Effective operators on an attractive magnetic edge

The semiclassical Laplacian with discontinuous magnetic field is considered in two dimensions. The magnetic field is sign changing with exactly two distinct values and is discontinuous along a smooth closed curve, thereby producing an attractive magnetic edge. Various accurate spectral asymptotics are established by means of a dimensional reduction involving a microlocal phase space localization allowing to deal with the discontinuity of the field.

with magnetic potential A := (A 1 , A 2 ) ∈ H 1 loc (R 2 ; R 2 ), generating the piecewise constant magnetic field where −1 ⩽ a ⩽ a 0 and a 0 is a fixed negative constant.Here h > 0 is a small parameter (the semiclassical parameter).Throughout this paper, we assume that and we refer to Γ as the magnetic edge (see Figure 1).We will denote the length of Γ by |Γ| = 2L.The operator P a h is self-adjoint in L 2 (R 2 ) with domain (1.4) Dom(P a h ) = {u ∈ L 2 (R 2 ) : (−ih∇ + A) j u ∈ L 2 (R 2 ), j = 1, 2}.Its essential spectrum is determined by the magnetic field at infinity (in our case it is equal to a).More precisely, by Persson's lemma, we have inf sp ess (P a h ) ⩾ |a|h.The purpose of this paper is to study the spectrum of P a h in the energy window J h = [0, Eh] with E ∈ (0, |a|) a fixed constant (thus, we analyze the spectrum below J.É.P. -M., 2023, tome 10 the essential spectrum) and in the semiclassical limit h → 0. We denote by λ n (P a h ) the n'th eigenvalue of P a h , and have sp(P a h ) ∩ [0, Eh] = {λ n (P a h )} N n=1 , (1.5) with N = N (h).
Let us stress that our spectral analysis will be uniform with respect to a ∈ [−1, a 0 ] and that the condition on the sign of a is crucial since we will see that it implies a localization of the eigenfunctions associated with eigenvalues in J h near the edge Γ.That is why we will say that the edge is attractive.
1.2.Heuristics, earlier results, and motivation 1.2.1.Analogy with an electric well and mini-wells.-The problem investigated in this paper shares common features with the semiclassical asymptotics of the Schrödinger operator, −h 2 ∆ + V , with an electric potential V , in the full plane, see [16,17,24,18].In this context, the "well" is the set Γ V := {x ∈ R 2 : V (x) = min R 2 V }, which attracts the bound states in the limit h → 0. The well is said to be nondegenerate if Γ V is a regular manifold, in which case the bound states might be localized near some points of Γ V , the mini-wells.This phenomenon of mini-wells is a manifestation of a multi-scale localization of the bound states.Interestingly, this phenomenon occurs also in the setting of the magnetic Laplacian, with a Neumann boundary condition, or with a magnetic field having a step-discontinuity as in the present article.In particular, if we consider the Neumann Laplacian with a constant magnetic field in a bounded, smooth domain, the boundary of the domain acts as the "well" and the set of points of the boundary with maximum curvature acts as the "mini-well" (see [15,8]).
1.2.2.Some known results.-Recently in [1,2], the operator P a h was considered in L 2 (Ω) with Dirichlet boundary condition on ∂Ω, Ω 1 ⊂ Ω and Γ a smooth curve that meets ∂Ω transversely.The edge Γ acts as the "well" and the set of points of Γ with maximum curvature acts as the "mini-well".Moreover, when the curvature has a unique non-degenerate maximum along the edge Γ, an accurate eigenvalue asymptotics displaying the splitting of the individual eigenvalues of P a h has been derived in [1, Th. 1.2], when −1 < a < 0. This result is clearly reminiscent of [8].
1.2.3.Motivation.-In the present article, we propose another perspective on the problem.Our spectral analysis will be uniform in various ways.Firstly, it will allow to derive, given some E ∈ (0, |a|), an effective operator in the whole energy window J h = [0, Eh] with h ∈ (0, h 0 ].In particular, the same strategy will provide us with Weyl estimates (estimating the number of eigenvalues in J h ) and the behavior of the individual eigenvalues.Secondly, it will also be uniform with respect to the parameter a ∈ [−1, a 0 ].This uniformity is the key to the understanding of the transition between the regimes a ∈ (−1, 0) and a = −1.This is all the more motivating since the miniwell phenomenon does not occur when a = −1.It is indeed rather satisfactory to have a point of view encompassing quite different phenomena and showing their unity.
1.3.The band functions.-The statement of our main results involves a family of 1D Schrödinger operators and their lowest eigenvalues, namely the operators obtained when the magnetic step is along a straight line, in which case a dimensional reduction is possible.This family has been the object of recent works (see [2,19]).Let us briefly recall some of its basic properties.Straightening the edge Γ locally, it is natural to consider the following "tangent" operator on R 2 with magnetic field where a ∈ [−1, a 0 ] is a fixed constant. (1)This operator is explicitly given by (1.6) . By using a rescaling and a partial Fourier transformation along the straight edge t = 0, we are led to consider the analytic family of Schrödinger operators with domain and has compact resolvent.We denote by (µ [n] a (σ)) n⩾1 the non-decreasing sequence of the eigenvalues (repeated according to their multiplicity) of h a [σ].For shortness, we let µ a (σ) = µ [1]  a (σ) = inf sp(h a [σ]).(1.9)By the Sturm-Liouville theory, we have the following proposition.
Proposition 1.1.-All the eigenvalues of h a [σ] are simple.The eigenfunction associated with µ [n] a (σ) has exactly n − 1 simple zeroes on R.

The functions µ
[n] a (σ), are called the band functions.When a = 1, we are reduced to the harmonic oscillator and µ a (σ) are no more constant functions, see [19].The lowest band function, µ a (σ) is studied in [2].
a is analytic as a function of σ.Moreover, the lowest band function satisfies and µ a has a unique critical point, which is a non-degenerate minimum β a ∈ (0, |a|), attained at σ(a) > 0.
(1) Our investigation concerns the attractive magnetic edge, which is the case when a < 0. In the opposite case, a ∈ (0, 1), the magnetic edge will no longer attract the bound states, since µa(σ) (defined in (1.9)) becomes a monotone decreasing function with inf σ∈R µa(σ) = a.
1.4.Main results.-Our analysis will reveal that the semiclassical spectral asymptotics of P a h in the interval [0, Eh] is governed by that of an effective operator acting on the edge Γ.In particular, we obtain accurate asymptotics for the low-lying eigenvalues of P a h highlighting a significant difference between the cases where −1 < a < 0 and a = −1.
(ii) The asymptotics in Theorem 1.3 is consistent with the phenomenon observed in surface superconductivity (see [8] and references therein) and the semiclassical analysis for the Schrödinger operator with a degenerate well in [17].In this comparison, the well corresponds here to Γ and the mini-wells correspond to the points of maximal curvature.
(iii) Actually, the proof of Theorem 1.3 provides us with a uniform description of the spectrum in [0, Eh] and could also help determining the behavior of the eigenvalues close to Eh when E is non-critical for µ a , i.e., when E ̸ = β a .In the context of the Robin Laplacian, such considerations are the object of the ongoing work [7].Note also that there are some results high up in the spectrum in the recent work [14], where Dirichlet conditions are considered.
(iv) It might happen that k does not have a unique minimum and even that Γ has some symmetry properties.In this case, tunneling occurs and the eigenvalue splitting is exponentially small (see [10]).The proof is similar to the case of the Laplacian with a constant magnetic field and Neumann boundary condition in a symmetric domain [6].
When a = −1, we will prove that C(a) = 0 and thus the second and third terms in the asymptotics formally vanish.We still get accurate estimates for the low-lying eigenvalues of P a h when a = −1, which involves an operator on the edge Γ ≃ [−L, L), whose half-length is denoted by L.
Theorem 1.5 (Case a = −1).-There exists C 0 < 0 such that, for every n ∈ N, we have as h → 0, λ n (P where γ n (h) is the non-decreasing sequence of the eigenvalues of the differential operator acting on [−L, L) with periodic boundary conditions, and (1.12) Here The quantity α h in (1.12) involves the circulation of the magnetic potential along Γ.In fact, by Stokes' Theorem, the circulation satisfies At the first glance, Theorems 1.3 and 1.5 seem independent.However, they both result from the analysis of the effective operator of P a h (see Theorem 1.6 below), which provides us with an accurate spectral description for −1 ⩽ a < 0.
This effective operator can be described as an ℏ-pseudodifferential operator on R with a 2L-periodic symbol with respect to the space variable, and acting on 2L-periodic functions.Here and along the whole paper the parameter (1.13) ℏ := h 1/2 is called the effective semiclassical parameter.Let us describe the shape of our effective operator.For a given symbol p ℏ (s, σ) ∈ S R 2 (1), (2) we consider the Weyl quantization, i.e., the operator defined by For an introduction to pseudo-differential operators, the reader is referred for instance to [25], where rigorous definitions are given and several fundamental properties are established.These operators being well defined on S(R), they can be extended by duality as operators on S ′ (R).We now underline that, if p ℏ (s + 2L, σ) = p ℏ (s, σ), then Op w ℏ (p ℏ ) transforms all the 2L-periodic distributions into 2L-periodic distributions.In fact, Op w ℏ (p ℏ ) also preserves the space of 2L-periodic functions that are in L 2 loc , denoted by L 2 2L (R) (see Section 4.1).Such an induced operator will give us our effective operator and we will call it a pseudodifferential operator on the edge, s representing the coordinate on Γ (parametrized by arc-length).
(2) that is, a smooth bounded function on R 2 such that its derivatives at any order are also bounded, uniformly in ℏ ∈ (0, 1].

J.É.P. -M., 2023, tome 10
The main result in this article is the following.
Theorem 1.6 (Spectral reduction to the edge).-There exists a self-adjoint ℏ-pseudodifferential operator (with symbol p eff ℏ ∈ S R 2 (1)) on the edge, whose principal symbol coincides with µ a below E, such that the spectrum of Moreover, for all n ∈ N such that λ n (P a h ) ∈ J h = [0, Eh], we have as h → 0, The discreteness of the spectrum of such an ℏ-pseudodifferential operator, for ℏ small enough, is rather classical.Indeed, fixing E + ∈ (E, |a|), we shall see that the principal symbol of p eff ℏ coincides with µ a below E + and thus, since µ a has a unique minimum, we can consider a smooth function of σ with compact support, denoted by χ, such that p eff ℏ (s, σ) + χ(σ) ⩾ E + .Since Op w ℏ χ is a compact operator on L 2 2L (R), we get that the essential spectra of Op w ℏ (p eff ℏ )+Op w ℏ χ and Op w ℏ (p eff ℏ ) coincide.By using the Gårding inequality, this essential spectrum is contained in (E, +∞).
The power of Theorem 1.6 is that it yields the two different asymptotics in Theorems 1.3 and 1.5.The analysis in [1] only works for −1 < a < 0, in which case the eigenfunctions are localized near the edge point(s) of maximal curvature, while in the perfectly symmetric situation when a = −1, the localization near the edge is displayed via an effective operator essentially independent of h (and thus the corresponding eigenfunctions are not particularly localized near specific points on the edge, even in the limit h → 0).
Of course, the present statement of Theorem 1.6 is not very informative if we do not describe the effective operator (see (7.1) for the expression of p eff ℏ , involving the curvature k along the edge Γ, viewed as a function of the arc-length s).However, it already gives an idea of the dimensional reduction approach using the tools developed in [21] and inspired by [13,22].
Besides the accurate asymptotics of the low-lying eigenvalues obtained in Theorems 1.3 and 1.5, another interesting result that follows from Theorem 1.6 is a Weyl estimate.
Theorem 1.7 (Asymptotic number of edge states).-We have The above Weyl estimate is similar to the one for the Neumann Laplacian with a magnetic field obtained by purely variational methods not involving pseudodifferential techniques in [12,11,20].Remark 1.8 (i) Our work does not cover the case when Γ has corners, in which case a strategy of dimensional reduction might be inefficient (as in the case for the Neumann magnetic Laplacian on corner domains, see [4,5]).
(ii) Another interesting question is to analyze the behavior of the spectrum near the Landau level |a|h, where we loose the uniformity in our estimates and we can expect that another regime occurs.
1.5.Organization.-In Section 2, we discuss and recall some elementary properties of the model in R 2 with a flat edge.Section 3 is devoted to the description of the Frenet coordinates along the edge Γ and the reduction of our problem to the study of an operator in a neighborhood of Γ.In Section 4, we express the operator obtained in Section 3 as an ℏ-pseudodifferential operator with operator symbol and expand this operator in powers of ℏ.In Section 5, we use a Grushin problem to construct a parametrix (that is an approximate inverse) for the operator introduced in Section 3. In Section 7, we deduce accurate eigenvalue estimates from the Grushin reduction, finish the proof of Theorem 1.6, and show how it yields the other theorems announced in the introduction.

The flat edge model
This section is devoted to the study of the flat edge model (1.6) and more precisely to the properties of the fibered family (1.7).We recall that our analysis holds for [−1, a 0 ], with −1 < a 0 < 0.
2.1.More on the band functions.-We will use the following lemma.
Proof.-Let us consider the L 2 -normalized eigenfunction u := u a,σ associated with µ [2] a (σ).We have a (σ)u(t).By the Sturm-Liouville theory, u has exactly one simple zero t 0 .Assume first that t 0 ⩾ 0.Then, for all t ⩾ 0, In the same way, we infer that µ For later use, we can consider a smooth bounded increasing function χ 1 on R such that χ 1 (σ) = σ on a neighborhood of the interval [σ − (a, E + ), σ + (a, E + ) , see (1.11).In particular µ a • χ 1 has still a unique minimum at σ a , which is not degenerate and not attained at infinity (since lim inf |σ|→+∞ µ a (χ 1 (σ)) > µ a (σ a )).The functions a .We denote by • u σ the positive and normalized ground state of (2.1) where h a is defined in (1.7).
We can express the projection on span( where where we emphasize that Π * (σ) ∈ L(C, L 2 (R)).Thanks to Lemma 2.1 (and the spectral theorem), for all z ∈ [0, E + ], we can consider the regularized resolvent Example 2.2.-As mentioned in the introduction, we will work with pseudodifferential operators in the s-variable (parallel to the boundary).A key example is given by Π above.We view (s, σ) → Π(σ) ∈ L (L 2 (R), C) as an operator-valued symbol.Thereby we get, using the Weyl quantization of (1.14) in the introduction, for φ = φ(s, t), ) is an operator-valued symbol and for ψ = ψ(s), we have -For all z ∈ [0, E + ] (or more generally Rez ⩽ E + ), the matrix operator is bijective for all σ ∈ R and Moreover, the operator symbols (s, σ) → P 0,z (σ) and (s, σ) a (σ)}, • n 0 (σ) − z can be inverted on the orthogonal complement of We recall that S(R 2 , F ) is the set of smooth functions on R 2 , valued in F with bounded derivatives (at any order).
Proof.-By straightforward computations, we can verify the identities 2. Some useful formulas.-Let us recall some formulas and results from [2].Let ϕ a be the positive and L 2 -normalized ground state of the operator for n ∈ N. It has been proved in [2] that The case a = −1 is special because Finally, we will also need the following two identities [1, Rem.
Then, by [8], we have and

Decay of bound states and spectral reduction
In this section, we consider the eigenfunctions of the operator P a h = P h with eigenvalues in the energy window (3.1) We prove that the eigenfunctions associated with eigenvalues in J + h are exponentially localized near Γ, see Corollary 3.3.To describe the effect of the edge on the localization, it is natural to use the classical tubular coordinates near Γ, whose definition will be recalled in Subsection 3.1.In order to prove Corollary 3.3, we will have to combine Agmon estimates and a rough estimate on the number of eigenvalues in J + h (polynomially in h −1 )), which will be discussed in Subsection 3.2.

Tubular coordinates.
-For all ε > 0, consider the ε-neighborhood of Γ Consider a parameterization M (s) of the edge Γ by the arc-length coordinate s ∈ [−L, L), where L = |Γ|/2.Consider the unit normal n(s) to Γ pointing inward to Ω 1 , and the unit oriented tangent t(s) = ṅ(s) so that (t(s), n(s)) is a direct frame, i.e., det(t(s), n(s)) = 1.We can now introduce the curvature k(s) at the point M (s), defined by n(s) = k(s)n(s).
Let us represent the torus (R/2LZ) by the interval [−L, L) and pick ε 0 > 0 so that The Hilbert space L 2 (Γ(ε 0 )) is transformed into the weighted space and the operator P h is (locally near the edge) transformed into the following operator (see [9, App.F])): where b a is defined in (1.7) and (3.4) 3.2.Number of eigenvalues.-We give a preliminary, rough bound on the number of eigenvalues in J + h .As we will see, this first estimate will be enough to deduce a stronger one at the end of our analysis.
Proof.-Let us introduce a fixed partition of the unity and supp(χ e ) ⊂ Γ(ε 0 ).The quadratic form associated with P h is given by The usual localization formula (see, for instance, [23, §4.1.1])gives the existence of a constant C > 0 such that By noticing that ψ → (χ out ψ, χ e ψ, χ in ψ) is injective, and thanks to the min-max theorem, we find that , where the operators P out h , P e h and P in h are the Dirichlet realizations of (−ih∇ + A) 2 on Ω 2 , Γ(ε 0 ) and Ω 1 , respectively.We recall that E + < |a| ⩽ |a 0 | < 1 and notice that P out h ⩾ |a|h and P in h ⩾ h.When h is small enough, E + h + Ch 2 < |a|h < h, so we must have N (P out h , E + h + Ch 2 ) = N (P in h , E + h + Ch 2 ) = 0. Thus, N (P h , E + h) ⩽ N (P e h , E + h + Ch 2 ).Therefore, we are reduced to estimate the number of eigenvalues of the operator with compact resolvent P e h below E + h + Ch 2 .For that purpose, we can use the tubular coordinates and notice that, for all ψ ∈ H 1 0 (Γ(ε 0 )), This gives the following rough estimate, for some c 0 , C 0 > 0, Thanks to the min-max theorem, this implies the upper bound where −∆ Dir is the Dirichlet Laplacian on the cylinder (R/2LZ) × (−ε 0 , ε 0 ).The spectrum of this operator can be computed explicit thanks to Fourier series, and we get the rough estimate □ Since E + < |a|, the eigenfunctions of P h associated with eigenvalues in the allowed energy window J + h are localized near the edge, see [2].
Proposition 3.2.-There exist constants α, h 0 , C 0 > 0 such that, if h ∈ (0, h 0 ] and u h is an eigenfunction of P h associated with an eigenvalue in J + h , then the following holds, Combining Propositions 3.1 and 3.2, we get the following estimate. Corollary 3.3.-Let η ∈ (0, 1/2).There exists h 0 > 0 such that for all h ∈ (0, h 0 ) Corollary 3.3 suggests to use the rescaling t = ℏ t.We also consider a smooth cutoff function where c ∈ C ∞ 0 (R) is even and satisfies c = 1 on [−1, 1] and c = 0 on R ∖ (−2, 2).This cutoff function is convenient to define the new operator on the Hilbert space acting on the domain As in Proposition 3.2, we can prove that the eigenfunctions of N h associated with eigenvalues in J h are localized near t = 0.
Proposition 3.4.-The spectra of P h and N h in Therefore, we are reduced to the spectral analysis of N h .For shortness, we drop the tildes.Up to a change of gauge, we are reduced to the operator and domain where m ∈ Z is chosen so that (3.8b) θ(ℏ) ∈ [0, ℏπL −1 ).
Before going ahead, we have to deal with the inconvenience of working in a Hilbert space with a weighted measure, which also depends on ℏ.Thus, let us use the canonical conjugation and work in the fixed Hilbert space with flat measure L 2 ((R/2LZ) × R, ds dt): where Note that We restate the Proposition 3.4 in terms of the new notation.
Proposition 3.5.-The spectra of P h and N ℏ,θ in J + h coincide modulo O(h ∞ ).Let us explain this for θ = 0. From the composition theorem for pseudodifferential operators (see [25,Th. 4.18]), we see that ⟨x⟩ −1 Op w ℏ (p ℏ )⟨x⟩ is a pseudodifferential operator with symbol in S(1) (and thus it is bounded on L 2 (R) thanks to the Calderón-Vaillancourt theorem, see [25,Th. 4.23]).This shows that Op w ℏ (p ℏ ) is bounded from

A pseudodifferential operator with operator valued symbol
) .Now the operator N ℏ,θ introduced in (3.12) (with T ℏ,θ introduced in (3.10)) can be seen as the action of an ℏ-pseudodifferential operator N ℏ with operator symbol n ℏ on F ℏ,θ(ℏ) where θ(ℏ) is defined in (3.8b).We have Note that, by using the Floquet-Bloch transform, N ℏ is unitarily equivalent to the direct integral of the N ℏ,θ .We recall the classical notation for the Weyl quantization Let us explain why the operator N ℏ can be written under the form (4.3).Note already that, at a formal level, we expect that 2 .This formal principal symbol suggests to consider a set of operator-valued symbols (containing n 0 ).We will need to introduce the space B 2 (s,σ) (R).As a vector space we have B 2 (s,σ) (R) = B 2 (R) (as defined in (1.8)) and the index (s, σ) refers to the norm, given by ∥ψ∥ 2 Here we used the standard notation It is inconvenient for a pseudodifferential calculus that the norm above depends on σ.Therefore, to have uniformity in σ we will later introduce a localization in σ, In the following proposition, we describe the (symmetric) differential operators n j .
p 0 = ℏD s − b a t, Proof.-Let us provide a Taylor expansion of (3.12).(3.10) can be rewritten in the form Straightforward computations yield, Now we expand m −1 ℏ in powers of ℏ and get We have the following expansion The previous expression can be rearranged as follows Recalling (3.11), we can also expand the operator in the transversal variable and get where the functions v ℏ and w ℏ satisfy, uniformly with respect to s, and ℏ, We get the expansion of the operator in (4.7) and the remainder term is expressed via R ℏ in (4.10) as follows We see that the remainder R ℏ satisfies (ii).□ We can now establish an expansion of the form (4.6) by considering the Weyl symbols of the p j in (4.9) (and the composition of pseudodifferential operators).We get the decomposition (4.12a) where , and w ℏ is introduced in (4.11b).

The Grushin reduction
Instead of the operator N ℏ , we consider its truncated version defined by (5.1) This localization effectively makes σ bounded in all estimates.Therefore, we avoid the use of the (s, σ) dependent spaces B j (s,σ) (R) (defined in (4.5)) but can work in their uniform versions B j (R).In particular, the remainder terms appearing in applications of the symbolic calculus such as (4.7) will therefore be uniform in σ.Consider the operator symbol, for all z ∈ [0, E] and E < E + < |a|, (5.2) where, Π σ = ⟨•, • u σ ⟩ and for all j ⩾ 1, (5.3) The operator P 0,z is introduced in Proposition 2.3.Recall that it is bijective (since z ∈ [0, E]) and (5.4) .
We let . Then, , where E ℏ,l/r is a pseudodifferential operator, whose operator-valued symbol belongs to the class S R 2 , L L 2 (R) × C, L 2 (R, ⟨t⟩ −N dt) × C , uniformly in ℏ, for some N ∈ N independent of ℏ.
The coefficients appearing in Proposition 5.1 can be computed explicitly.Of particular importance to us is (5.5) we have where R + ℏ , R ± ℏ are pseudodifferential operators whose symbols belong to the class S R 2 , L L 2 (R) × C, L 2 (R, ⟨t⟩ −N dt) × C , and where E − ℏ , E + ℏ are pseudodifferential operators whose symbols belong to the class S R 2 , L L 2 (R) × C, L 2 (R) × C , uniformly in ℏ.

Spectral applications
6.1.Localization of the eigenfunctions of N ℏ,θ .-In order to perform the spectral analysis of N ℏ,θ , we need to prove that its eigenfunctions (associated with eigenvalues in [0, E + ]) are ℏ-microlocalized, with respect to σ + θ in This can be formulated in terms of the semiclassical wavefront/frequency set (see [25, §8.4.2, p. 188]), however we write a stronger estimate in Proposition 6.1 below which holds uniformly with respect to θ ∈ R.This is a consequence of the behavior of the principal operator symbol n 0,θ = −∂ 2 t + (σ + θ + b a t) 2 (which appears after the Bloch-Floquet transform), which is bounded from below by µ a (σ + θ).
Let us consider the operator N c ℏ,θ (with periodic boundary conditions) defined as the operator induced by N c ℏ on F ℏ,θ (defined in (4.1)).By using Proposition 6.1 and the min-max theorem, we get the following.there exist ℏ 0 , C > 0 such that, for all θ ∈ R and all ℏ ∈ (0, ℏ 0 ) and all k ⩾ 1 such that λ k (N ℏ,θ ) ⩽ E + , we have 6.2.Weyl estimate.-A remarkable consequence of Proposition 5.1 and its corollary is the following Weyl estimate, which improves Proposition 3.1.
Note that Q ± = Op(q ± h (z)) with q ± h (z) given in (5.5).Let us now test (5.8)(with z = 0) and (5.9) with functions of F ℏ,θ of the form u = e iθs/ℏ ψ, with ψ in the domain of the operator N c ℏ,θ (with periodic conditions).We get (6.2) where the index θ refers to the conjugation by e iθs/ℏ (or the translation by θ of the symbol in σ).Then, we take the inner product with P ℏ,θ ψ.To deal with the term involving Q − ℏ,θ , we use the first equality in (5.9), and this gives We apply this inequality to ψ being a linear combination of eigenfunctions of N c ℏ,θ associated with eigenvalues less than E and thus, thanks to the Agmon estimates (with respect to t), we can write, for some C 0 > 0, η ∈ (0, 1) and for ℏ small enough, ℏ,θ ψ∥(∥P * ℏ,θ P ℏ,θ ψ∥ + Cℏ∥ψ∥) + Cℏ∥ψ∥∥P ℏ,θ ψ∥.By definition of P ℏ,θ , we see that the principal symbol of P * ℏ,θ P ℏ,θ is a projection so that ℏ,θ ψ∥∥ψ∥ + Cℏ∥ψ∥∥P ℏ,θ ψ∥.Applying this inequality to functions in the space spanned by the k first eigenfunctions (5)  5) associated with eigenvalues repeated according to the multiplicity.
J.É.P. -M., 2023, tome 10 and also (6.4) We have now to check that, when ψ runs over our k-dimensional space, P ℏ,θ ψ runs over a k-dimensional space.Using the first equality in (5.8) with the j-th eigenfunction ψ = ψ j and z = λ j (N c ℏ ), and by using the Agmon estimates, we see that, there exists C > 0 such that for all j, ℓ, Then, writing ψ = k j=1 α j ψ j , we have Recalling (6.4) and using the min-max theorem, this shows that there exist C, ℏ 0 > 0 such that for all ℏ ∈ (0, ℏ 0 ), ) ⩽ E. By using Proposition 5.1 and similar arguments, we get the reversed inequality.Let us only sketch the proof.Thanks to Proposition 5.1, we get, for all f ∈ L 2 loc (R) that is 2L-periodic, By taking z = 0 and by using the Calderón-Vaillancourt theorem to deal with the right-hand-side, we get We can check that ∥Q + ℏ,θ f ∥ ⩾ c∥f ∥ for some c > 0. From the min-max theorem, we infer that There exist C, ℏ 0 > 0 such that for all k ⩾ 1 and all ℏ ∈ (0, ℏ 0 ), as soon as λ k (N c ℏ,θ ) ⩽ E. It remains to apply the usual Weyl estimate available for a ℏ-pseudodifferential operator whose principal symbol is • µ a (σ + θ) and remember that θ → 0 when ℏ → 0 and that the symbol is 2L-periodic with respect to s. □ + ℏ 2 q ± 2,βa (s, σ a ) + (∂ σ q ± 1 (s, σ a ) 2 2µ ′′ a (σ a ) .
Note that Op w ℏ b ℏ,θ is a differential operator of order 2 and that it shares common features with that of [3, (8.10)].The difference is the presence of the a priori non-zero term ℏ q ± 1 (s, σ a ).In Lemmas 7.2 and 7.3, we describe the terms appearing in (7.3).Here w = 2t(ξ 0 − t) 2 + t 2 (ξ 0 − t) f 0 (t) and v is the unique solution of We will prove by a somewhat lengthy but elementary calculation that From the definitions M 2 , M 4 > 0 and ξ 0 > 0. It follows from (2.7) that M 3 < 0. Consequently it is immediate (from the first expression for G) that G < 0. So to finish the proof of Lemma 7.3 it only remains to prove (7.5).
Let p, q be two polynomial functions such that