SEMI-CLASSICAL EIGENVALUE ESTIMATES UNDER MAGNETIC STEPS

. We establish accurate eigenvalue asymptotics and, as a by-product, sharp estimates of the splitting between two consecutive eigenvalues, for the Dirichlet magnetic Laplacian with a non-uniform magnetic ﬁeld having a jump discontinuity along a smooth curve. The asymptotics hold in the semiclassical limit which also corresponds to a large magnetic ﬁeld limit, and is valid under a geometric assumption on the curvature of the discontinuity curve.


Introduction
The paper studies a semiclassical Schrödinger operator with a step magnetic field and Dirichlet boundary conditions, in a smooth bounded domain.The aim is to give accurate estimates of the lower eigenvalues in the semiclassical limit.
Let Ω be an open, bounded, and simply connected subset of R 2 with smooth C 1 boundary.We consider a simple smooth curve Γ ⊂ R 2 that splits R 2 into two disjoint unbounded open sets, P 1 and P 2 , and such that Γ is a semi-straight line when |x| tends to +∞.We assume that Γ decomposes Ω into two sets Ω 1 and Ω 2 as follows (see Figure 1): (1) Γ intersects transversally ∂Ω at two distinct points.
(1.2) Note that the curve Γ separates the two regions Ω 1 and Ω 2 which are assigned with different values of the magnetic field.For this reason, we refer to Γ as the magnetic edge.We  This quadratic form is closed on the form domain H 1 0 (Ω).By the Friedrichs extension procedure, we can associate its Dirichlet realization in Ω (1.4) whose domain is (1.5) The operator P h is self-adjoint, has compact resolvent, and its spectrum is an increasing sequence, (λ n (h)) n∈N , of real eigenvalues listed with multiplicities.
In this contribution, we aim at giving the asymptotic expansion of the low-lying eigenvalues of P h , in the semiclassical limit, i.e. when h tends to 0.
Schrödinger operators with a discontinuous magnetic field, like P h , appear in many models in nanophysics such as in quantum transport while studying the transport properties of a bidimensional electron gas [RP00,PM93].In that context, the magnetic edge is straight and bound states interestingly feature currents flowing along the magnetic edge.
The present contribution addresses another appealing question on the influence of the shape of the magnetic edge on the energy of the bound states.We give an affirmative answer by providing sharp semiclassical eigenvalue asymptotics under a single 'well' hypothesis on the curvature of the magnetic edge (cf.Assumption 1.1 and Theorem 1.2 below).Loosely speaking, our hypothesis says that we perform a local deformation of the magnetic edge so that its curvature has a unique non-degenerate maximum.
Another important occurrence of magnetic Laplace operators is in the Ginzburg-Landau model of superconductivity [SJG63].In bounded domains, the spectral properties of these operators can describe interesting physical situations.In the context of superconductivity, an accurate information about the lowest eigenvalues is important for giving a precise description of the concentration of superconductivity in a type-II superconductor.Moreover, it improves the estimates of the third critical field, H C 3 , that marks the onset of superconductivity in the domain.We refer the reader to [AK20,Ass20] for discontinuous field cases, and to [FH06, HP03, LP00, LuP99, LP99, BNF07, BND06, BS98, TT00] for a further discussion in smooth fields cases.In the present paper, the Dirichlet realization of P h in the bounded domain Ω can physically correspond to a superconductor which is set in the normal (non superconducting) state at its boundary.
Using symmetry and scaling arguments, one can reduce the problem to the study of cases of a = (a 1 , a 2 ), where a 1 = 1 and a 2 = a ∈ [−1, 1).Moreover, we will soon make a more restrictive choice of cases of a (see (1.11) below).Towards justifying the upcoming choice of a, we introduce the effective operator h a [ξ] with a discontinuous field, defined on R and parametrized by ξ ∈ R: where b a (τ ) = 1 R + (τ ) + a1 R − (τ ) . (1.7) This operator arises from the approximation by the case where Ω = R 2 and Γ = {x 2 = 0}, τ corresponding to the variable x 2 and ξ being the dual variable of x 1 .The known spectral properties of h a [ξ], obtained earlier in [HPRS16, AKPS19, AK20], are recalled in Subsection 2.1.Here, we only present some features of this operator that are useful to this introduction.The bottom of the spectrum of h a [ξ], denoted by µ a (ξ), is a simple eigenvalue for a = 0, usually called band function in the literature.Minimizing the band function leads us to introduce β a = inf ξ∈R µ a (ξ) . (1.8) We list the following properties of β a , depending on the values of a: Case a = −1.
In the case where Ω = R 2 and Γ = {x 2 = 0}, this case is called the 'symmetric trapping magnetic steps', and is well-understood in the literature (see e.g.[HPRS16]).In this case, the study of h a [ξ] can be reduced to that of the de Gennes operator (a harmonic oscillator on the half-axis with Neumann condition at the origin).We refer the reader to [FH10] and the references therein for the spectral properties of this operator.Here, Θ 0 := β −1 0.59 (1.9) is attained by µ −1 (•) at a unique and non-degenerate minimum This case is called the 'asymmetric trapping magnetic steps', and is studied in many works (see [AK20,AKPS19,HPRS16]).We have |a|Θ 0 < β a < min(|a|, Θ 0 ) and β a is attained by µ a (•) at a unique ζ a < 0 [AK20] µ a (ζ a ) = β a .
A key-ingredient in establishing the asymptotics of the eigenvalues λ n (h) is that β a is an eigenvalue of h a [ξ], for some ξ ∈ R. We will use the corresponding eigenfunction in constructing quasi-modes of the operator P h .The above discussion shows that β a is an eigenvalue only when a ∈ [−1, 0).The case a = −1 is excluded from our study, despite the fact that β −1 is an eigenvalue of h −1 [ξ 0 ].Except if Γ is an axis of symmetry of Ω as in [HPRS16], the situation is more difficult and the curvature will play a more important role.We hope to treat this case in a future work.This explains our choice to work under the following assumption on a (thus on the magnetic field curl F) throughout the paper: a = (1, a), with − 1 < a < 0 . (1.11) Under assumption (1.11), we introduce two spectral invariants: where µ a and ζ a are introduced in (1.8) and (1.10), and φ a is the positive L 2 -normalized eigenfunction of h a [ζ a ] corresponding to β a .Furthermore, we work under the following assumption: Assumption 1.1.The curvature Γ s → k(s) at the magnetic edge has a unique maximum k(s) < k(s 0 ) =: k max , for s = s 0 .
This maximum is attained in Γ ∩ Ω and is non-degenerate The goal of this paper is to prove the following theorem: where β a , c 2 (a) and M 3 (a) are the spectral quantities introduced in (1.8) and (1.12).
Remark 1.3.This theorem extends [AK20, Theorem 4.5]) where the first two terms in the expansion of the first eigenvalue were determined with a remainder in O(h 3 ) .The proof of Theorem 1.2 partially relies on decay estimates of the eigenfunctions with the right scale (see Sec. 6 and [AK20]).In fact, away from the edge Γ, the eigenfunctions decay exponentially at the scale h −1/2 of the distance to Γ, while, along Γ, they decay exponentially with a scale of h −1/8 of the tangential distance on Γ to the point with maximum curvature.
Comparison with earlier situations.It is useful to compare the asymptotics of λ n (h) in Theorem 1.2 with those obtained in the literature, for regular domains submitted to uniform magnetic fields.In bounded planar domains with smooth boundary, subject to unit magnetic fields and when the Neumann boundary condition is imposed, the low-lying eigenvalues of the linear operator, analogous to P h , admit the following asymptotics as h tends to 0 (see e.g.[FH06]) where Θ 0 is as in (1.9), C 1 > 0 is some spectral value, and kmax and k2 are positive constants introduced in what follows.In this uniform field/Neumann condition situation, the eigenstates localize near the boundary of the domain.More precisely, they localize near the point s with maximum curvature k(s) of this boundary, assuming the uniqueness and non-degeneracy of this point.We define kmax = k(s) and k2 = −k (s) > 0. In [FH06], the foregoing localization of eigenstates restricted the study to the boundary, involving a family of 1D effective operators which act in the normal direction to the boundary.These are the de Gennes operators , defined on R + with Neumann boundary condition at τ = 0, and parametrized by ξ ∈ R. We recover the value Θ 0 as an effective energy associated to (h Back to our discontinuous field case with Dirichlet boundary condition, we prove that our eigenstates are localized near the magnetic edge Γ, and more particularly, near the point with maximum curvature of this edge (see Section 6).Analogously to the aforementioned uniform field/Neumann condition situation, our study near Γ involves the family of 1D effective operators (h a [ξ]) ξ∈R which act in the normal direction to the edge Γ, along with the associated effective energy β a .
At this stage, it is natural to discuss our problem when the Dirichlet boundary conditions are replaced by Neumann boundary ones.In this situation, one can prove the concentration of the eigenstates of the operator P h near the points of intersection between the edge Γ and the boundary ∂Ω.This was shown in [Ass20] at least for the lowest eigenstate (see Theorem 6.1 in this reference).In such settings, a geometric condition is usually imposed related to the angles formed at the intersection Γ ∩ ∂Ω (see [Ass20, Assumption 1.3 and Remark 1.4]).The localization of the eigenstates near Γ ∩ ∂Ω will involve effective models that are genuinely 2D, i.e. they can not be fibered to 1D operators (see [Ass20, Section 3]).Studying this case may show similarity features with the case of piece-wise smooth bounded domains with corners submitted to uniform magnetic fields, treated in [BND06] (see also [BNDMV07,BNF07,Bon05,Bon03] for studies on corner domains).Such similarities were first revealed in [Ass20] (see Subsection 1.3 in this reference).More precisely, one expects the result in the discontinuous field/Neumann condition situation to be similar to that in [BND06, Theorem 7.1].Such a result is worth to be established in a future work.
Theorem 1.2 permits to deduce the splitting between the ground-state energy (lowest eigenvalue) and the energy of the first excited state of P h .More precisely, introducing the spectral gap ∆(h we get by Theorem 1.2: Corollary 1.4.Under the conditions in Theorem 1.2, we have as h → 0 ).
Apart from its own interest, estimating the foregoing spectral gap has potential applications in non-linear bifurcation problems, for instance, in the context of the Ginzburg-Landau model of superconductivity (cf.[FH10, Sec.13.5.1]).
Remark 1.5.Altering the regularity/geometry of the edge Γ may lead to radical changes in Theorem 1.2.
• If Γ is a piecewise smooth curve (a broken edge) then we have to analyze a new model in the full plane (reminiscent of a model in [Ass20]).We expect analogies with domains with corners in a uniform magnetic field [Bon03].• If we relax Assumption 1.1 by allowing the curvature k to have two symmetric maxima, then a tunnel effect may occur and the splitting in Theorem 1.2 becomes of exponential order.This is recently analyzed in [FHK] based on the analysis of this paper and the recent work [BHR].• If the curvature along Γ or a part of Γ is constant, then we expect that the magnitude of the splitting in Theorem 1.2 will change too, probably leading to multiple eigenvalues.It would be desirable to get accurate estimates in this setting.We expect analogies with disc domains in a non-uniform magnetic field [FP].
Heuristics of the proofs.Our proof of Theorem 1.2 is purely variational.The derivation of the eigenvalue upper bound is rather standard.It is obtained by computing the energy of a well chosen trial state, v app h,n , constructed by expressing the operator in a Frenet frame near the point of maximum curvature and doing WKB like expansions (for the operator and the trial state).
Proving the eigenvalue lower bound is more involved.The idea is to project the actual bound state, v h,n , on the trial state v app h,n , and to prove that this provides us with a well chosen trial state for a 1D effective operator, 2 σ 2 .To validate this method, we need sharp estimates of the tangential derivative of the actual bound state, which we derive via a simple, but lengthy and quite technical method involving Agmon estimates and other implementations from 1D model operators.At this stage, one advantage of our approach seems its applicability with weaker regularity assumptions on the magnetic edge or the magnetic field, which could be useful in other situations as well, like the study of the 3D problem in [HM2].
Outline of the paper.The paper is organized as follows.Sections 2 and 3 contain the necessary material on the model 1D problems, for flat and curved magnetic edges, respectively.Section 4 is devoted to the eigenvalue upper bounds matching with the asymptotics of Theorem 1.2.Here, we give the construction of the aforementioned trial state v app h,n .In Sections 5 and 6, we estimate the tangential derivative of the actual bound states, after being truncated and properly expressed in rescaled variables.The tangential derivative estimate of the L 2 norm will follow straightforwardly from the main result of Section 5.However, a higher regularity estimate will require additional work in Section 6.
In Section 7, using the actual bound states, we construct trial states for the effective 1D operator, and eventually prove the eigenvalue lower bounds of Theorem 1.2.Finally, we give two appendices, Appendix A on the Frenet coordinates near the magnetic edge, and Appendix B on the control of a remainder term that we meet in Section 7.

Fiber operators
2.1.Band functions.Let a ∈ [−1, 0).We first introduce some constants whose definition involves the following family of fiber operators in L 2 (R) where ξ ∈ R is a parameter, and the domain of h a [ξ] is given by : Here the space B n (I) is defined for a positive integer n and an open interval I ⊂ R as follows The operator h a [ξ] is essentially self-adjoint and has compact resolvent.Actually, it can also be defined as the Friedrichs realization starting from the closed quadratic form For (a, ξ) ∈ [−1, 0) × R, the ground-state energy (bottom of the spectrum) µ a (ξ) of h a [ξ] can be characterized by and ξ → µ a (ξ) will be called the band function.
(2) There exists a unique ζ a ∈ R such that β a = µ a (ζ a ) .
In particular, using (2.10) for ξ = ζ a , we observe that the functions φ a and (ζ a + b a (τ )τ )φ a are orthogonal Moreover, the ground-state φ a satisfies the following decay estimates Proposition 2.1.Let a ∈ [−1, 0).For any γ > 0, there exists a positive constant Consequently, for all n ∈ N * there exists (2.12) The proof is classical by using Agmon's approach for proving decay estimates.We omit it and refer the reader to [FH10, Theorem 7.2.2] or to the proof of Lemma 2.4 below.

Moments.
Later in the paper, we will encounter the following moments which are finite according to (2.12).
For n ∈ {1, 2, 3}, they were computed in [AK20] and we have: Remark 2.2.From the properties of the band function recalled in Subsection 2.2, we get that M 3 (a) is negative for −1 < a < 0, and vanishes for a = −1.
Remark 2.3.The next identities follow in a straightforward manner from the foregoing formulae of the moments: We will also encounter the moment: involving the resolvent R a , which is an operator defined on L 2 (R) by means of the following lemma: (extended by linearity).Then, for any γ ≥ 0, R a and d dτ • R a are two bounded operators on L 2 (R, exp(γ|τ |) dτ ).
Integration by parts yields Using the Cauchy-Schwarz inequality, we get further Rearranging the terms in (2.19) and using Cauchy's inequality we get We end up with the following estimate where we note that the right hand side is independent of N .Since Φ γ,N is non negative and monotone increasing with respect to N , we get by monotone convergence that e Φγ v and e Φγ v belong to L 2 (R) and satisfy To finish the proof, we note that, since the regularized resolvent is bounded and Φ γ ≥ 0, Proposition 2.5.For any a ∈ (−1, 0), it holds (2.20) , and E a (z) be the lowest eigenvalue of the operator H a (z), defined on L 2 (R) as follows We adopt the same proof of [FH06, Proposition A.3] (replacing P 0 by H a (0) − β a there) to get the identity in (2.20).Finally, by [AK20], µ (ζ a ) > 0.

1D model involving the curvature
We consider a new family of fiber operators which are obtained by adding to the fiber operators in Section 2 new terms that will be related to the geometry of the magnetic edge.This family was introduced earlier in [AK20] and their definition is reminiscent of the weighted operators introduced in the context of the Neumann Laplacian with a uniform magnetic field [HM1].
We introduce the following parameters that satisfy and will be fixed throughout this section.Consider on (−h −δ , h −δ ), the positive function a κ,h (τ ) = (1 − κh and for ξ ∈ R, the following operator where b a is the function in (2.2) and The operator H a,ξ,κ,h is a self-adjoint operator in L 2 (−h −δ , h −δ ); a κ,h dτ with compact resolvent.We denote by λ n (H a,ξ,κ,h ) n≥1 its sequence of min-max eigenvalues.The first eigenvalue can be expressed as follows (3.4) By Cauchy's inequality, we write for any ε ∈ (0, 1), Noticing that hτ 4 ≤ h 1−4δ for τ ∈ (h −δ , h δ ) and optimizing with respect to ε, we choose ε = h We plug (3.5) in (3.4) to get, for some C 0 > 0, where q a [ξ] is the quadratic form in (2.4).The min-max principle ensures that Since β a > 0, (3.6) and (3.7) imply ) and the min-max principle we deduce the lower bounds in Lemma 3.1 below (see [AK20, Subsection 4.2] for details).
Proof.The approach is similar to the one used in the literature in establishing upper bounds for the low-lying eigenvalues of operators defined on smooth bounded domains, like Schrödinger operators with uniform magnetic fields (and Neumann boundary conditions) or the Laplacian (with Robin boundary conditions).For instance, one can see [BS98,FH06,HK17].The proof relies on the construction of quasi-modes localized near the point of maximal curvature on Γ.Let h ∈ (0, 1).Working near Γ, we start by expressing the operator P h in the adapted (s, t)-coordinates there (see Appendix A): Recall that we assume that the maximum is attained for s = 0, hence k max = k(0), and having Lemma A.1, we perform a global change of gauge ω such that the magnetic potential F satisfies in Ω near the edge Γ, when expressed in the (s, t) coordinates Performing the following change of variables: the operator Ph becomes in the (σ, τ )-coordinates 5) It is convenient to introduce the following operator where ζ a is introduced in Subsection 2.2 and we get Using the boundedness and the smoothness of k, and the fact that k (0) = 0 and k (0) < 0, we write where (e i,h ) i=1,••• ,4 are functions of σ and τ having the property that there exist C and h 0 such that 1 , for h 1 The following conditions on the length scales of τ and σ (namely that σ ∈ (−h −δ , h −δ ) and τ ∈ (−h −ρ , h −ρ )), as well as (4.7) and (4.8) below are set for a later use in the paper. and Hence, and (4.11)Here the terms (E i,h ) i=1,••• ,4 are functions in σ and τ having the property, that there exist C and h 0 such that, for h In what follows, we will construct, for each n ∈ N * , a trial function φ n ∈ Dom P new h satisfying the following , (4.13) The result in (4.13), once established, will imply by the spectral theorem the existence of an eigenvalue λ new n (h) of P new h such that Furthermore, by the definition of P new h in (4.6) we have: ).Thus, (4.14) will yield the result in (4.1).Hence, the discussion above shows that establishing (4.13) is sufficient to complete the proof of the theorem.
Let u 0 = φ a be the positive normalized eigenfunction of the operator h a [ζ a ] (in (2.1)) corresponding to the lowest eigenvalue β a .
We implement this choice of (µ 0 , g 0 ) in (e 1 ) and write ) and Remark 2.2), and the pair is a solution of (e 1 ).Similarly, is a solution of Equation (e 2 ).Finally, we consider Equation (e 3 ): We will search for µ 3 and f satisfying for every fixed σ.This orthogonality result will allow us to choose in order to satisfy (e 3 ).To that end, the aforementioned choice of g 0 , g 1 and g 2 gives for any fixed σ where I 2 (a) is introduced in (2.17) and (2.20), and c 2 (a) is introduced in (1.12).We consider the harmonic oscillator on R For each n ∈ N * , let f n ∈ S(R) be the n th normalized eigenfunction of H harm a corresponding to the eigenvalue (2n − 1) 2 . The choice For h sufficiently small, using the properties of Q h in (4.11) and (4.12), the fact that f ∈ S(R), the decay properties of φ a in Proposition 2.1 and those of the resolvent R a in (2.18), the foregoing choice of g and µ implies (4.18).Now, we consider the trial function (see (4.15)) associated with g (n) .Using again the decay properties of u 0 and f , and Lemma 2.4 for getting the same properties for the g j , one can neglect the effect of the cut-off functions in the computation while concluding from (4.18) the desired result in (4.13).We omit further details of the computation, and refer the reader to [FH06, Sections 2&3].
Remark 4.2.The formal construction of the pairs (µ i , g i ) i=0,••• ,3 in the proof of Theorem 4.1 can be pushed to any order, assuming that the curve Γ is C ∞ smooth.Using the same approach we can construct pairs (µ i , g i ) i∈N * for defining quasimodes yielding an accurate upper bound of the eigenvalue λ n (h), which is an infinite expansion of powers of h 1 8 .This upper bound will agree with the one in Theorem 4.1 up to the order h 7 4 (see [BS98,FH06,HK17]).
Remark 4.3.In the derivation of the lower bound in Section 7, the operator H harm a introduced in (4.25) plays the role of an effective operator in the tangential variable.In light of (4.16), (4.19), (4.20), (4.21) and (4.26), the quasi-mode ) , is a candidate for the profile of an actual eigenfunction of the operator P h , after rescaling and a gauge transformation.

Functions localized near the magnetic edge
In this section, we consider functions satisfying the energy bound2 in (5.1), which are consequently localized near the maximum of the curvature of the magnetic edge Γ.We will be able to estimate the tangential derivative of such functions.
As we shall see in Subsection 5.1, bound states and their first order tangential derivatives are examples of the functions we discuss in this section.5.1.Localization hypotheses.We fix t 0 > 0 so that the Frenet coordinates recalled in Appendix A are valid in {d(x, Γ) < t 0 }.We recall our assumption that the curvature of Γ attains its maximum at a unique point defined by the tangential coordinate s = 0.
Let θ ∈ (0, 3 8 ) be a fixed constant.Consider a family of functions (g h ) h∈(0,h 0 ] in H 1 (Ω) for which there exist positive constants C 1 , C 2 such that for h ∈ (0, h 0 ], where Q h is the quadratic form introduced in (1.3).Suppose also that there exist constants α, C > 0 and a family (r h ) h∈(0,h 0 ] ⊂ R + such that lim sup and the following two estimates hold, (5.4) We can derive from the decay estimates in (5.3) and (5.4) four estimates.

Rescaled functions and tangential estimates.
Let δ ∈ (0, 1 12 ) and η ∈ (0, 1 8 ) be two fixed constants.Consider the function w h defined as follows (5.9) where gh is the function assigned to g h by the Frenet coordinates as in (A.3) namely gh (s, t) = g h (x) , Note that, due to our conditions on δ and η, w h can be seen as a function on R 2 , and its L 2 -norm can be estimated by using (A.7) and (5.5) as follows (5.10) Under our hypotheses on the function g h (particularly (5.1) for θ ∈ (0, 3 8 ) and (5.3)-(5.4)),we can estimate the tangential derivative of the function w h .Proposition 5.1.For all θ ∈ (0, 3 8 ), there exist constants C θ , h θ > 0 such that, if h ∈ (0, h θ ], and g h satisfies (5.1) θ , (5.3) and (5.4), then the function w h introduced in (5.9) satisfies the following estimate (5.11) Proof.The proof is split into four steps.
We localize the integrals defining the L 2 -norm and the quadratic form of g h to the neighborhood, of the point of maximal curvature, s = 0.In fact, by the decay estimates in (5.7) and (5.8), We refine the localization of these integrals by using the decay estimates in (5.5) and (5.6), the change of variable formulas in (A.7) and the following expansions where we set κ = k max .More precisely, To estimate the second term in the right hand side we use the Cauchy-Schwarz inequality to obtain Hence by (5.5) (with N = 2) and (5.6) (with N = 4) we get Implementing the above, we have (5.12) and where Proceeding as above for the treatment of R 2 s 4 t 4 |g h | 2 dsdt, we infer from (5.1), (5.5) and (5.6) that Now, coming back to (5.1), we get after performing a change of variable and dividing by h where (5.15) In the sequel, we set (5.16) Next we perform a Fourier transform with respect to σ and denote the transform of w h by It results then immediately from (5.14) and (5.15) the following, and m h introduced in (5.15) now satisfies (5.18) Step 2. We introduce where q a,ζ,κ,h is the quadratic form introduced in (3.4).We rewrite (5.17) as follows (5.20) Fix a positive constant ε < 1.Then by Proposition 3.2, (5.21) Inserting this into (5.20)we get from which we infer the following two estimates (5.23) Step 3.

Localization of bound states
In this section, we fix a labeling n ≥ 1 and denote by ψ h,n a normalized eigenfunction of the operator P h with eigenvalue λ n (h).By Theorem 4.1, it holds where Q h is the quadratic form introduced in (1.3).The decay estimates in Subsections 6.1 and 6.2 follow by standard semiclassical Agmon estimates.We refer to [HM1,FH06] for details in the case of the Laplacian with a smooth magnetic field, and to [AK20] for adaptations in the piecewise constant field discussed here.
Using the aforementioned decay estimates, the bound state ψ h,n satisfies the hypotheses in Section 5. Namely the estimates in (5.1) θ , (5.3) and (5.4) hold with g h = ψ h,n , r h = 1 and for any θ ∈ (0, 3 8 ).Consequently, we will be able to estimate its tangential derivative (see Proposition 6.2).Estimating the second order tangential derivative of ψ h,n (as in Proposition 6.3) requires the analysis of the decay of its first order tangential derivative in order to verify the hypotheses of Section 5.

Decay away from the edge.
The derivation of an Agmon decay estimate relies on the following useful lower bound of the quadratic form [AK20, Sec.4.3].For every R 0 > 1, there exists a positive constant C 0 and h 0 > 0 such that, for h ∈ (0, h 0 ], where Q h is introduced in (1.3) and Note that the decay property is a consequence of β a < |a|.Following [FH10, Thm.8.2.4], it results from the foregoing lower bound that the eigenfunction ψ h,n decays roughly like exp −α 0 h −1/2 d(x, Γ) , for some constant α 0 > 0.More precisely, the following holds 6.2.Decay along the edge.Here we discuss tangential estimates along the edge Γ.
Recall that s = 0 corresponds to the (unique) point of maximal curvature.The starting point is the following refined lower bound of the quadratic form [AK20, Sec.4.3] where, with x = Φ(s, t), .
We introduce the following function where ψh,n is the function assigned to ψ h,n by the Frenet coordinates as in . Note that u h,n can be seen as a function on R 2 , and by (5.10) (applied with g h = ψ h,n ), its L 2 -norm satisfies Using Proposition 5.1, we can estimate the tangential derivative of u h,n .More precisely, we apply this proposition with g h = ψ h,n , r h = 1 and any 0 < θ < 3 8 (see Remark 6.1).In this case, the function introduced in (5.9) is given by w h = u h,n .Proposition 6.2.For all θ ∈ (0, 3 8 ), there exist constants C θ , h θ > 0 such that, for all h ∈ (0, h θ ], ( We can estimate higher order tangential derivatives of u h,n .
Before proceeding with the proof of Proposition 6.3, we introduce the notation, r h = Õ(h γ ) for a positive number γ, to mean the following (6.9) Proof of Proposition 6.3.We will apply Proposition 5.1 with an adequate choice of the function g h defining the function w h in (5.9).We introduce the function ϕ h on Ω as follows where f (x) = (1−χ dist(x, ∂Ω)/t 1 ) χ dist(x, Γ)/t 0 , t 1 and t 0 are constants so that the set {x ∈ Ω : dist(x, ∂Ω) > t 1 } contains the point of maximum curvature and the transforma- where φh is the function assigned to ϕ h by (A.3).Notice that, using the notation in (6.9), the conclusion of Proposition 6.2 can be written as ) .(6.12) We will show that g h satisfies (5.1) θ for any θ ∈ (0, 3 8 ), and that (5.3) and (5.4) hold with This will be done in several steps outlined below.
• In Step 1, we establish rough decay estimates for g h in the normal and tangential directions (see (6.20)).These estimates are nevertheless weaker than the estimates in (5.3) and (5.4) that we wish to prove.• In Step 2, we show that g h is in the domain of the operator P h introduced in (1.4).
• In Step 3, using the rough estimates obtained in Steps 1 and 2, we can verify that (5.1) holds for any θ ∈ (0, 3 8 ).• In Step 4, using the estimates obtained in Steps 1 and 3, and the Agmon method, we derive the decay estimates for g h as in (5.3) and (5.4) with r h given in (6.13).• In Step 5, we can apply the conclusion of Proposition 5.1 and conclude the proof of Proposition 6.3.
We show that the function g h decays exponentially in the normal and tangential directions.We select the constant t 0 so that the two functions x → dist(x, Γ) and x → s(x) are smooth in the neighborhood, Γ 2t 0 , of the edge Γ.Consequently, the transformation in (A.1) is valid in Γ 2t 0 .Since we encounter integrals of the function g h , which is supported in Γ t 0 ∩ Ω, we select the gauge given in Lemma A.1.In particular, by (A.4), we have Let α 2 ∈ (0, 1 2 min(α 0 , α 1 )), where α 0 , α 1 are the positive constants in (6.3) and (6.5).We introduce on Ω the weight functions and Φ tan (x) = exp α 2 s(x) h 1/8 .(6.15)By Remark 6.1, we can use (5.5) for ψ h,n .It results from (6.5), (6.14), the Hölder inequality, and our choice of α 2 , that, for j ∈ {1, 2}, where A 4j (•) is defined in (5.5) and In a similar fashion, we estimate the L 2 (Ω)-norms of Fψ h,n Φ norm , (F • F)ψ h,n Φ norm and Φ norm F • (h∇ − iF)ψ h,n using (6.3).Eventually, we get the following estimates (6.17) Furthermore, the following two estimates hold Notice that for w # := ψ h,n Φ # , (# ∈ {norm, tan}), we have, with P h the operator introduced in (1.4) Hence, noting that P h = −h 2 ∆ + 2ihF • ∇ + ih divF + |F| 2 , we find by (4.1), (6.16) and (6.17), By the L 2 -elliptic estimates for the Dirichlet problem in Γ 2t 0 ∩ Ω, and noting that w # satisfies the Dirichlet condition, Consequently, we get the following estimate Now we can derive decay estimates of the function g h introduced in (6.11).Controlling the decay of the magnetic gradient of g h requires a decay estimate of ψ h,n in the H 2 norm.Actually, collecting (6.18) and (6.19), we observe that Step 2. By the definition of g h in (6.11), this function is compactly supported in Ω ∩ Γ t 0 .Hence, there exists a regular open set ω such that, for h ∈ (0, h 0 ], supp g h ⊂ ω ⊂ ω ⊂ Ω ∩ Γ 2t 0 .Consequently g h satisfies the Dirichlet boundary condition on ∂ω.To prove that g h is in the domain of the operator P h , it suffices to establish that To that end, we consider the spectral equation satisfied by the eigenfunction ψ h,n Using (A.5) with the potential F in (4.3), (6.22) reads in the (s, t)-coordinates as follows where We differentiate with respect to s in (6.24), and get Having s → k(s) smooth, a = 1 − tk(s) for t ∈ (−2t 0 , 2t 0 ), and ψ n,h ∈ Dom P h ensure that the function in the RHS of (6.25) is in (6.26) Hence (6.21) follows from (6.26) using the interior elliptic estimates associated with the differential operator L := (a −2 ∂ 2 s + ∂ 2 t ).
Step 3. We prove that where Q h is the the quadratic form introduced in (1.3) .
With the notation introduced in (6.9), the estimates in (4.1) and (6.27) yield (5.1) for any θ ∈ (0, 3 8 ).We start by noticing that where ϕ h is defined in (6.10) and Recall that ϕ h and G h are compactly supported in Ω ∩ Γ t 0 so that we can use the Frenet coordinates valid near the edge Γ.By (6.19) we have and by (6.20) G h L 2 (Ω) = O(1).(6.30)By Hölder's inequality, we infer from (6.29) and (6.30) Furthermore, computing the integrals in the Frenet coordinates and integrating by parts, we find (6.32)Here we get the O(h 9/8 ) remainder by using that ∂ s a = O(ts), the Hölder inequality and Remark 6.1 on the decay estimates in (5.5) and (5.6) for ψ h,n as follows By (4.1) and (6.12), we infer from (6.32) Therefore, inserting the estimates in (6.33) and (6.31) into (6.28),we find Now, by Lemma A.2 (used with φ = 0), we get where the function R h is defined via (A.3) as follows, (6.36) Our choice of gauge in Lemma A.1 ensures that F2 = 0 and F1 = O(t).By Remark 6.1 and (A.7), we have Now we can estimate Rh in (6.36), by expressing it as follows Rh = m 1 (h∂ We get then that the norm of R h satisfies, (6.37)By Hölder's inequality, we infer from (6.37) and (6.12) the following estimate Consequently, (6.34) and (6.35) yield (6.27).

Lower bound
We fix a labeling n ≥ 1 corresponding to the eigenvalue λ n (h) of the operator P h introduced in (1.4).The purpose of this section is to obtain an accurate lower bound for λ n (h).This will be done by doing a spectral reduction via various auxiliary operators.7.1.Useful operators.We introduce operators, on the real line and in the plane, which will be useful to carry out a spectral reduction for the operator P h and deduce the eigenvalue lower bounds that match with the established eigenvalue asymptotics in Theorem 1.2.
These new operators are defined via the spectral characteristics of the model operator introduced in Subsection 2.2, namely, the spectral constants β a > 0 and ζ a < 0 introduced in (1.10) and (1.12), and the positive normalized eigenfunction φ a ∈ L 2 (R) corresponding to β a .We introduce the following two operators and . It is easy to check that the operator norms of R ± 0 are equal to 1, hence, for any f ∈ L 2 (R) and ψ ∈ L 2 (R 2 ), we have 3) If we denote by π a the projector in L 2 (R τ ) on the vector space generated by φ a , we notice that 7.2.Structure of bound states.Our aim is to determine a rough approximation of the bound state ψ h,n of P h , satisfying this approximation being valid near the point of maximum curvature and reading as follows in the Frenet coordinates ψh,n (s, t) ≈ h −5/16 e iζas/h 1/2 φ a h −1/2 t .
Associated with ψ h,n , we introduced in (6.6) the function u h,n which can be seen as a function on R 2 with L 2 -norm satisfying (6.7).We recall that where ψh,n is the function assigned to ψ h,n by ( We consider the function defined as follows is the aim of the next proposition, which also yields an approximation of the bound state ψ h,n by the previous considerations. Proposition 7.1.Let P new h be the operator in (4.6).It holds the following. (1) Proof.

Proof of item (1).
Let z h be the function supported near Γ and defined in the Frenet coordinates by means of (A.3) as follows zh (s, t) = χ(h − 1 8 +η s)χ(h − 1 2 +δ t) . (7.7) We introduce the function involving the commutator of P h and z h acting on ψ h,n , By Remark 6.1, we may use the localization estimates in (5.7) and (5.8) with g h = ψ h,n and r h = 1.Consequently, where fh which is assigned to the function We infer from (7.5), (4.2), (4.4) and (6.6), Consequently, after performing the change of variable By (4.6) and (7.6), we observe that Ph u h,n = he iζaσ/h 3/8 (P new h + β a )v h,n , which after being inserted into (7.9),yields the estimate in item (1).

Proof of item (2).
By the normalization of ψ h,n and Remark 6.1, we have and We notice that the function z h introduced above in (7.7) equals 1 in {|s(x Similarly we get Consequently, returning to (7.6), doing a change of variables and noticing that zh is supported in

Proof of items (3) and (4).
Step 1.We recall that the Õ notation was introduced in (6.9).Note that Proposition 6.2 yields ) .(7.10)By Remark 6.1, we can use (5.13) and (5.14) with g h = ψ h,n , r h = 1 (and w h = ǔh,n ).In the same vein, we can use (5.5) and (5.6) too.Since u h,n = e iζaσ/h 3/8 v h,n , we get (7.11)By Cauchy's inequality and (7.10), we obtain for any > 0, We choose = h 3/8 and insert the resulting inequality into (7.11) to get: (7.12) Step 2. In light of (7.4), let us introduce Using the last relation, and since the orthogonal projection π a commutes with the operator h a [ζ a ], we have the following two identities, for almost every We deduce from (7.12) and the first item in Proposition 7.1 ) , (7.17) and Step 3. Coming back to the definition of r ⊥ in (7.13), we still have to improve the error term in (7.16) to get the estimate of the third item in Proposition 7.1.
7.3.Projection on a refined quasi-mode.We would like to improve the approximation v h,n ∼ χ(h η σ)χ(h δ τ )φ a (τ ) obtained in Proposition 7.1 by two ways which eventually are correlated: (i) displaying the curvature effects in v h,n and (ii) getting better estimates of the errors.Along the proof of Proposition 7.1, curvature effects were neglected and absorbed in the error terms.Not neglecting the curvature, we get the approximation v h,n ∼ χ(h η σ)χ(h δ τ )φ a,h (τ ) where φ a,h (τ ) corrects φ a (τ ) via curvature dependent terms (see (7.31)).This is precisely stated in Proposition 7.3 after introducing the necessary preliminaries.
7.3.1.Preliminaries.In this subsection, we write κ = k(0) = k max and k 2 = k (0).We consider the weighted L 2 space endowed with the Hilbertian norm which is self-adjoint on the space X h,δ .This operator can be decomposed as follows where h[ζ a ] is introduced in (2.1) and and where C 1 , C 2 are positive constants independent of h, τ .We introduce the following quasi-mode in the space X h,δ , We now explain the construction of φ cor a .By (7.28), starting from some φ cor a to be determined, where Note that, by Remark 2.3, h (1) [ζ a ]φ a − M 3 (a)φ a is orthogonal to φ a in L 2 (R) .Hence we can choose so that the coefficient of h 1/2 in (7.32) vanishes.In this way, we infer from (7.32), Notice that φ a,h is constructed so that it has compact support in (−h −δ , h −δ ) hence satisfies the Dirichlet conditions at τ = ±h −δ .Since, φ a and φ cor a decay exponentially at infinity by Lemma 2.4, we deduce (7.34) We denote by φ gs a,h the normalized ground state of the Dirichlet realization of H a,κ,h in the weighted space X h,δ (i.e. in L 2 ((−h −δ , h −δ ); (1 − h 1/2 κτ )dτ )).By (3.8), the min-max principle and Proposition 3.2, we have 1) , (7.35) so we infer from (7.34) and the Hölder inequality Thus, by the spectral theorem, (7.36) 7.3.2.New projections.We fix h 0 > 0 so that 1 − h 1 2 −δ 0 κ > 1 2 .In the sequel, the parameter h varies in the interval (0, h 0 ).Consider the space endowed with the weighted norm which is equivalent to the usual norm of L 2 R × (−h −δ , h δ ) .
We introduce the following two operators and where π a,h is the orthogonal projection, in the weighted Hilbert space X h,δ , on the space span φ a,h .With this projection in hand, we can approximate the truncated bound state v h,n , introduced in (7.6), with better error terms, thereby improving Proposition 7.1.
Proposition 7.3.The following holds where Π h is the projection in (7.40).
Remark 7.4.By (7.31) and (7.32), we observe that, where Π 0 is the projection introduced in (7.4).Since the norm of X 2 h,δ is equivalent to the usual norm of L 2 , Proposition 7.3 yields the following improvement of Proposition 7.1, ) , (7.41) where Π 0 is the projection in (7.4).This remark will be useful in the next subsection.
Proof of Proposition 7.3.
We give here preliminary estimates that we will use in Step 3 below.Firstly, by Remark 6.1, Secondly, we will prove that ) , (7.43)By (7.10) and Proposition 7.1, Similarly, using (7.19) and Hölder's inequality, we write ) and collecting the foregoing estimates, we get ) [by integration by parts] .
Step 2: We introduce operators involving the ground state φ gs a,h as follows.First we introduce the operators, R− ) , (7.47) then we deduce the estimate in Proposition 7.3.
Adapting the proof of Proposition 7.1, we prove now (7.47).By Remark 6.1, we can use (5.14) with w Since u h,n = e iζaσ/h 3/8 v h,n (by (7.6)), we get . (7.49) Using (7.10), (7.43) and (7.42), we deduce the following estimate from (7.49), where we used also that v h,n where c 2 (a) > 0 is introduced in (1.12).Moreover, by Remark 4.3, it is natural to consider the following quasi-mode where R a is the regularized resolvent introduced in (2.18), φ cor a is the function in (7.33), and f n is the normalized nth eigenfunction of the operator H harm a .Denoting by Π app h,n the orthogonal projection, in L 2 (R 2 ), on the space generated by v app h,n , we observe formally, by neglecting the terms with coefficients having order lower than where Π new n is the projection, in L 2 (R 2 ), on the space generated by the function ϕ a (τ )f n (σ), and Guided by these heuristic observations, we will use the truncated bound state v h,n in (7.6) to construct quasi-modes of the operator H harm a , by projecting v h,n on the vector space generated by the function ϕ a introduced in (7.52).To that end, we introduce the following operator (7.53) We will prove the following proposition.
Proposition 7.5.Let n ∈ N be fixed.The following holds: ( = O(h 1/4 ) where R − 0 is the operator in (7.1) and I 2 (a) is introduced in (2.17). ( where , and H harm a is the operator introduced in (7.51).

Proof of item (3).
With item (2) in hand, we get the conclusion of item (3).

Proof of item (4).
Step 1.We introduce the following operator Rnew Thus, by Proposition 7.1 and Remark 7.2, where P new h is the operator in (4.6).
(7.60)We first observe that it results from (7.1), (7.10), (7.56), and (7.57), For the sake of simplicity, we write κ = k(0) = k max .We introduce the following functions in L 2 (R), Recall the operators P 0 , P 1 , P 2 , P 3 , Q h introduced in (4.10) and (4.11).Noticing the decomposition in (4.9), we write, for any function v with compact support in R We now compute the first three terms on the right side of (7.64).
( * ) For the first term, since P 0 is self-adjoint in L 2 (R), we have We estimate W 2 v h,n by writing v h,n = Π 0 v h,n + (v h,n − Π 0 v h,n ), with Π 0 the projection introduced in (7.4), and by using (7.41).Eventually, since P 0 Π 0 = 0 and φ a , φ a L 2 (R) = 0, we get by Remark 2.3, ) .(7.68) We still have to estimate the terms involving W 3 and R h,n in (7.66) when v = v h,n .By choosing η small enough, the following error term = o(h 3/4 ) R new 0 v h,n L 2 (R) .By (7.66) and (7.70), we get from the above estimate where Finally, by item (1) and Proposition 2.5, we get (7.60).
The inner product of the remainder, r n (σ, h) in (7.69), and the function, R new 0 v h,n in (7.53), can be expressed as the inner product of a linear combination of functions having the forms in Lemma B.1 and B.2.The polynomials we encounter are of degree 6 at most.More precisely, r n (•, h), R new 0 v h,n L 2 (R) = h 7/8 A 1 + h 7/8 A 2 + hA 3 + h 9/8 A 4 + h 5/4 A 5 , where , and Here, Q h is the operator introduced in (4.11), P 0 , P 1 , P 2 , P 3 are the operators introduced in (4.10), f 1 , f 2 are the functions introduced in (7.62)-(7.63), the functions g 1 , g 2 , g 3 and g are defined as follows (see (7.57) and (7.53))By choosing η < 1 64 , we get (7.70).
We can now gather the above results.For each n ∈ N * , we choose µ in (4.17) and g = g (n) in (4.16) such that µ i , g i and f are as in (4.19)-(4.21),(4.23) and (4.26).

,
(7.14) by the min-max principle, where µ 2 (ζ a ) is the second eigenvalue of the operator h a [ζ a ], satisfying µ 2 (ζ a ) > β a (see Section 2.1).Integrating with respect to σ, we get The function φ a is the positive ground state of h[ζ a ] with corresponding ground state energy β a Let us start with some heuristic considerations.The derivation of the eigenvalue upper bound of Theorem 4.1 suggested in the tangent variable the following one dimensional effective operator (see (4.25)) by (6.7) and (7.6).Now we get (7.47) by decomposing v h,n in X 2 h,δ in the formv h,n = rh + rh,⊥ , rh := Πh v h,n , rh,⊥ = (I − Πh )v h,n ,and by using the spectral asymptotics for the operator H h,a,κ , recalled in (7.35).7.4.Quasi-modes for the effective operator.