Philippe Briet
email: briet@cpt.univ-mrs.fr

Hynek Kovarik
email: hynek.kovarik@ing.unibs.it

Georgi Raikov
email: graikov@mat.puc.cl

Hynek Kova

H Kovařík

SCATTERING IN TWISTED WAVEGUIDES

Keywords: AMS 2000 Mathematics Subject Classification: 35P05, 35P25, 47A10, 81Q10 Twisted waveguides, wave operators, Mourre estimates

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

Introduction

Let ω ⊂ R 2 be a bounded domain with boundary ∂ω ∈ C 2 . Denote by Ω := ω × R the straight tube in R 3 . For a given θ ∈ C 1 (R, R) we define the twisted tube Ω θ by Ω θ = r θ (x 3 ) x ∈ R 3 | x = (x 1 , x 2 , x 3 ) ∈ R 3 , x ω := (x 1 , x 2 ) ∈ ω , where r θ (x 3 ) =   cos θ(x 3 ) sin θ(x 3 ) 0 sin θ(x 3 ) cos θ(x 3 ) 0 0 0 1   .

We define the Dirichlet Laplacian H θ as the unique self-adjoint operator generated in L 2 (Ω θ ) by the closed quadratic form

Q θ [u] := Ω θ |∇u| 2 dx, u ∈ D(Q θ ) := H 1 0 (Ω θ ). (1.1)
In fact, we do not work directly with H θ , but rather with a unitarily equivalent operator H θ ′ acting in the straight tube Ω, see (2.4). The related unitary transformation is generated by a change of variables which maps the twisted tube Ω θ onto the straight tube Ω, see equation (2.3). The goal of the present article is to study the nature of the essential spectrum of the operator H θ under appropriate assumptions about the twisting angle θ. Although the spectral properties of a twisted waveguide have been intensively studied in recent years, attention has been paid mostly to the discrete spectrum of H θ , [START_REF] Briet | Eigenvalue asymptotics in a twisted waveguide[END_REF][START_REF] Exner | Spectrum of the Schrödinger operator in a perturbed periodically twisted tube[END_REF][START_REF] Grushin | On the eigenvalues of finitely perturbed Laplace operators in infinite cylindrical domains[END_REF], or to the Hardy inequality for H θ , [START_REF] Ekholm | A Hardy inequality in twisted waveguides[END_REF].

In this article we discuss the influence of twisting on the nature of the essential spectrum of H θ . First, we show that if the difference θ ′ 1θ ′ 2 decays fast enough as |x 3 | → ∞, then the wave operators for the operator pair (H θ ′ 1 , H θ ′ 2 ) exist and are complete, and in particular, the absolutely continuous spectra of H θ ′ 1 and H θ ′ 2 coincide. Further, we observe that if θ ′ = β is constant, then the operator H β is analytically fibered, cf. (2.9), and therefore its singular continuous spectrum is empty, [START_REF] Filonov | Absence of the singular continuous component in spectra of analytic direct integrals[END_REF][START_REF] Gérard | The Mourre theory for analytically fibered operators[END_REF]. Assuming that θ ′ (x 3 ) = β-ε(x 3 ) with ε ∈ C 1 (R, R), we then show that if ε decays fast enough at infinity, then H θ ′ has no singular continuous spectrum, see Theorem 2.7. The proof of Theorem 2.7 is based on the Mourre commutator method, [START_REF] Mourre | Absence of singular continuous spectrum for certain selfadjoint operators[END_REF][START_REF] Perry | Spectral analysis of N-body Schrödinger operators[END_REF][START_REF] Amrein | C0-Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians[END_REF]. We construct a suitable conjugate operator A and show that the commutator [H θ ′ , iA] satisfies a Mourre estimate on sufficiently small intervals outside a discrete subset of R, Theorem 8.2. The construction of the conjugate operator is based on a careful analysis of the band functions E n (k) of the unperturbed operator H β , k ∈ R being the Fourier variable dual to x 3 . A similar strategy was used in [START_REF] Gérard | Multiparticle quantum scattering in constant magnetic fields[END_REF][START_REF] Astaburuaga | Dynamical resonances and SSF singularities for a magnetic Schrödinger operator[END_REF][START_REF] Briet | Spectral properties of a magnetic quantum Hamiltonian on a strip[END_REF][START_REF] Krejčiřík | The nature of the essential spectrum in curved quantum waveguides[END_REF], where the generator of dilations in the longitudinal direction of the waveguide was used as a conjugate operator. However, in the situations studied in these works the associated band functions have a non zero derivative everywhere except for the origin. In our model, contrary to [START_REF] Gérard | Multiparticle quantum scattering in constant magnetic fields[END_REF][START_REF] Astaburuaga | Dynamical resonances and SSF singularities for a magnetic Schrödinger operator[END_REF][START_REF] Briet | Spectral properties of a magnetic quantum Hamiltonian on a strip[END_REF][START_REF] Krejčiřík | The nature of the essential spectrum in curved quantum waveguides[END_REF], the band functions E n may have many stationary points. In addition, we have to take into account possible crossing points between different band functions. The generator of dilations therefore cannot be used as a conjugate operator in our case, and a different approach is needed. Our conjugate operator acts in the fibered space as

i 2 (γ(k) ∂ k + ∂ k γ(k)) (1.2)
where γ ∈ C ∞ 0 (R; R) is a suitably chosen function, whose particular form depends on the interval on which the Mourre estimate is established, see Theorem 7.2. We would like to mention that a general theory of Mourre estimates for analytically fibered operators and their appropriate perturbations was developed in [START_REF] Gérard | The Mourre theory for analytically fibered operators[END_REF]. The situation with the twisted waveguide analyzed in the present article is much more specific than the general abstract scheme studied in [START_REF] Gérard | The Mourre theory for analytically fibered operators[END_REF]. Hence, although the construction in (1.2) is influenced in some extent by [START_REF] Gérard | The Mourre theory for analytically fibered operators[END_REF], our conjugate operator is essentially different from the one used in [START_REF] Gérard | The Mourre theory for analytically fibered operators[END_REF], and is considerably more useful for our purposes. In particular, the construction of this quite explicit conjugate operator allows us to handle the specific second-order differential perturbation which arises in the context of the twisted waveguide, and to verify all the regularity conditions for e itA , [H θ ′ , iA] and [[H θ ′ , iA], iA] needed for the passage from the Mourre estimate to the absence of the singular continuous spectrum, see Proposition 8.3. We have thus been able to apply the Mourre theory to the perturbed operator H θ ′ , and to find simple and efficient sufficient conditions on ε under which the singular continuous spectrum of H θ ′ is empty. We therefore believe that our construction of the conjugate operator might be of independent interest. The article is organized as follows. In Section 2 we state our main results. In Section 3 we prove Proposition 2.1 describing the domain of the operator H θ . In Section 4 we prove Theorem 2.3 which entails the existence and the completeness of the wave operators for the operator pair (

H θ ′ 1 , H θ ′ 2 ) for appropriate θ ′ 1 -θ ′ 2 ,
and hence the coincidence of σ ac (H θ ′ 1 ) and σ ac (H θ ′ 2 ). In Section 5 we assume that the twisting is constant, i.e. θ ′ = β and examine the spectral and analytical properties of the fiber family h β (k), k ∈ R. In Section 6 we construct the conjugate operator needed for the subsequent Mourre estimates. In Section 7 we obtain Mourre estimates for the case of a constant twisting. Finally, in Section 8 we extend these estimates to the case of θ ′ = βε where β ∈ R, and ε decays fast enough together with its first derivative.

Main results

2.1.

Notation. Let us fix some notation. Given a measure space (M, A, µ), we denote by 1 M the identity operator in L 2 (M ) = L 2 (M ; dµ). Further, we will denote by (u, v) L 2 (M ) = M ū vdµ the scalar product in L 2 (M ) and by u L p (M ) , p ∈ [1, ∞], the L p -norm of u. If there is no risk of confusion we will drop the indication to the set M and write (u, v) and u p instead in order to simplify the notation. Given a set M and two functions

f 1 , f 2 : M → R, we write f 1 (m) ≍ f 2 (m), m ∈ M , if there exists a constant c ∈ (0, ∞) such that for each m ∈ M we have c -1 f 1 (m) ≤ f 2 (m) ≤ c f 1 (m)
. Given a separable Hilbert space X, we denote by L(X) (resp., S ∞ (X)) the class of bounded (resp., compact) linear operators acting in X. Similarly, by S p (X), p ∈ [1, ∞), we denote the Schatten-von Neumann classes of compact operators acting in X; we recall that the norm in S p (X) is defined as T Sp := Tr (T * T ) p/2 1/p , T ∈ S p (X). In particular, S 1 is the trace class, and S 2 is the Hilbert-Schmidt class. Moreover, if T is a self-adjoint operator acting in X, we denote by D(T ) the operator domain of T . Finally, for α ∈ R define the function

φ α (s) := (1 + s 2 ) -α/2 , s ∈ R. ( 2
(Ω θ ) ∩ H 1 0 (Ω θ ). Proposition 2.1. Assume that ω ⊂ R 2 is a bounded domain with boundary ∂ω ∈ C 2 , and θ ∈ C 2 (R) with θ ′ , θ ′′ ∈ L ∞ (R). Then D (H θ ) = H 2 (Ω θ ) ∩ H 1 0 (Ω θ ). (2.2)
Proposition 2.1 could be considered a fairly standard result but since we have not been able to find in the literature a version suitable for our purposes (most of the references available treat bounded domains or the complements of compact sets), we include a detailed sketch of the proof in Section 3. Next, we introduce the operator U θ : L 2 (Ω θ ) → L 2 (Ω) generated by the change of variables

Ω ∋ x → r θ (x 3 ) x ∈ Ω θ . (2.3) Namely, for w ∈ L 2 (Ω θ ) set (U θ w)(x) = w (r θ (x 3 ) x) , x ∈ Ω. Evidently, U θ : L 2 (Ω θ ) → L 2 (Ω) is unitary since (2.
3) defines a diffeomorphism whose Jacobian is identically equal to one. Now assume g ∈ C(R; R) ∩ L ∞ (R) and introduce the quadratic form

Q g [u] = Ω |∇ ω u| 2 + |∂ 3 u + g ∂ τ u| 2 dx, u ∈ D(Q g ) = H 1 0 (Ω),
where

∇ ω := (∂ 1 , ∂ 2 ) T , and ∂ τ := x 1 ∂ 2 -x 2 ∂ 1 .
Denote by H g the self-adjoint operator generated in L 2 (Ω) by the closed quadratic form Q g . The transformation U θ also maps H 1 0 (Ω θ ) bijectively onto H 1 0 (Ω). Hence, for g = θ ′ we get

Q[w] = Q θ ′ [U θ w], w ∈ H 1 0 (Ω θ ), which implies H θ ′ = U θ H θ U -1 θ . (2.4) Assume now that g ∈ C 1 (R) with g, g ′ ∈ L ∞ (R). Set G(x 3 ) := x 3 0 g(s)ds, x 3 ∈ R. Then U G maps bijectively H 2 (Ω G ) onto H 2 (Ω). Therefore, Proposition 2.1 and the unitarity U G : L 2 (Ω G ) → L 2 (Ω) implies the following Corollary 2.2. Assume that ω ⊂ R 2 is a bounded domain with boundary ∂ω ∈ C 2 , and g ∈ C 1 (R) with g, g ′ ∈ L ∞ (R). Then the domain of the operator H g coincides with H 2 (Ω) ∩ H 1 0 (Ω). Furthermore, if g ∈ C 1 (R) with g, g ′ ∈ L ∞ (R) we have H g u = -∂ 2 1 -∂ 2 2 -(∂ 3 + g ∂ τ ) 2 u, u ∈ H 2 (Ω) ∩ H 1 0 (Ω), (2.5)
since

H G ϕ = -∆ϕ, ϕ ∈ H 2 (Ω G ) ∩ H 1 0 (Ω G ).
2.3. Existence and completeness of the wave operators. Next we show that under appropriate assumptions on the difference g 1g 2 , the wave operators for the operator pair (H g 1 , H g 2 ) exist and are complete, and hence the absolutely continuous spectra of the operators H g 1 and H g 2 coincide.

Theorem 2.3. Assume that ω ⊂ R 2 is a bounded domain with C 2 -boundary. Let g j ∈ C 1 (R; R) with g j , g ′ j ∈ L ∞ (R), j = 1, 2. Suppose that there exists α > 1 such that φ -α (g 1 -g 2 ) L ∞ (R) < ∞, (2.6)
the function φ α being defined in (2.1). Then we have

H -2 g 1 -H -2 g 2 ∈ S 1 (L 2 (Ω)). ( 2
Corollary 2.5. Assume that ω ⊂ R 2 is a bounded domain with C 2 -boundary. Let θ j ∈ C 2 (R; R) with θ j , θ ′ j , θ ′′ j ∈ L ∞ (R), j = 1, 2. Suppose that there exists α > 1 such that φ -α (θ ′ 1 -θ ′ 2 ) L ∞ (R) < ∞, Then the wave operators s -lim t→±∞ e itH θ 1 J e -itH θ 2 P ac (H θ 2 ), J := U -1 θ 1 U θ 2 ,
for the operator pair (H θ 1 , H θ 2 ) exist and are complete. Therefore, the absolutely continuous parts of H θ 1 and H θ 2 are unitarily equivalent, and, in particular, σ ac (H θ 1 ) = σ ac (H θ 2 ).

2.4. Constant twisting. In our remaining results, we concentrate on the case of appropriate perturbations of a constant twisting, i.e. the case where θ ′ is equal to a constant β ∈ R. First, we consider the unperturbed operator H β We define the partial Fourier transform F, unitary in L 2 (Ω), by

(F u)(x ω , k) = (2π) -1/2 R e -ikx 3 u(x ω , x 3 ) dx 3 , k ∈ R, x ω ∈ ω.
Then we have

Ĥβ = F H β F * = ⊕ R h β (k) dk, (2.9)
where, by (2.5) with g = β, the operator h β (k) acts on its domain D (h

β (k)) = H 2 (ω) ∩ H 1 0 (ω) as h β (k) = -∆ ω + (βi∂ τ + k) 2 , -∆ ω being the self-adjoint operator generated in L 2 (ω) by the closed quadratic form ω |∇v| 2 dx ω , v ∈ H 1 0 (ω).
Note that for all k ∈ R the resolvent h β (k) -1 is compact, and h β (k) has a purely discrete spectrum. Let 0

< E 1 (k) ≤ E 2 (k) ≤ • • • ≤ E n (k) ≤ . . . , k ∈ R, (2.10
) be the non-decreasing sequence of the eigenvalues of h β (k). Denote by p n (k) the orthogonal projection onto Ker(h β (k) -E n (k)), k ∈ R and n ∈ N. By [START_REF] Briet | Eigenvalue asymptotics in a twisted waveguide[END_REF][START_REF] Exner | Spectrum of the Schrödinger operator in a perturbed periodically twisted tube[END_REF] we have

σ(H β ) = σ ac (H β ) = [E 1 (0), ∞). (2.11)
A detailed discussion of the properties of E n (k) is given in Section 5. It turns out that the functions E n (k) are piecewise analytic, and that for any given k 0 ∈ R, the function E n (k) can be analytically extended into an open neighborhood of k 0 . We denote such an extension by Ẽn,k 0 (k). If k 0 is a point where E n (k) is analytic, then of course Ẽn,k 0 (•) = E n (•). With this notation at hand, we introduce the following subsets of R:

E 1 := E ∈ R : ∃ n ∈ N, ∃ k 0 ∈ R : E n (k 0 ) = E ∧ ∂ k Ẽn,k 0 (k 0 ) = 0 , E 2 := E ∈ R : ∃ k 0 ∈ R, ∃ n, m ∈ N, n = m : E n (k 0 ) = E m (k 0 ) = E ∧ ∧ ∂ k Ẽn,k 0 (k 0 ) ∂ k Ẽm,k 0 (k 0 ) < 0 .
We then define the set E of critical levels as follows:

E = E 1 ∪ E 2 .
(2.12)

Lemma 2.6. The set E is locally finite.

The proof of Lemma 2.6 is given in Section 5, immediately after Lemma 5.4.

Absence of singular continuous spectrum of H

β-ε . Theorem 2.7. Let θ ′ (x 3 ) = β -ε(x 3 ), where ε ∈ C 1 (R, R) is such that ε φ -2 ∞ + ε ′ φ -2 ∞ < ∞, ( 2
σ ac (H θ ′ ) = σ ac (H β ) = [E 1 (0), ∞).
Note that in order to prove the absence of singular continuous spectrum of H θ ′ we need stronger hypothesis on ε and ε ′ , see equation (2.13).

By [22,

(Ω θ ) the class of functions u ∈ C ∞ (Ω θ ), compactly supported in Ω θ . Set Ċ∞ (Ω θ ) := u ∈ C ∞ 0 (Ω θ ) | u |∂Ω θ = 0 . Lemma 3.1.
Under the assumptions of Proposition 2.1 there exists a constant c ∈ (0, ∞) such that

u 2 H 2 (Ω θ ) ≤ c Ω θ (|∆u| 2 + |u| 2 )dx (3.1)
for any u ∈ Ċ∞ (Ω θ ).

Proof. Our argument will follow closely the proof of [18, Chapter 3, Lemma 8.1]. We have

Ω θ (|∆u| 2 + c 0 |u| 2 )dx = Ω θ   3 j,k=1 |∂ j ∂ k u| 2 + c 0 |u| 2   dx + 2 ∂Ω θ K ∂u ∂ν 2 dS, u ∈ Ċ∞ (Ω θ ), (3.2
) (see [START_REF] Ladyzhenskaya | Linear and Quasilinear Elliptic Equations[END_REF] or [25, Chapter 5, Section 5, Problem 6]) where c 0 ∈ (0, ∞) is an arbitrary constant which is to be specified later, K is the mean curvature, and ν is the exterior normal unit vector at ∂Ω θ . Our assumptions on ∂ω and θ imply that for any u ∈ Ċ∞ (Ω θ ) we have 2

∂Ω θ K ∂u ∂ν 2 dS ≥ -c 1 ∂Ω θ |∇u| 2 dS. (3.3)
with

c 1 := 2 sup x∈∂Ω θ |K(x)| ≤ 2 sup (xω,x 3 )∈∂ω×R (|θ ′′ (x 3 )| + θ ′ (x 3 ) 2 )|x ω | + (1 + θ ′ (x 3 ) 2 |x ω | 2 )|κ(x ω )| ,
where κ(x ω ) is the curvature of ∂ω at the point x ω ∈ ∂ω. Let us check that for any ε > 0 there exists a constant c 2 (ε) such that for any v ∈ C ∞ 0 (Ω θ ) we have

∂Ω θ |v| 2 dS ≤ Ω θ ε|∇ ω v| 2 + c 2 (ε)|v| 2 dx (3.4)
where, as above, ∇ ω := (∂ 1 , ∂ 2 ) T . In order to prove this, we note the inequality

∂Ω θ |v| 2 dS ≤ c 3 R ∂ω θ(x 3 ) |v| 2 ds dx 3 (3.5)
where

c 3 := sup (xω,x 3 )∈∂ω×R 1 + θ ′ (x 3 ) 2 |x ω | 2 1/2 ,
and ω θ(a) is the cross-section of Ω θ with the plane {x 3 = a}, a ∈ R.

Next, since ω is a bounded domain with sufficiently regular boundary, we find that for any ε > 0 there exists a constant c 4 (ε) such that for any x 3 ∈ R and any w ∈ C ∞ (ω θ(x 3 ) ) we have

∂ω θ(x 3 ) |w| 2 ds ≤ ω θ(x 3 ) ε|∇w| 2 + c 4 (ε)|w| 2 dx ω (3.6)
(see e.g. [18, Chapter 2, Eq. (2.25)]). Choosing w = v(•, x 3 ) in (3.6), integrating with respect to x 3 , and bearing in mind (3.5), we get

∂Ω θ |v| 2 dS ≤ Ω θ c 3 ε|∇ ω v| 2 + c 3 c 4 (ε)|v| 2 dx which implies (3.4) with c 2 (ε) = c 3 c 4 (ε/c 3 ). Now the combination of (3.3) and (3.4) implies 2 ∂Ω θ K ∂u ∂ν 2 dS ≥ -c 1 Ω θ   ε 3 j,k=1 |∂ j ∂ k u| 2 + c 2 (ε)|∇u| 2   dx. (3.7)
Further, we have

Ω θ |∇u| 2 dx = -Re Ω θ ∆uudx ≤ 1 2 Ω θ |∆u| 2 + |u| 2 dx. (3.8)
Combining (3.2), (3.7), and (3.8), we find that for any ε > 0 we have

Ω θ (1 + c 1 c 2 (ε)/2) |∆u| 2 + c 0 |u| 2 dx ≥ Ω θ   (1 -c 1 ε) 3 j,k=1 |∂ j ∂ k u| 2 + (c 0 -c 1 c 2 (ε)/2) |u| 2   dx
which yields (3.1) under appropriate choice of c 0 , c and ε.

Denote by H2 (Ω

θ ) the Hilbert space u ∈ H 1 0 (Ω θ ) | ∆u ∈ L 2 (Ω θ ) with scalar product gener- ated by the quadratic form Ω θ (|∆u| 2 + |u| 2 )dx. Lemma 3.2. Under the assumptions of Proposition 2.1 we have u ∈ H2 (Ω θ ) if and only if u ∈ H 2 (Ω θ ) ∩ H 1 0 (Ω θ ). Proof. By Ω θ (|∆u| 2 + |u| 2 )dx ≤ 3 u 2 H 2 (Ω θ ) , u ∈ H 2 (Ω θ ), (3.9)
and (3.1), we have

Ω θ (|∆u| 2 + |u| 2 )dx ≍ u 2 H 2 (Ω θ ) , u ∈ Ċ∞ (Ω θ ). (3.10) Evidently, the class Ċ∞ (Ω θ ) is dense in H 2 (Ω θ ) ∩ H 1 0 (Ω θ )
. Then (3.9) easily implies that Ċ∞ (Ω θ ) is dense in H2 (Ω θ ) as well. Now the claim of the lemma follows from (3.10).

Proof of Proposition 2.1. Let L be the operator -∆ with domain C ∞ 0 (Ω θ ), and

L * be the adjoint of L in L 2 (Ω θ ). If v ∈ D(L * ), then a standard argument from the theory of distributions over C ∞ 0 (Ω θ ) shows that L * v = -∆v ∈ L 2 (Ω θ ). Since H θ is a restriction of L * , we find that u ∈ D (H θ ) implies that H θ u = -∆u ∈ L 2 (Ω θ ). On the other hand, u ∈ D (H θ ) implies u ∈ D (Q θ ) = H 1 0 (Ω θ ). By Lemma 3.2 we have u ∈ H 2 (Ω θ ) ∩ H 1 0 (Ω θ ), i.e. D (H θ ) ⊆ H 2 (Ω θ ) ∩ H 1 0 (Ω θ ). (3.11) If we now suppose that D (H θ ) = H 2 (Ω θ ) ∩ H 1 0 (Ω θ ), (3.12
) then (3.11) and (3.12) would imply that the operator H θ has a proper symmetric extension, namely the operator -∆ with domain H 2 (Ω θ ) ∩ H 1 0 (Ω θ ), which contradicts the self-adjointness of H θ . Therefore, (2.2) holds true, and the proof of Proposition 2.1 is complete.

Proof of Theorem 2.3

For the proof of Theorem 2.3 we need an auxiliary result, Lemma 4.1, preceded by some necessary notation. Let {µ j } j∈N be the the non-decreasing sequence of the eigenvalues of the operator -∆ ω . Since H g ≥ µ 1 1 Ω , and µ 1 > 0, the operator H g is invertible.

Lemma 4.1. Let g ∈ C 1 (R; R) with g, g ′ ∈ L ∞ (R). (i) Assume f ∈ L 2 (R). Then we have f (x 3 )H -1 g ∈ S 2 (L 2 (Ω)). (4.1)
(ii) Assume h ∈ L 4 (R). Then we have

h(x 3 )∂ j H -1 g ∈ S 4 (L 2 (Ω)), j = 1, 2, 3. (4.2)
Proof. By Corollary 2.2 the operator H 0 H -1 g is bounded, so that it suffices to prove (4.1) -(4.2) for g = 0. Evidently,

f H -1 0 2 S 2 (L 2 (Ω)) = j∈N f (-∂ 2 3 + µ j ) -1 2 S 2 (L 2 (R)) = = (2π) -1 j∈N R |f (s)| 2 ds R dξ (ξ 2 + µ j ) 2 = (2π) -1 j∈N µ -3/2 j R |f (s)| 2 ds R dξ (ξ 2 + 1) 2 . (4.3) Set N (λ) := #{j ∈ N | µ j < λ}, λ > 0. By the celebrated Weyl law, we have N (λ) = |ω| 4π λ(1 + o(1)) as λ → ∞
where |ω| is the area of ω (see the original work [START_REF] Weyl | Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)[END_REF] or [START_REF] Reed | Methods of Modern of Mathematical Physics, IV: Analysis of Operators[END_REF]Theorem XIII.78]). Therefore, the series j∈N µ

-γ j = γ ∞ µ 1 λ -γ-1 N (λ)dλ is convergent if and only if γ > 1. In particular, j∈N µ -3/2 j < ∞, (4.4)
so that the r.h.s. of (4.3) is finite which implies (4.1) with g = 0.

Let us now prove (4.2) with g = 0 and j = 1, 2. We have

h∂ j H -1 0 = ∂ j ((-∆ ω ) ⊗ 1 R ) -1/2 h((-∆ ω ) ⊗ 1 R ) 1/2 H -1 0 . Since the operators ∂ j ((-∆ ω ) ⊗ 1 R ) -1/2 , j = 1, 2, are bounded, it suffices to show that h((-∆ ω ) ⊗ 1 R ) 1/2 H -1 0 ∈ S 4 (L 2 (Ω)). ( 4
h((-∆ ω ) ⊗ 1 R ) 1/2 H -1 0 4 S 4 (L 2 (Ω)) = j∈N µ 2 j h(-∂ 2 3 + µ j ) -1 4 S 4 (L 2 (R)) ≤ ≤ (2π) -1 j∈N µ 2 j R |h(s)| 4 ds R dξ (ξ 2 + µ j ) 4 = (2π) -1 j∈N µ -3/2 j R |h(s)| 4 ds R dξ (ξ 2 + 1) 4 < ∞
which implies (4.5). Finally, we prove (4.2) with g = 0 and j = 3. To this end it suffices to apply again [24, Theorem 4.1] and (4.4), and get

h∂ 3 H -1 0 4 S 4 (L 2 (Ω)) = j∈N h∂ 3 (-∂ 2 3 + µ j ) -1 4 S 4 (L 2 (R)) ≤ ≤ (2π) -1 j∈N R |h(s)| 4 ds R ξ 4 dξ (ξ 2 + µ j ) 4 = (2π) -1 j∈N µ -3/2 j R |h(s)| 4 ds R ξ 4 dξ (ξ 2 + 1) 4 < ∞. Proof of Theorem 2.3. For z ∈ ρ(H g 1 ) ∩ ρ(H g 2 ) we have (H g 1 -z) -2 -(H g 2 -z) -2 = ∂ ∂z (H g 1 -z) -1 W (H g 2 -z) -1 = (H g 1 -z) -2 W (H g 2 -z) -1 + (H g 1 -z) -1 W (H g 2 -z) -2 (4.6) with W := ∂ τ (g 2 1 -g 2 2 )∂ τ + ∂ 3 (g 1 -g 2 )∂ τ + ∂ τ (g 1 -g 2 )∂ 3 . Choosing z = 0, we obtain H -2 g 1 -H -2 g 2 = -φ 3α/4 ∂ τ H -2 g 1 * (g 1 -g 2 )φ -α φ α/4 ∂ 3 H -1 g 2 + (g 2 1 -g 2 2 )φ -α φ α/4 ∂ τ H -1 g 2 - φ 3α/4 ∂ 3 H -2 g 1 * (g 1 -g 2 )φ -α φ α/4 ∂ τ H -1 g 2 - φ α/4 ∂ τ H -1 g 1 * (g 1 -g 2 )φ -α φ 3α/4 ∂ 3 H -2 g 2 + (g 2 1 -g 2 2 )φ -α φ 3α/4 ∂ τ H -2 g 2 - φ α/4 ∂ 3 H -1 g 1 * (g 1 -g 2 )φ -α φ 3α/4 ∂ τ H -2 g 2 .
Since the multipliers by (g 1g 2 )φ -α and (g 2 1g 2 2 )φ -α are bounded operators by (2.6) and

g j ∈ L ∞ (R), j = 1, 2, while φ α/4 ∂ ℓ H -1 g j ∈ S 4 (L 2 (Ω)), ℓ = τ, 3, j = 1, 2, by Lemma 4.1 (ii), it suffices to show that φ 3α/4 ∂ ℓ H -2 g j ∈ S 4/3 (L 2 (Ω)), ℓ = τ, 3, j = 1, 2. (4.7)
In what follows we write g instead of g j , j = 1, 2. Commuting multipliers by functions φ which depend only on x 3 and belong to appropriate Hörmander classes, with the resolvent H -1 g , and bearing in mind that

[φ, H -1 g ] = -H -1 g (φ ′′ + 2φ ′ (∂ 3 + g∂ τ ))H -1 g ,
we obtain

φ 3α/4 ∂ τ H -2 g = φ α/4 ∂ τ H -1 g φ α/2 H -1 g -φ α/4 ∂ τ H -1 g φ ′′ α/2 H -2 g + 2φ α/4 ∂ τ H -1 g (φ ′ α/2 ) 2 φ -α/2 H -2 g - 2φ α/4 ∂ τ H -1 g φ ′ α/2 φ -α/2 (∂ 3 + g∂ τ ) H -1 g φ α/2 H -1 g -2 φ ′′ α/2 + φ ′ α/2 (∂ 3 + g∂ τ ) H -2 g , (4.8)
φ 3α/4 ∂ 3 H -2 g = φ α/4 ∂ 3 H -1 g φ α/2 H -1 g -φ α/4 ∂ 3 H -1 g φ ′′ α/2 H -2 g + 2φ α/4 ∂ 3 H -1 g (φ ′ α/2 ) 2 φ -α/2 H -2 g -2φ α/4 ∂ 3 H -1 g φ ′ α/2 φ -α/2 (∂ 3 + g∂ τ ) H -1 g φ α/2 H -1 g -2 φ ′′ α/2 + φ ′ α/2 (∂ 3 + g∂ τ ) H -2 g +φ ′ α/2 φ -α/4 H -1 g φ α/2 H -1 g -φ ′′ α/2 + 2 (∂ 3 + g∂ τ ) H -2 g . (4.9)
Bearing in mind that S p ⊂ S q if p < q, and that H -1 g is a bounded operator, we find that Lemma 4.1 implies that all the terms at the r.h.s. of (4.8) and (4.9) can be presented either as a product of an operator in S 2 and an operator in S 4 , or as a product of three operators in S 4 , which yields (4.7), and the proof of Theorem 2.3 is complete.

Kato theory for a constant twisting

In this section we assume that θ ′ = β is constant. Then by (2.9) the operator

H β is unitarily equivalent to ⊕ R h β (k) dk with h β (k) = -∆ ω + (iβ∂ τ + k) 2 , k ∈ R.
The goal of the section is to establish various properties of the fiber operator h β (k), which will be used later in Section 7 for the Mourre estimates involving the commutator [H β , iA] with a suitable conjugate operator A described in Section 6.

Lemma 5.1. The operators h β (k), k ∈ R, with common domain H 2 (ω) ∩ H 1 0 (ω), form a self-adjoint holomorphic family of type (A) in the sense of Kato. Proof. Note that h β (k) = h β (0) + 2 Re β i k∂ τ + k 2 ,
and that h β (0) is self-adjoint on H 2 (ω) ∩ H 1 0 (ω). Let u ∈ H 1 0 (ω). Then for any ε > 0 we have

β i∂ τ u 2 2 ≤ (u, h β (0) u) L 2 (ω) ≤ u 2 h β (0) u 2 ≤ ε -1 u 2 2 + ε h β (0) u 2 2 .
Hence βi∂ τ is relatively bounded with respect to h β (0) with relative bound zero and the assertion follows from [16, Theorem VII.2.6].

From Lemma 5.1 and the Rellich Theorem, [16, Theorem VII.3.9], it follows that all the eigenvalues of h β (k) can be represented by a family of functions

{λ ℓ (k)} ℓ∈L , L ⊂ N, k ∈ R, (5.1)
which are analytic on R. Each eigenvalue λ ℓ (k) has a finite multiplicity which is constant in

k ∈ R. Moreover, if ℓ = ℓ ′ , then λ ℓ (k) = λ ℓ ′ (k) may hold only on a discrete subset of R.
Lemma 5.2. Let λ ℓ (k) be one of the analytic eigenvalues (5.1). Let k 0 ∈ R be given. Then

λ ℓ (k) -λ ℓ (k 0 ) ≤ |k -k 0 |, k ∈ R. (5.2)
Proof. By [16, Theorem VII.3.9] there exists an analytic normalized eigenvector ψ ℓ (k) associated to λ ℓ (k). From the Feynman-Hellmann formula, see e.g. [16, Section VII.3.4], we obtain

|∂ k λ ℓ (k)| 2 = 4 |((iβ∂ τ + k) ψ ℓ (k), ψ ℓ (k)) L 2 (ω) | 2 ≤ 4 (iβ∂ τ + k) ψ ℓ (k) 2 L 2 (ω) ≤ 4 (ψ ℓ (k), h β (k) ψ ℓ (k)) L 2 (ω) = 4 λ ℓ (k), k ∈ R. Hence |∂ k λ ℓ (k)| ≤ 2 λ ℓ (k) k ∈ R. (5.3)
By integrating this differential inequality we arrive at (5.2).

Remark 5.3. The eigenvalues E n (k) given in (2.10) might be degenerate. For example if β = 0 and if the operator -∆ ω has a degenerate eigenvalue

µ n = µ m = µ, then E n (k) = E m (k) = µ 2 + k 2 , ∀k ∈ R.
On the other hand, since every E n (k) coincides with one of the functions λ ℓ (k) locally on intervals between the crossing points of {λ ℓ (k)} ℓ , its multiplicity on these intervals is constant.

Let us define the set

E c :={E ∈ R : ∃ k ∈ R, ∃ ℓ, ℓ ′ ∈ L, ℓ = ℓ ′ : λ ℓ (k) = λ ℓ ′ (k) = E} ∪ {E ∈ R : ∃ k ∈ R, ∃ ℓ ∈ L : λ ℓ (k) = E ∧ ∂ k λ ℓ (k) = 0}. Lemma 5.4. Let R ∈ R. Then the set (-∞, R] ∩ E c is finite. Moreover, there exists an N R ∈ N such that for all n > N R and all k ∈ R we have E n (k) > R.
Proof. We know that

inf σ(h β (k)) = E 1 (k) ≥ E 1 (0) + c k 2 , k ∈ R, (5.4)
for some c ∈ (0, 1), see [START_REF] Briet | Eigenvalue asymptotics in a twisted waveguide[END_REF]Theorem 3.1]. This means that there exists some k R > 0 such that

E 1 (k) > R, k : |k| > k R . (5.5)
Let us denote

I R = [-k R , k R ]. Hence for any ℓ ∈ L we have λ ℓ (k) ≥ E 1 (k) > R on R \ I R . We claim that the set L R := {ℓ ∈ L : ∃ k ∈ I R : λ ℓ (k) ≤ R } is finite. Indeed, if #L R = ∞,
then, in view of (5.5), there is an infinite sequence {k j } ⊂ I R such that λ j (k j ) = R for all j ∈ L R . By inequalities (5.2) and (5.3) it follows that

sup j∈L R max k∈I R |∂ k λ j (k)| ≤ 4 k R + 2 √ R.
Let k ∞ ∈ I R be an accumulation point of the sequence {k j }. Hence, for any ε > 0 there exists an infinite set J ε ⊂ L R such that |λ j (k ∞ ) -R| ≤ ε for all j ∈ J ε . This means that R is an accumulation point of the spectrum of h β (k ∞ ) which contradicts the fact that σ(h β (k ∞ )) is discrete. We thus conclude that the set L R is finite. Since λ ℓ (k)λ ℓ ′ (k) is an analytic function for any ℓ, ℓ ′ ∈ L R , it has finitely many zeros in the interval I R . Next, by (5.4) it follows that none of the eigenvalues λ ℓ (k), ℓ ∈ L, is constant and therefore, by analyticity, every ∂ k λ ℓ (k) has finitely many zeros in I R . Hence the sets

∪ ℓ =ℓ ′ ,ℓ,ℓ ′ ∈L R {k ∈ I R : λ ℓ (k) = λ ℓ ′ (k)} and ∪ ℓ∈L R {k ∈ I R : ∂ k λ ℓ (k) = 0}
are finite and therefore (-∞, R] ∩ E c is finite too. As for the second statement of the Lemma, note that, by (5.

p n (k) ∂ k E n (k) = 2 p n (k) (iβ∂ τ + k) p n (k), k ∈ I. (5.6)
Proof. Since E n (k) is analytic on I, it coincides there with one of the analytic functions (5.1).

Hence by the Rellich Theorem, [16, Theorem VII.3.9], there exists a family of orthonormal eigenvectors φ j n (k), j = 1, . . . , q(n, I), analytic on I, associated with E n (k). Here q(n, I) denotes the multiplicity of E n (k) on I. Since (φ j n (k), φ i n (k)) L 2 (ω) = δ i,j for all k ∈ I, where δ i,j is the Kronecker symbol, we have

(h β (k) φ j n (k), φ i n (k)) L 2 (ω) = E n (k) δ i,j k ∈ I.
By differentiating this identity with respect to k, we easily obtain

2 ((iβ∂ τ + k) φ j n (k), φ i n (k)) L 2 (ω) = ∂ k E n (k) δ i,j k ∈ R, (5.7)
Hence for any u ∈ L 2 (ω) (5.9)

2 p n (k) (iβ∂ τ + k) p n (k) u = 2 q(n,I) i,j=1 φ i n (k) (φ i n (k), (iβ∂ τ + k) φ j n (k)) L 2 (ω) (φ j n (k), u) L 2 (ω) = ∂ k E n (k) q(n,I) j=1 φ j n (k) (φ j n (k), u) L 2 (ω) = ∂ k E n (k) p n (k) u.
Lemma 5.6. Suppose that I ⊂ R is an open interval. Let λ(k) and µ(k) be two analytic functions from the family (5.1) and assume that there is exactly one point k 0 ∈ I such that λ(k 0 ) = µ(k 0 ), and λ(k) = µ(k) for k 0 = k ∈ I. Let π λ (k) and π µ (k) be the eigenprojections associated with λ(k) and µ(k). Then in the sense of quadratic forms on L 2 (ω) we have

χ I (λ(k)) π λ (k) (iβ∂ τ + k) π µ (k) χ I (µ(k)) ≤ (5.10) ≤ b λ,µ |λ(k) -µ(k)| (χ 2 I (λ(k)) π λ (k) + χ 2 I (µ(k)) π µ (k)) for all k ∈ I, k = k 0 ,
where b λ,µ > 0 is a constant which depends only on λ, µ and I.

Proof. Let q(λ) and q(µ) denote the multiplicities of λ(k) and µ(k). Let ψ i λ (k), i = 1, . . . , q(λ) and ψ j µ (k), i = 1, . . . , q(µ) be sets of mutually orthonormal eigenvectors associated to λ(k) and µ(k). By the Rellich Theorem, [16, Theorem VII.3.9], these vectors can be chosen analytic in k. Hence, by differentiating the equation

(h β (k) ψ i λ (k), ψ j µ (k)) L 2 (ω) = 0 k ∈ I, k = k 0 with respect to k we arrive at 2((k +iβ, ∂ τ ) ψ i λ (k), ψ j µ (k)) L 2 (ω) = (λ(k)-µ(k))(∂ k ψ i λ (k), ψ j µ (k)) L 2 (ω) k ∈ I, k = k 0 . (5.11)
Note that for all k = k 0 , k ∈ I we have

π λ (k) = q(λ) j=1 ψ j λ (k) (ψ j λ (k), • ) L 2 (ω) , π µ (k) = q(µ) i=1 ψ i µ (k) (ψ i µ (k), • ) L 2 (ω) .
Let u ∈ L 2 (ω) and let max

1≤j≤q(µ) max 1≤i≤q(λ) sup k∈I |(∂ k ψ i λ (k), ψ j µ (k)) L 2 (ω) | =: bλ,µ .
(5.12)

From (5.11) we obtain

(u, χ I (λ(k)) π λ (k) (iβ∂ τ + k) π µ (k) χ I (µ(k)) u) L 2 (ω) = = 1 2 q(λ) j=1 q(µ) i=1 χ I (λ(k)) χ I (µ(k))(u, ψ i λ (k)) (ψ j µ (k), u) (λ(k) -µ(k))(∂ k ψ j λ (k), ψ i µ (k)) ≤ bλ,µ |λ(k) -µ(k)| q(λ) j=1 q(µ) i=1 χ 2 I (λ(k))|(u, ψ j λ (k))| 2 + χ 2 I (µ(k)) |(ψ i µ (k), u)| 2 ≤ b λ,µ |λ(k) -µ(k)| χ 2 I (λ(k))(u, π λ (k) u) + χ 2 I (µ(k)) (u, π µ (k) u) ,
for all k = k 0 , k ∈ I, where b λ,µ = bλ,µ max{q(λ), q(µ)}.

The conjugate operator

This section is devoted to the construction of the conjugate operator A occurring in the Mourre estimates obtained in the subsequent two sections.

Pick γ ∈ C ∞ 0 (R; R), and introduce the operator

Â0 = i 2 (γ ∂ k + ∂ k γ), D( Â0 ) = S(R), (6.1)
with S(R) being the Schwartz class on R. Proposition 6.1. Let γ ∈ C ∞ 0 (R; R). Then the operator Â0 defined in (6.1) is essentially self-adjoint in L 2 (R).

Proof. Without loss of generality we may assume that there exist a < b such that γ(a) = γ(b) = 0 and γ(k) > 0 for k ∈ (a, b). Consider solutions u ± to the equations

( Â * 0 u)(k) = i 2 (γ(k)∂ k + ∂ k γ(k)) u(k) = ± i u(k). (6.2)
A direct calculation gives This, combined with (6.3), yields

u ± (k) = exp k k 0 ±2 -γ ′ (r) 2γ(r) dr , k ∈ (a, b) (6.
u + (k) = γ(k 0 ) γ(k) 1/2 exp k k 0 dr γ(r) ≥ γ(k 0 ) γ(k) 1/2 exp k b-ε dr d b (b -r) ≥ c ε (b -k) -1 2 -1 d b , k ∈ (b -ε, b),
for some c ε > 0. Hence u + ∈ L 2 (R). The same argument shows that

u -(k) ≥ cε (k -a) -1 2 -1 da , ∀ k ∈ (a, a + ε), cε > 0,
which implies u -∈ L 2 (R). We thus conclude that Â0 has deficiency indices (0, 0) and therefore is essentially self-adjoint.

We define the self-adjoint operator  as the closure of Â0 in L 2 (R).

Further, we describe explicitly the action of the unitary group generated by Â. Given a k ∈ R and a function γ ∈ C ∞ 0 (R), we consider the initial value problem

d dt ϕ(t, k) = -γ(ϕ(t, k)), ϕ(0, k) = k. (6.4)
Proposition 6.2. The mapping

(W (t)f )(k) = |∂ k ϕ(t, k)| 1/2 f (ϕ(t, k)) (6.5)
defines a strongly continuous one-parameter unitary group on L 2 (R). Moreover, Â is the generator of W (t).

Proof. Since γ is globally Lipschitz, the Cauchy problem (6.4) has a unique global solution.

By the regularity of γ and [15, Corollary V. [START_REF] Birman | Existence conditions for wave operators (Russian)[END_REF].1], it follows that ϕ ∈ C ∞ (R 2 ). Moreover,

∂ k ϕ(t, k) = exp - t 0 γ ′ (ϕ(s, k)) ds ∀ t ≥ 0, ∀ k ∈ R, (6.6)
[15, Corollary V.3.1]. Hence ∂ k ϕ(t, k) > 0. Since ϕ(t + t ′ , k) = ϕ(t, ϕ(t ′ , k)), we have

W (t) W (t ′ ) = W (t + t ′ ).
Next, from (6.4) and (6.6) we deduce that for k ∈ supp γ we have ϕ(t, k) = k for all t ≥ 0. In order to verify that W (t) is strongly continuous on L 2 (R), let that f ∈ L 2 (R). We then have

W (t)f -f 2 L 2 (R) ≤ 2 ∂ k ϕ(t, k) 1/2 (f • ϕ(t, k) -f ) 2 L 2 (R) + 2 (∂ k ϕ(t, k) 1/2 -1) f 2 L 2 (R) (6.7) ≤ c supp γ |f (ϕ(t, k)) -f (k)| 2 + |∂ k ϕ(t, k) 1/2 -1| 2 |f (k)| 2 dk.
From (6.6) and from the fact that γ ′ ∈ L ∞ (R) it is easily seen that ϕ(t, k) → k and ∂ k ϕ(t, k) → 1 as t → 0 uniformly in k on compact subsets of R. Since supp γ is compact, (6.7) implies that

W (t)f -f L 2 (R) → 0, t → 0.
Moreover, using (6.4), a direct calculation gives

d dt (W (t)f )(k) t=0 = - 1 2 γ ′ (k)f (k) -γ(k)f ′ (k) = (i  f )(k), f ∈ S(R).
Hence by [START_REF] Reed | Methods of Modern of Mathematical Physics, I: Functional Analysis[END_REF]Theorem VIII.10] it follows that  generates the unitary group W (t).

Let γ be as in Theorem 7.2. By Proposition 6.1 and [21, Theorem VIII.33] it follows that the operator 1 ω ⊗ Â is essentially self-adjoint on C ∞ 0 (ω) ⊗ S(R). The same is true for the operator F (1 ω ⊗ Â) F * . We define the conjugate operator A in L 2 (Ω) as its closure:

A = Ā0 , A 0 = F (1 ω ⊗ Â) F * , D(A 0 ) = C ∞ 0 (ω) ⊗ S(R). (6.8)
Let Γ be the operator in L 2 (R) acting as

(Γ ψ)(x 3 ) := (2π) -1/2 R γ(x 3 -t) ψ(t) dt, γ := F 1 γ, (6.9)
where F 1 denotes the Fourier transform from L 2 (R) onto L 2 (R):

(F 1 f )(k) = (2π) -1/2 R e -iks f (s) ds, f ∈ L 2 (R).
A direct calculation then shows that

A 0 = - 1 2 1 ω ⊗ (Γ x 3 + x 3 Γ).

Mourre estimates for a constant twisting

In this section we establish a Mourre estimate for the commutator [H β , iA] with β constant and A defined in (6.8).

In the sequel, we use the following notation. Given a self-adjoint positive operator S, invertible in L 2 (Ω), we denote by D(S ν ) * , ν > 0, the completion of L 2 (Ω) with respect to the norm S -ν u L 2 (Ω) . Proof. Let u ∈ C ∞ 0 (Ω). A simple calculation then gives

( Ĥβ u, i(1 ω ⊗ Â) u) L 2 (Ω) -(i(1 ω ⊗ Â) u, Ĥβ u) L 2 (Ω) = 2(u, γ (k + iβ∂ τ ) u) L 2 (Ω) .
Hence (7.1) follows. Moreover, from the above equation we easily obtain Proof. First of all we chose δ small enough such that dist(E, (E c \ E)) > δ, (7.4) which is possible in view of Lemma 5.4. Recall that E ⊂ E c . Next we define

|([ Ĥβ , i(1 ω ⊗ Â)] u, u) L 2 (Ω) | ≤ C Ĥβ u 2 L 2 (Ω) + u 2 L 2 (Ω) . (7.2) So [ Ĥβ , i(1 ω ⊗ Â)] is a bounded operator from D( Ĥβ ) into L 2 (Ω),
K(n, E) = k ∈ R : E n (k) = E .
Note that by Lemma 5.4 K(n, E) is finite for every n and K(n, E) = ∅ for all n > N E+δ . In the rest of the proof we use the notation N = N E+δ . Let E), and define

K(E) = ∪ ∞ n=1 K(n, E) = ∪ N n=1 K(n,
K 0 (E) = {k ∈ R : there exists a unique n such that E n (k) = E}, K 1 (E) = K(E) \ K 0 (E).

Now we introduce the sets

B(n, E) = {k ∈ R : E n (k) ∈ (E -δ, E + δ)}.
By Lemma 5.4 we have B(n, E) = ∅ for all n > N . From (7.4) it follows that each B(n, E) is given by a union of finitely many non-degenerate disjoint open intervals:

B(n, E) = ∪ Gn j=1 Q(j, n), Q(j, n) ∩ Q(i, n) = ∅ if i = j.
Moreover, every Q(j, n) contains exactly one element of K(n, E). We will label the intervals Q(n, E) as follows:

Q 0 (j, n) := Q(j, n) if Q(j, n) ∩ K(n, E) ⊂ K 0 (E) Q 1 (j, n) := Q(j, n) if Q(j, n) ∩ K(n, E) ⊂ K 1 (E).
By the hypothesis on E we can take δ small enough so that

Q 0 (j, n) ∩ Q 0 (j, m) = ∅ n = m,
and at the same time

Q 1 (j, n) ∩ K 1 (E) = Q 1 (i, m) ∩ K 1 (E) implies Q 1 (j, n) ∩ Q 1 (i, m) = ∅.
Hence, for δ sufficiently small, we can construct intervals J 0,l with l = 1, . . . , L(E), and J 1,p with p = 1, . . . , P (E), such that

J 0,l ∩ J 0,l ′ = ∅ l = l ′ , J 1,p ∩ J 1,p ′ = ∅ p = p ′ , J 0,l ∩ J 1,p = ∅ ∀ l, p, (7.5)
and such that

M 0 (E) := N n=1 ∪ j Q 0 (j, n) = L(E) l=1 J 0,l , M 1 (E) := N n=1 ∪ j Q 1 (j, n) = P (E) p=1 J 1,p .
Moreover, each J 0,l contains exactly one element k 0,l of K 0 (E) and each J 1,p contains exactly one element k 1,p of K 1 (E). By construction, we have

M(E) := M 0 (E) ∪ M 1 (E) = ∪ N n=1 B(n, E), M 0 (E) ∩ M 1 (E) = ∅.
With these preliminaries we can proceed with the estimation of the commutator. From Lemma 7.1 we find that

χ I ( Ĥβ ) [ Ĥβ , i(1 ω ⊗ Â)] χ I ( Ĥβ ) = (7.6) = 2 ∞ n,m=1 ⊕ R χ I (E n (k))p n (k)γ(k)(k + iβ∂ τ )χ I (E m (k))p m (k) dk = 2 N n,m=1 ⊕ M 0 (E) χ I (E n (k))p n (k)γ(k)(k + iβ∂ τ )χ I (E m (k))p m (k) dk + 2 N n,m=1 ⊕ M 1 (E) χ I (E n (k))p n (k)γ(k)(k + iβ∂ τ )χ I (E m (k))p m (k) dk.
To estimate the first term on the right hand side of (7.6) we note that by construction of M 0 (E), for each l = 1, . . . , L(E) there exists exactly one n(l) ≤ N such that

N n,m=1 ⊕ J 0,l χ I (E n (k))p n (k)γ(k)(k + iβ∂ τ )χ I (E m (k))p m (k) dk = = ⊕ J 0,l χ I (E n(l) (k))p n(l) (k)γ(k)(k + iβ∂ τ )χ I (E n(l) (k))p n(l) (k) dk.
Moreover, since E n(l) (k) does not cross any other eigenvalue of h β (k) on J 0,l , it is analytic on J 0,l . Hence by Lemma 5.5 we obtain

⊕ J 0,l χ I (E n(l) (k))p n(l) (k)γ(k)(k + iβ∂ τ )χ I (E n(l) (k))p n(l) (k) dk = = ⊕ J 0,l χ 2 I (E n(l) (k)) p n(l) (k) γ(k) ∂ k E n(l) dk.
In view of (7.5) we can choose the function γ such that

γ(k) ∂ k E n(l) (k) = |∂ k E n(l) (k)| ∀ k ∈ J 0,l , ∀ l = 1, . . . , L(E). (7.7)
Note that |∂ k E n(l) (k)| is strictly positive on J 0,l . Therefore we have

d 0 := min 1≤l≤L(E) inf k∈J 0,l |∂ k E n(l) (k)| > 0. Hence, N n,m=1 ⊕ M 0 (E) χ I (E n (k)) p n (k) γ(k)(k + iβ∂ τ ) χ I (E m (k))p m (k) dk ≥ (7.8) ≥ d 0 N n=1 ⊕ M 0 (E) χ 2 I (E n (k))p n (k) dk.
Let us now estimate the second term on the right hand side of (7.6). On every interval J 1,p we have

N n,m=1 ⊕ J 1,p χ I (E n (k))p n (k)γ(k)(k + iβ∂ τ )χ I (E m (k))p m (k) dk = = ⊕ J 1,p r,r ′ ∈R(p) χ I (E r (k)) p r (k) γ(k)(k + iβ∂ τ )χ I (E r ′ (k)) p r ′ (k) dk
for some R(p) ⊂ {1, . . . , N }. Moreover, from the construction of the intervals J 1,p it follows that there exists a family of analytic eigenfunctions λ s (k), s ∈ S(p), with S(p) being a finite subset of N, such that each E r (k) coincides with some λ s (k) on J 1,p ∩ (-∞, k 1,p ) and with some

λ s ′ (k) on J 1,p ∩ (k 1,p , ∞),
where k 1,p is the only element of K 1 (E) contained in J 1,p . Let π s (k) be the eigenprojection associated with λ s (k). With the help of Lemma 5.6 we obtain

⊕ J 1,p r,r ′ ∈R(p) χ I (E r (k)) p r (k)γ(k)(k + iβ∂ τ )χ I (E r ′ (k))p r ′ (k) dk = = ⊕ J 1,p s,s ′ ∈S(p) χ I (λ s (k)) π s (k)γ(k)(k + iβ∂ τ )χ I (λ s ′ (k)) π s ′ (k) dk = = ⊕ J 1,p s∈S(p) χ 2 I (λ s (k)) γ(k) ∂ k λ s (k) π s (k) dk+ + ⊕ J 1,p s =s ′ ∈S(p) χ I (λ s (k)) π s (k)γ(k)(k + iβ∂ τ )χ I (λ s ′ (k)) π s ′ (k) dk. (7.9)
Since the intervals J 1,p are mutually disjoint and also disjoint from the set M 0 (E), see (7.5), and since the functions ∂ k λ s (k) are either all strictly negative or all strictly positive on every interval J 1,p , by the construction of J 1,p , we can choose γ such that, in addition to (7.7), it holds

γ(k) ∂ k λ s (k) = |∂ k λ s (k)| ∀ k ∈ J 1,p , ∀ s ∈ S(p), ∀ p = 1, . . . , P (E). (7.10)
Moreover,

d 1 := min 1≤p≤P (E) min s∈S(p) inf k∈J 1,p |∂ k λ s (k)| > 0.
Now, to control the last term in (7.9) assume that s = s ′ and let b λs,λ s ′ be the constant given in Lemma 5.6 with I = J 1,p . Note that |J 1,p | decreases as δ → 0. From the explicit expression for b λs,λ s ′ , see (5.12), it is then easily seen that there exists b p > 0, independent of δ, such that max

s,s ′ ∈S(p),s =s ′ b λs,λ s ′ ≤ b p .
Hence, (7.9), in combination with Lemmata 5.5 and 5.6, yields

⊕ J 1,p r,r ′ ∈R(p) χ I (E r (k)) p r (k)γ(k)(k + iβ∂ τ )χ I (E r ′ (k))p r ′ (k) dk = ≥ (d 1 -c p b p δ) ⊕ J 1,p s∈S(p) χ 2 I (λ s (k)) π s (k) dk = = (d 1 -c p b p δ) ⊕ J 1,p r∈R(p) χ 2 I (E r (k)) p r (k) dk,
where c p > 0 depends only on p. Therefore we obtain

N n,m=1 ⊕ M 1 (E) χ I (E n (k)) p n (k) γ(k)(k + iβ∂ τ ) χ I (E m (k))p m (k) dk ≥ (7.11) ≥ (d 1 -C E δ) N n=1 ⊕ M 1 (E) χ 2 I (E n (k))p n (k) dk,
with C E = max 1≤p≤P (E) c p b p . Taking into account (7.8), we thus conclude that for δ small enough there exists some c > 0 such that

N n,m=1 ⊕ M(E) χ I (E n (k))p n (k)γ(k)(k + iβ∂ τ )χ I (E m (k))p m (k) dk ≥ c N n=1 ⊕ M(E) χ 2 I (E n (k))p n (k) dk = c ∞ n=1 ⊕ R χ 2 I (E n (k))p n (k) dk = c χ 2 I ( Ĥβ ).
In view of (7.6) this proves the theorem.

Corollary 7.3. Let E ∈ R \ E and I = (Eδ, E + δ) be given as in Theorem 7.2. Then

χ I (H β ) [H β , iA] χ I (H β ) ≥ c χ 2 I (H β ), (7.12)
where [H β , iA] is understood as a bounded operator from D(H β ) into D(H β ) * , and the conjugate operator is defined by (6.1) and (6.8).

Proof. This follows from (2.9), (6.8) and Theorem 7.2.

8. Perturbation of the constant twisting

8.1. Mourre estimate for [H θ ′ , iA].
In the sequel we will suppose that

θ ′ (x 3 ) = β -ε(x 3 ).
In this section we will prove a Mourre estimate for the commutator [H θ ′ , iA], see below Theorem 8.2. Notice that H θ ′ acts as

H θ ′ = H β + W, W = (2εβ -ε 2 )∂ 2 τ + 2 ε ∂ τ ∂ 3 + +ε ′ ∂ τ (8.1)
on H 1 0 (Ω) ∩ H 2 (Ω), cf. Corollary 2.2. Together with (8.1) we will also use the decomposition

H θ ′ = H 0 + U, U = W -β 2 ∂ 2 τ -2 β ∂ τ ∂ 3 . (8.2)
Lemma 8.1. Let χ I ∈ C ∞ 0 (R) be given by (5.9). Then the operator χ

I (H θ ′ ) -χ I (H β ) is compact in L 2 (Ω).
Proof. The Helffer-Sjöstrand formula, [START_REF] Davies | Spectral Theory and Differential Equations[END_REF][START_REF] Dimassi | Spectral Asymptotics in the Semi-Classical Limit[END_REF], gives

χ I (H θ ′ ) -χ I (H β ) = - 1 π R 2 ∂ χ ∂ z (H θ ′ -z) -1 W (H β -z) -1 dx dy, (8.3)
where z = x + iy, and χ is a compactly supported quasi-analytic extension of χ I I in R 2 which satisfies

sup x∈R ∂ χ ∂z (x + iy) ≤ const y 4 , |y| ≤ 1. (8.4)
Since (H θ ′z) -1 W (H 0z) -1 is compact whenever y = 0, see [START_REF] Briet | Eigenvalue asymptotics in a twisted waveguide[END_REF], it follows that

∂ χ ∂ z (H θ ′ - z) -1 W (H β -z) -1
is compact for all (x, y) ∈ R 2 with y = 0. Moreover, by the resolvent equation the norm of (H θ ′z) -1 W (H 0z) -1 is bounded by a constant times y -2 . In view of (8.4) the integrand on the right hand side of (8.3) where P I(E,δ) is the spectral projection for the interval

I(E, δ) := (E -δ/2, E + δ/2), associated to H θ ′ .
Proof. Let I = (Eδ, E + δ). We proceed in several steps. First we show that there exists c > 0 and a compact operator K 1 in L 2 (Ω) such that

χ I (H θ ′ )[H β , iA]χ I (H θ ′ ) ≥ c χ 2 I (H β ) + K 1 . (8.6)
We write

χ I (H θ ′ )[H β , iA]χ I (H θ ′ ) = χ I (H β )[H β , iA]χ I (H β ) + χ I (H β )[H β , iA](χ I (H θ ′ ) -χ I (H β )) + (χ I (H θ ′ ) -χ I (H β ))[H β , iA]χ I (H θ ′ ).
By Corollary 7.3 there exist c > 0 such that

χ I (H β )[H β , iA]χ I (H β ) ≥ cχ 2 I (H β ).
It can be verified by a simple calculation that the operator Γ defined in (6.9) commutes with H β . Hence

χ I (H β )[H β , iA](χ I (H θ ′ ) -χ I (H β )) = 2χ I (H β )(i∂ 3 + βi∂ τ )Γ(χ I (H θ ′ ) -χ I (H β )).
We know that Γ(χ I (H θ ′ )χ I (H β )) is compact (see e.g. Lemma 8.1). The operators (H θ ′ + 1) -1 (i∂ 3 + βi∂ τ ) and χ I (H θ ′ )(H θ ′ + 1) are bounded so χ I (H β )(i∂ 3 + βi∂ τ ) is bounded too and

K 11 := χ I (H β )(i∂ 3 + βi∂ τ )Γ(χ I (H θ ′ ) -χ I (H β )) is compact. The same arguments show that K 12 := (χ I (H θ ′ ) -χ I (H β ))[H β , iA]χ I (H θ ′ ) := 2(χ I (H θ ′ ) -χ I (H β ))Γ(i∂ 3 -βi∂ τ )χ I (H θ ′ ) is compact. Putting K 1 = K 11 + K 12 concludes
(s) =: (T φ)(s) = η(s) R γ(s -s ′ )s ′ φ(s ′ ) ds ′ + η(s) R sγ(s -s ′ )φ(s ′ ) ds ′ - R γ(s -s ′ )η(s ′ )s ′ φ(s ′ ) ds ′ - R sγ(s -s ′ )η(s ′ )φ(s ′ ) ds ′ .
Hence T is an integral operator on L 2 (R) with the kernel

T (s, s ′ ) = η(s)γ(s -s ′ )s ′ + η(s)sγ(s -s ′ ) -γ(s -s ′ )η(s ′ )s ′ -sγ(s -s ′ )η(s ′ ).
To control the s-dependence we rewrite the kernel as

T (s, s ′ ) = η(s)γ(s -s ′ )(s ′ -s) + 2η(s) s γ(s -s ′ ) (8.8) -2γ(s -s ′ )η(s ′ ) s ′ -(s -s ′ )γ(s -s ′ )η(s ′ ). Next we recall that if f ∈ L q (R), g ∈ L p (R), q ∈ [2, ∞), 1/q + 1/p = 1, then the Hausdorff- Young inequality ĝ L q (R) ≤ (2π) 1 2 -1 p g L p (R)
and the interpolation result which we already used in the proof of Lemma 4.1 (see [START_REF] Simon | Trace Ideals and Their Applications[END_REF]Theorem 4.1] or [START_REF] Bergh | Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften[END_REF]Section 4.4]), imply that the integral operator with a kernel of the form f (s) g(ss ′ ), s, s ′ ∈ R, belongs to the class S q , and hence is compact on L 2 (R). By (2.13), both functions η(s) and sη(s) are in L q (R) for q large enough. Since γ ∈ C ∞ 0 (R), its Fourier transform γ is in the Schwartz class on R and therefore in any L p (R) with p ≥ 1. Therefore, the operator [η, Γs + sΓ] is compact on L 2 (R). In order to ensure the compactness of χ I (H θ ′ ) T ∂ 2 τ χ I (H θ ′ ) on L 2 (Ω), we note that by Corollary 2.2 and the closed graph theorem the operators H -1 β H θ ′ and H -1 θ ′ H β are bounded on L 2 (Ω). Since H θ ′ χ I (H θ ′ ) is bounded too, it suffices to prove that

H -1 β T ∂ 2 τ H -1 β (8.9)
is compact on L 2 (Ω). To this end we point out that H β ≥ 1 ω ⊗ (-∆ ω ) and that the operators ∂ 2 τ (-∆ ω ) -1 and (-∆ ω ) -1 are respectively bounded and compact on L 2 (ω). Hence (

1 ω ⊗ (-∆ ω )) -1 T ∂ 2 τ (1 ω ⊗ (-∆ ω )) -1
is a product of a bounded and a compact operator and hence is compact on L 2 (Ω). This yields the compactness of the operator (8.9).

In the same way we deal with the remaining terms on the right hand side of (8.7). As for the the operator

[∂ τ ε∂ s , iA] = - i 2 ∂ τ [ε∂ s , Γs + sΓ],
with the help of the integration by parts we find that

([ε∂ s , Γs + sΓ]φ)(s) = (2π) -1/2 (R 1 φ)(s) + (2π) -1/2 (R 2 φ)(s) = (2π) -1/2 R R 1 (s, s ′ ) φ(s ′ ) ds ′ + (2π) -1/2 R R 2 (s, s ′ ) φ(s ′ ) ds ′ ,
where the integral kernels R 1 (s, s ′ ) and R 2 (s, s ′ ) of the operators R 1 and R 2 are given by

R 1 (s, s ′ ) = ε(s) γ′ (s -s ′ )(s ′ -s) + γ(s -s ′ ) + 2 s γ′ (s -s ′ ) (8.10) R 2 (s, s ′ ) = ε(s ′ ) -γ ′ (s -s ′ )s ′ + γ(s -s ′ ) -s γ(s -s ′ ) + ε ′ (s ′ ) γ(s -s ′ )s ′ + sγ(s -s ′ ) ,
and γ′ denotes the derivative of γ. As above we need to write also the kernel T 2 (s, s ′ ) as a sum of the terms of the form f (s) g(ss ′ ) and f (s ′ ) g(ss ′ ):

R 2 (s, s ′ ) = ε(s ′ ) -2γ ′ (s -s ′ )s ′ + γ(s -s ′ ) -(s -s ′ ) γ(s -s ′ ) + ε ′ (s ′ ) 2γ(s -s ′ )s ′ + (s -s ′ )γ(s -s ′ ) . (8.11)
Using the assumptions of Theorem 2.7 and the fact that γ′ is the Schwartz class on R, we conclude as before that R 1 and R 2 are compact on L 2 (R) and therefore

χ I (H θ ′ ) ∂ τ [ε ∂ s , Γs + sΓ] χ I (H θ ′ ) is compact on L 2 (Ω). The compactness of χ I (H θ ′ ) [∂ s ε ∂ τ , iA] χ I (H θ ′ )
follows in a completely analogous way. Hence we obtain

χ I (H θ ′ )[H θ ′ , iA] χ I (H θ ′ ) ≥ c χ 2 I (H θ ′ ) + K (8. 12
)
where I = (Eδ, E + δ) and K is compact. Now we fix η = δ/2 in (5.9). The statement then follows by multiplying the last inequality from the left and from the right by P I(E,δ) .

:= H θ ′ u 2 L 2 (Ω) + u 2 L 2 (Ω) 1/2 , u ∈ H 2 (Ω) ∩ H 1 0 (
Ω), and recall that if ε satisfies (2.13), then in view of Corollary 2.2

u +2,θ ≍ u +2,0 ≍ u H 2 (Ω) , u ∈ H 2 (Ω) ∩ H 1 0 (Ω). ( 8
(A) ∩ D(H 0 ) is bounded on L 2 (Ω). (c) The operator B = [H θ ′ , iA] defined as a quadratic form on D(A) ∩ D(H θ ′ ) is bounded from D(H θ ′ ) into D(H 1/2 θ ′ ) * . (d) There is a common core C for A and H 0 so that A maps C into H 1 0 (Ω). Proof. Note that H 0 = -∆ and that D(H θ ′ ) = D(H 0 ) = H 1 0 (Ω) ∩ H 2 (Ω), in view of Corollary 2.2. To prove assertion (a), pick f ∈ D(H θ ′ ) and denote g = F * f . By Lemma 6.2, (e itA f )(x) = F (∂ k ϕ(t, k)) 1/2 g(x ω , ϕ(t, k)) . (8.14) Hence, ∆ (e itA f ) 2 L 2 (Ω) = e itA (∆ ω f ) + ∂ 2 3 (e itA f ) 2 L 2 (Ω) (8.15) ≤ ∆ ω f 2 L 2 (Ω) + k 2 (∂ k ϕ(t, k)) 1/2 g(x ω , ϕ(t, k)) 2 L 2 (Ω) ,
where we have used the fact that e itA : L 2 (Ω) → L 2 (Ω) and F : L 2 (Ω) → L 2 (Ω) are unitary. Assume that supp γ ⊂ [-k c , k c ] for some k c > 0. Then ϕ(t, k) = k and ∂ k ϕ(t, k) = 1 for all k with |k| > k c and all t ≥ 0, see the proof of Lemma 6.2. We thus obtain

k 2 ∂ k ϕ(t, k) g(x ω , ϕ(t, k)) 2 L 2 (Ω) ≤ k 4 c ω×[-kc,kc] ∂ k ϕ(t, k) |g(x ω , ϕ(t, k))| 2 dk dx ω + Ω k 4 |g(x ω , k)| 2 dk dx ω ≤ k 4 c Ω |g(x ω , z)| 2 dz dx ω + Ω k 4 |g(x ω , k)| 2 dk dx ω = k 4 c f 2 L 2 (Ω) + ∂ 2 3 f 2 L 2 (Ω)
, where in the first integral on right hand side we have used the change of variables z = ϕ(t, k) taking into account that ∂ k ϕ(t, k) > 0, see (6.6). In view of (8.13) and (8.15) we have

e itA f 2 +2,0 = ∆ (e itA f ) 2 L 2 (Ω) + f 2 L 2 (Ω) ≤ const f 2 H 2 (Ω)
. This implies that sup |t|≤1 e itA f +2,θ < ∞, see (8.13). Moreover, since e itA f = 0 on ∂Ω, see (8.14), we find that e itA f ∈ D(H θ ′ ). This proves (a). Next we note that by Lemma 7.1

[H 0 , iA] = 1 ω ⊗ F 1 (2 γ(k) k) F * 1
which is a bounded operator on L 2 (Ω). This proves (b).

As for assertion (c), note that B = [H β , iA] + [W, iA]. By inequality (7.2) we know that (H β + 1) -1 [H β , iA] is bounded on L 2 (Ω). On the other hand, from the proof of Theorem 8.2 it follows that the same is true for the operator (H As for the term B 1 , a direct calculation gives Since γ and γ ′ are bounded, the last inequality implies that also (H β + 1) -1 B 1 (H β + 1) -1 is bounded. From Proposition 2.1 it then follows that (H θ ′ + 1) -1 B 1 (H θ ′ + 1) -1 is bounded too. As for the remaining part of the double commutator, we first note that in view of (8.7) and of the fact that the operators ∂ τ (H θ ′ + 1) -1 and ∂ 2 τ (H θ ′ + 1) -1 are bounded, it suffices to show that

β + 1) -1 [W, iA]. Since (H β + 1)(H θ ′ + 1) -1 is bounded, by Corollary 2.2, we conclude that (H θ ′ + 1) -1 B is bounded on L 2 (Ω)
B 1 = F [[ Ĥβ , i(1 ω ⊗ Â)], i(1 ω ⊗ Â)] F * = F γ(k) 2 + γ(k)γ ′ (k)(k -iβ ∂ τ ) F * . ( 8
[η , F 1 Â F * 1 ] + [ε ′ , F 1 Â F * 1 ] + [ε ∂ s , F 1 Â F -1 1 ], F 1 ÂF * 1 (8.17)
is a bounded operator on L 2 (R). Let u ∈ L 2 (R) and recall that

(F 1 Â F * 1 u)(s) = - 1 2 √ 2π R γ(s -s ′ )s ′ u(s ′ ) ds ′ + s R γ(s -s ′ ) u(s ′ ) ds ′ .
It will be useful to introduce the shorthands γj (r) = r j γ(r).

Note that γj ∈ S(R) for all j ∈ N. We have Accordingly,

- √ 2π [T 1 , F 1 Â F * 1 ]u = η(s) R R γ(s -s ′ ) s ′2 γ(s ′ -s ′′ ) u(s ′′ ) ds ′′ ds ′ + η(s) R R γ(s -s ′ )s ′ s ′′ γ(s ′ -s ′′ ) u(s ′′ ) ds ′′ ds ′ - R R γ(s -s ′ ) η(s ′ ) γ(s ′ -s ′′ ) s ′ s ′′ u(s ′′ ) ds ′′ ds ′ -s R R
γ(ss ′ ) η(s ′ ) γ(s ′s ′′ ) u(s ′′ ) ds ′′ ds ′ =:

4 j=1
T 1,j u.

Note that

s ′2 γ(ss ′ )γ(s ′s ′′ ) = s 2 γ(ss ′ )γ(s ′s ′′ ) + γ2 (ss ′ )γ(s ′s ′′ ) -2s γ1 (ss ′ )γ(s ′s ′′ ), which implies T 1,1 u(s) = s 2 η(s) γ * (γ * u) + η(s) γ2 * (γ * u) -2s η(s) γ * (γ 1 * u).

Hence, by a repeated use of the Young inequality g * h p ≤ C g q h r ,

1 q + 1 r = 1 + 1 p , (8.19)
with p = q = 2 and r = 1, we get

T 1,1 u 2 ≤ C 1,1 s 2 η ∞ γ 2 1 + η ∞ γ 1 γ2 1 + s η ∞ γ 1 γ1 1 u 2 ,
for some C 1,1 < ∞. Moreover, since s ′ s ′′ γ(ss ′ )γ(s ′s ′′ ) = s ′2 γ(ss ′ )γ(s ′s ′′ )s γ(ss ′ )γ 1 (s ′s ′′ ) + γ1 (ss ′ )γ 1 (s ′s ′′ ), with the help of (8.19) we obtain

T 1,2 u 2 ≤ T 1,1 u 2 + C 1,2 η ∞ γ1 2 1 + s η ∞ γ 1 γ1 1 u 2 .
As for T 1,3 , we note that T 1,3 u = γ * (sη(γ 1 * u))γ * (s 2 η(γ * u)), which, in combination with (8.19), implies

T 1,3 u 2 ≤ C 1,3 sη ∞ γ1 1 γ 1 + s 2 η ∞ γ 1 γ 1 u 2 .
Next, for T 1,4 u we find Hence by (8.19)

[T 2 , F 1 Â F * 1 ] u 2 ≤ C 2 η ∞ ( γ1 2 1 + γ2 1 γ 1 ) + sη ∞ γ1 1 γ 1 u 2 .
Next we consider the last term on the right hand side of (8.18). A direct calculation gives By the Young inequality,

[T 4 , F 1 Â F * 1 ]u 2 ≤ C 4 γ1 1 γ 1 sη ∞ + γ1 2 1 η ∞ + γ 2 1 s 2 η ∞ u 2 ,
with some C 4 < ∞. The same argument applies to [T 3 , F 1 Â F * 1 ]. We thus conclude that the first term in (8.17) defines a bounded operator in L 2 (R). The same arguments apply to the second term in (8.17) replacing η by ε ′ . As for the last term in (8.17), integration by parts shows that

-2 √ 2π [ε∂ s , F 1 Â F * 1 ]u = ε(s) R γ′ (s -s ′ )s ′ u(s ′ )ds ′ + ε(s) R γ(s -s ′ )u(s ′ )ds ′ + ε(s)s R γ′ (s -s ′ )u(s ′ )ds ′ + R γ(s -s ′ )(ε(s ′ ) + s ′ ε ′ (s ′ ) -γ′ (s -s ′ )s ′ ε(s ′ ) u(s ′ )ds ′ + s R γ(s -s ′ )ε ′ (s ′ ) -γ′ (s -s ′ )ε(s ′ ) u(s ′ )ds ′ .
Note that each term on the right hand side of the above equation is of the same type as one of the terms that we have already treated above, with γ replaced by γ′ when necessary. Since r j γ′ ∈ S(R) for all j ∈ N, by following the same line of arguments as above we obtain

[ε ∂ s , F 1 Â F -1 1 ], F 1 Â F * 1 u 2 ≤ C ε (1 + s 2 ) ∞ + ε ′ (1 + s 2 ) ∞ u 2 .
for some constant C < ∞. This together with the previous estimates implies that (8.17) defines a bounded operator in L 2 (R).

Lemma 5 . 5 .

 55 5), E n (k) > R for all k ∈ I R and for all n ∈ N. If we now set N R = #L R + 1, then N R satisfies the claim. Proof of Lemma 2.6. Let -∞ < a < b < ∞ be given. By Lemma 5.4 we know that E c ∩ (a, b) is a finite set. Since the functions E n (k) are analytic away from the crossing points of the functions (5.1), it follows that E ⊂ E c . Hence E ∩ (a, b) is finite too. Let I ⊂ R be an open interval. Assume that E n (k) is analytic on I and let p n (k) be the associated eigenprojection. Then

For the next lemma

  we need the following definition. Let I ⊂ R be an open interval. Fix 0 < η < |I|/2 and define the interval I(η) := {r ∈ I : dist(r, R \ I) ≥ η}. (5.8) Let χ I be a C ∞ smooth function such that χ I (r) = 1 if r ∈ I(η) and χ I (r) = 0 if r / ∈ I.

3 )

 3 for some k 0 ∈ (a, b). The positivity of γ in (a, b) implies that γ ′ (a) ≥ 0 and γ ′ (b) ≤ 0. Hence by the Taylor expansion there exists an ε > 0 and positive constants d a , d b such that γ(r) ≤ d a (ra) for r ∈ (a, a + ε), γ(r) ≤ d b (br) for r ∈ (bε, b).

Lemma 7 . 1 .

 71 The commutator [ Ĥβ , i(1 ω ⊗ Â)] defined as a quadratic form on C ∞ 0 (Ω) extends to a bounded operator from D( Ĥβ ) into D( Ĥβ ) * . Moreover, [ Ĥβ , i(1 ω ⊗ Â)] = 2γ(k)(k + βi∂ τ ). (7.1)

Theorem 7 . 2 .

 72 and hence it is also bounded fromD( Ĥβ ) into D( Ĥβ ) * . Let E ∈ R \ E. Then there exist δ > 0, a function γ ∈ C ∞ 0 (R) and a positive constant c = c(E, δ) such that in the form sense on L 2 (Ω) we have χ I ( Ĥβ ) [ Ĥβ , i(1 ω ⊗ Â)] χ I ( Ĥβ ) ≥ c χ 2 I ( Ĥβ ),(7.3)where I = (E -δ, E +δ), χ I is given by (5.9) and the commutator [ Ĥβ , i(1 ω ⊗ Â)] is understood as a bounded operator from D( Ĥβ ) into D( Ĥβ ) * .

. 13 ) 8 . 3 .

 1383 Proposition Let ε satisfy the assumptions of Theorem 2.7. Then (a) The unitary group e itA leaves D(H θ ′ ) invariant. Moreover, for each u ∈ D(H θ ′ ), sup |t|≤1 e itA u +2,θ < ∞. (b) The operator B 0 = [H 0 , iA] defined as a quadratic form on D

  and (c) follows. To prove (d) we define C := D(-∆ ω ) ⊗ S(R). By definition of A, C is a core for A. On the other hand, C is also a core for H 0 . Since F : C → C is a bijection and since  : S → S, it follows that A : C → C ⊂ H 1 0 (Ω). Lemma 8.4. Let ε satisfy assumptions of Theorem 2.7. Then (H θ ′ + 1) -1 [B, A] (H θ ′ + 1) -1 is bounded as an operator on L 2 (Ω). Proof. Recall that B = [H θ ′ , iA]. We write B = B 1 + B 2 , where B 1 = [[H β , iA], iA], B 2 = [[W, iA], iA].

. 16 )

 16 Let u ∈ L 2 (Ω). Similarly as in (7.2) we find out that|(u, (kiβ ∂ τ ) u) L 2 (Ω) | ≤ (u, ( Ĥβ + 1) u) L 2 (Ω) .

- 2 √ 4 j=1T

 24 2π [η , F 1 ÂF * 1 ]u = 2η(s) R γ(ss ′ )s ′ u(s ′ ) ds ′ + η(s) R γ1 (ss ′ )u(s ′ ) ds ′ (8.18) -R γ(ss ′ )s ′ η(s ′ )u(s ′ ) ds ′s R γ(ss ′ )η(s ′ )u(s ′ ) ds ′ =: j u.

T 1 , 2 1 2 1 u 2 . 1 ] u 2 = 1 2 4 j=1T 1 ,j u 2 ≤ C 1 u 2 ,

 122214122 4 u = -γ 1 * (η(γ 1 * u)) + γ1 * (sη(γ * u))γ * (sη(γ 1 * u)) + γ * (s 2 η(γ * u)).By using again (8.19) we getT 1,4 u 2 ≤ C 1,4 η ∞ γ1 + 2 sη ∞ γ1 1 γ 1 + s 2 η ∞ γ This implies that [T 1 , F 1 Â F * for some C 1 < ∞. As for the term [T 2 , F 1 Â F * 1 ] u, we find out that -2 √ 2π [T 2 , F 1 Â F * 1 ]u = -η γ1 * (γ 1 * u) -2 η γ2 * (γ * u) + 2s η γ1 * (γ * u) -2γ * (sη(γ 1 * u)) -γ1 * (η(γ 1 * u)).

4 ,

 4 F 1 Â F * 1 ]u = 2 γ1 * (sη (γ * u)) -2 γ1 * (η (γ 1 * u)) + 2 γ * (s 2 η (γ * u))γ * (sη (γ 1 * u)).

  The point spectrum of H θ ′ has no accumulation points in R \ E; (c) The singular continuous spectrum of H θ ′ is empty.

	(b) Theorem 2.7 is proven in Subsection 8.2.
	Remark 2.8. If ε ∈ C 1 (R, R) is such that ε ′ is bounded and ε φ -α ∞ < ∞ for some α > 1, then Corollary 2.4 and equation (2.11) imply
	.13)
	the function φ α being defined in (2.1). Then:
	(a) Any compact subinterval of R \ E contains at most finitely many eigenvalues of H θ ′ , each having finite multiplicity;

  is then uniformly norm-bounded in R 2 and hence χ I (H θ ′ )χ I (H β ) is compact. Theorem 8.2. Let E ∈ R \ E and let ε satisfy (2.13). Choose δ > 0 and γ ∈ C ∞ 0 (R) as in Theorem 7.2. Then there exists a positive constant c and a compact operator K in L 2 (Ω) such that P I(E,δ) [H θ ′ , iA] P I(E,δ) ≥ c P 2 I(E,δ) + P I(E,δ) K P I(E,δ) , (8.5)

  the first step of the proof.Next we consider χI (H θ ′ )[W, iA]χ I (H θ ′ ).For the sake of simplicity we now write s instead of x 3 . Defining η(s):= 2ε(s)βε(s) 2 we get [W, iA] = [η , iA] ∂ 2 τ + [ε ∂ s , iA] ∂ τ + [∂ s ε, iA] ∂ τ .

		(8.7)
	We first deal with the term [η ∂ 2 τ , iA] = i[η, A]∂ 2 τ = -i 2 [η, Γs + sΓ] ∂ 2 τ . For an appropriate test function φ we obtain
	√	2π ([η, Γs + sΓ]φ)

Acknowledgements. The authors were partially supported by the Bernoulli Center, EPFL, Lausanne, within the framework of the Program "Spectral and Dynamical Properties of Quantum Hamiltonians", January -June 2010. H. Kovařík gratefully acknowledges also partial support of the MIUR-PRIN08 grant for the project "Trasporto ottimo di massa, disuguaglianze geometriche e funzionali e applicazioni", and would like to thank as well the Faculty of Mathematics of Pontificia Universidad Católica de Chile for the warm hospitality extended to him. G. Raikov was partially supported by by the Chilean Science Foundation Fondecyt under Grant 1130591, and by Núcleo Científico ICM P07-027-F.

With these prerequisites, we can finally state the result about the nature of the essential spectrum of H θ ′ : Corollary 8.5. Let ε satisfy the assumptions of Theorem 2.7. Let E ∈ R \ E be given and define the interval Proof of Theorem 2.7. Let J ⊂ R \ E be a compact interval. For each E ∈ J choose I(E, δ) as in Theorem 8.2. Then J ⊂ ∪ E∈J I(E, δ) and since J is compact, there exists a finite subcovering:

(8.20) By Corollary 8.5(a), each interval I(E n , δ n ) contains at most finitely many eigenvalues of H θ ′ , each of them having finite multiplicity. This proves assertion (a). Part (b) follows immediately from (a). To prove (c) assume that σ sc (H θ ′ ) ∩ (R \ E) = ∅. Since the set E is locally finite, see Lemma 2.6, it follows that there exists a compact interval J ⊂ R\E such that σ sc (H θ ′ )∩J = ∅. This is in contradiction with (8.20) and Corollary 8.5(c). Hence, σ sc (H θ ′ ) ∩ (R \ E) = ∅. Since E is discrete, this implies that σ sc (H θ ′ ) = ∅.