears that the order of magnitude h 13 6 is the biggest one among all powers of h appearing in the remainders. This completes the proof of Proposition 4.13.

[17] I. Semoradova and P. Siegl. Diverging eigenvalues in domain truncations of Schrödinger operators with complex potentials. SIAM J. Math. Anal., 54(4):5064-5101, 2022.

(F. Hérau) LMJL -UMR6629, Nantes University , CNRS, 2 rue de la Houssinière, BP 92208, F-44322 Nantes cedex 3, France Email address: herau@univ-nantes.fr (D. Krejčiřík) Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 12000 Prague, Czech Republic Email address: david.krejcirik@fjfi.cvut.cz (N. Raymond) Univ Angers, CNRS, LAREMA, Institut Universitaire de France, SFR MATHSTIC, F-49000 Angers, France

Email address: nicolas.raymond@univ-angers.fr


Dala Raad

Kamar Hamade

Damien Herfurth

Roland Molinié

Jean-Xavier Fontaine

Hassan Rammal

Akram Hijazi

Marc Ongena

François Mesnard

Ophélie Fliniaux

Profilage métabolique de cultures de pomme de terre infectées par le mildiou et traitées par un agent de biocontrôle

La culture de la pomme de terre (Solanum tuberosum) est souvent menacée par le mildiou qui est causé par Phytophtora infestans, un oomycete qui affecte les feuilles et les tubercules. La surfactine, un agent de biocontrôle, est un cyclopeptide biactif produit par différentes souches bactériennes et qui a montré l'induction d'une résistance systémique dans la pomme de terre [1]. Dans cette étude un profilage métabolique est effectué par RMN pour des vitroplants infectées par P.infestans et traitées par la surfactine afin d'identifier les métabolites impactés par l'emploi de cette substance active et de comprendre le métabolisme de la résistance induit chez la pomme de terre [2] [3]

Conclusion et Perspectives

Concentration relative des 7 métabolites discriminats présents dans chaque condition influençant le plus la séparation des variables au niveau des PLS-DA Les analyses multivariées réalisées en RMN pour des extraits de feuilles de pommes de terre cultivées in vitro ont montré une différence du profilage métabolique des plantes traitées par la surfactine et les plantes infectées par le champignon. Ces résultats pourront être combinés aux analyses des extraits de tiges qui peuvent expliquer la réponse de cette plante contre un stress biotique et clarifier le rôle de l'agent de biocontrôle qui induit le système de résistance de la pomme de terre. Ensuite, une analyse des extraits des organes de la plante sera realisée avec l'utilisation d'une bactérie qui produit de la surfactine pour avoir une vue plus large de la réponse globale de la plante à un microorgansime comme agent de biocontrôle.

Références:

[1]S.

L-Proline



  PLS-Da score plot des extractions de feuilles de pomme de terre dans 4 conditions: FC-feuilles contrôle, FS-feuilles traitées avec 12 µg/ml surfactine, FM-feuilles infectés avec le champignon Phytophtora infestans qui cause le mildiou, FMS-feuilles infectées par le champignon et traitées avec la surfactine. Les CP1 et CP2 de l'analyse expliquent 62.3% de la variabilité des données. Selon CP1, FM et FMS sont séparés du groupe FC et FS; la présence du pathogène explique la différence des métabolites secondaires impliqués dans le système de résistance de la plante.

Introduction Matériel et Méthode Résultats

			Matériel végétal
					100
					mg
					MS
	Culture in vitro
	dans 4 conditions	Lyophilisation	Séparation feuilles/tiges	Broyage	Pesée
		30 min 20min	Extraction
		10 min
	10min, 4°C,1400-rpm		10 min, 60°C,2000 rpm
				700µlH2O/MeOH
				50:50
	Centrifuge	Bain ultrason	Thermomixeur		Extraction par robot
				Tubes RMN
	Extrait du
	surnageant 400 µl		1200µl surnageant	Transfert 750µl	Séchage sous flux d'azote
				pH=6
					Ajout du TMSP
					pH=6
			Données spectrales		RMN 600 MHz
					TXI-5mm Bruker
	Analyse statistique	Identification des métabolites
	i. Phasage des spectres 1H	I.	Acquisition 1D et 2D
	(TopSpin 3.2)
			II.	Base de données
	ii. Alignement des spectres
	(Matlab 2014b)		I.	Profilage métabolique des
				feuilles de pomme de terre
	iii. Découpage en bins
			II.	Identification des métabolites
	iv. Intégration			discriminants des 4
				conditions mentionnées
	v. Analyse multivariée
	PLS-DA score plot
	(SIMCA-P software)