Scattering in twisted waveguides
aa r X i v : . [ m a t h . SP ] O c t SCATTERING IN TWISTED WAVEGUIDES
PHILIPPE BRIET, HYNEK KOVA ˇR´IK, AND GEORGI RAIKOV
Abstract.
We consider a twisted quantum waveguide, i.e., a domain of the form Ω θ := r θ ω × R where ω ⊂ R is a bounded domain, and r θ = r θ ( x ) is a rotation by the angle θ ( x )depending on the longitudinal variable x . We investigate the nature of the essential spectrumof the Dirichlet Laplacian H θ , self-adjoint in L (Ω θ ), and consider related scattering problems.First, we show that if the derivative of the difference θ − θ decays fast enough as | x | → ∞ ,then the wave operators for the operator pair ( H θ , H θ ) exist and are complete. Further, weconcentrate on appropriate perturbations of constant twisting, i.e. θ ′ = β − ε with constant β ∈ R , and ε which decays fast enough at infinity together with its first derivative. In thatcase the unperturbed operator corresponding to ε is an analytically fibered Hamiltonian withpurely absolutely continuous spectrum. Obtaining Mourre estimates with a suitable conjugateoperator, we prove, in particular, that the singular continuous spectrum of H θ is empty. AMS 2000 Mathematics Subject Classification:
Keywords:
Twisted waveguides, wave operators, Mourre estimates1.
Introduction
Let ω ⊂ R be a bounded domain with boundary ∂ω ∈ C . Denote by Ω := ω × R thestraight tube in R . For a given θ ∈ C ( R , R ) we define the twisted tube Ω θ byΩ θ = (cid:8) r θ ( x ) x ∈ R | x = ( x , x , x ) ∈ R , x ω := ( x , x ) ∈ ω (cid:9) , where r θ ( x ) = cos θ ( x ) sin θ ( x ) 0 − sin θ ( x ) cos θ ( x ) 00 0 1 . We define the Dirichlet Laplacian H θ as the unique self-adjoint operator generated in L (Ω θ )by the closed quadratic form Q θ [ u ] := Z Ω θ |∇ u | d x , u ∈ D( Q θ ) := H (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 bya 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 theoperator H θ under appropriate assumptions about the twisting angle θ . Although the spectralproperties of a twisted waveguide have been intensively studied in recent years, attention hasbeen paid mostly to the discrete spectrum of H θ , [5, 10, 14], or to the Hardy inequality for H θ , [9]. 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 ′ − θ ′ decays fast enough as | x | → ∞ , then thewave operators for the operator pair ( H θ ′ , H θ ′ ) exist and are complete, and in particular, theabsolutely continuous spectra of H θ ′ and H θ ′ coincide. Further, we observe that if θ ′ = β is constant, then the operator H β is analytically fibered, cf. (2.9), and therefore its singularcontinuous spectrum is empty, [11, 13]. Assuming that θ ′ ( x ) = β − ε ( x ) with ε ∈ C ( R , R ), wethen 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,[19, 20, 1]. 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 ofthe band functions E n ( k ) of the unperturbed operator H β , k ∈ R being the Fourier variabledual to x . A similar strategy was used in [12, 2, 6, 17], where the generator of dilations inthe longitudinal direction of the waveguide was used as a conjugate operator. However, inthe situations studied in these works the associated band functions have a non zero derivativeeverywhere except for the origin. In our model, contrary to [12, 2, 6, 17], the band functions E n may have many stationary points. In addition, we have to take into account possiblecrossing points between different band functions. The generator of dilations therefore cannotbe used as a conjugate operator in our case, and a different approach is needed. Our conjugateoperator acts in the fibered space as i γ ( k ) ∂ k + ∂ k γ ( k )) (1.2)where γ ∈ C ∞ ( R ; R ) is a suitably chosen function, whose particular form depends on theinterval 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 fiberedoperators and their appropriate perturbations was developed in [13]. The situation with thetwisted waveguide analyzed in the present article is much more specific than the generalabstract scheme studied in [13]. Hence, although the construction in (1.2) is influenced in someextent by [13], our conjugate operator is essentially different from the one used in [13], and isconsiderably more useful for our purposes. In particular, the construction of this quite explicitconjugate operator allows us to handle the specific second-order differential perturbation whicharises 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 absenceof the singular continuous spectrum, see Proposition 8.3. We have thus been able to applythe Mourre theory to the perturbed operator H θ ′ , and to find simple and efficient sufficientconditions on ε under which the singular continuous spectrum of H θ ′ is empty. We thereforebelieve 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 3we prove Proposition 2.1 describing the domain of the operator H θ . In Section 4 we proveTheorem 2.3 which entails the existence and the completeness of the wave operators for theoperator pair ( H θ ′ , H θ ′ ) for appropriate θ ′ − θ ′ , and hence the coincidence of σ ac ( H θ ′ ) and σ ac ( H θ ′ ). In Section 5 we assume that the twisting is constant, i.e. θ ′ = β and examine thespectral and analytical properties of the fiber family h β ( k ), k ∈ R . In Section 6 we constructthe conjugate operator needed for the subsequent Mourre estimates. In Section 7 we obtainMourre estimates for the case of a constant twisting. Finally, in Section 8 we extend these CATTERING IN TWISTED WAVEGUIDES 3 estimates to the case of θ ′ = β − ε where β ∈ R , and ε decays fast enough together with itsfirst derivative. 2. Main results
Notation.
Let us fix some notation. Given a measure space ( M, A , µ ), we denote by M the identity operator in L ( M ) = L ( M ; dµ ). Further, we will denote by ( u, v ) L ( M ) = R M ¯ u vdµ the scalar product in L ( M ) and by k u k L p ( M ) , p ∈ [1 , ∞ ], the L p -norm of u . If thereis no risk of confusion we will drop the indication to the set M and write ( u, v ) and k u k p instead in order to simplify the notation. Given a set M and two functions f , f : M → R ,we write f ( m ) ≍ f ( m ) , m ∈ M , if there exists a constant c ∈ (0 , ∞ ) such that for each m ∈ M we have c − f ( m ) ≤ f ( m ) ≤ c f ( 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 theSchatten-von Neumann classes of compact operators acting in X ; we recall that the norm in S p ( X ) is defined as k T k S p := (cid:0) Tr ( T ∗ T ) p/ (cid:1) /p , T ∈ S p ( X ). In particular, S is the trace class,and S is the Hilbert-Schmidt class. Moreover, if T is a self-adjoint operator acting in X , wedenote by D( T ) the operator domain of T . Finally, for α ∈ R define the function φ α ( s ) := (1 + s ) − α/ , s ∈ R . (2.1)2.2. Domain issues.
Our first result shows that if both θ ′ and θ ′′ are continuous and bounded,then the domain of the operator H θ coincides with H (Ω θ ) ∩ H (Ω θ ). Proposition 2.1.
Assume that ω ⊂ R is a bounded domain with boundary ∂ω ∈ C , and θ ∈ C ( R ) with θ ′ , θ ′′ ∈ L ∞ ( R ) . Then D ( H θ ) = H (Ω θ ) ∩ H (Ω θ ) . (2.2)Proposition 2.1 could be considered as a fairly standard result but since we have not been ableto find in the literature a version suitable for our purposes (most of the references availabletreat bounded domains or the complements of compact sets), we include a detailed sketch ofthe proof in Section 3.Next, we introduce the operator U θ : L (Ω θ ) → L (Ω) generated by the change of variablesΩ ∋ x r θ ( x ) x ∈ Ω θ . (2.3)Namely, for w ∈ L (Ω θ ) set ( U θ w )( x ) = w ( r θ ( x ) x ) , x ∈ Ω . Evidently, U θ : L (Ω θ ) → L (Ω) is unitary since (2.3) defines a diffeomorphism whose Jacobianis identically equal to one. Now assume g ∈ C ( R ; R ) ∩ L ∞ ( R ) and introduce the quadraticform Q g [ u ] = Z Ω (cid:0) |∇ ω u | + | ∂ u + g ∂ τ u | (cid:1) d x, u ∈ D( Q g ) = H (Ω) , where ∇ ω := ( ∂ , ∂ ) T , and ∂ τ := x ∂ − x ∂ . Denote by H g the self-adjoint operatorgenerated in L (Ω) by the closed quadratic form Q g . The transformation U θ also maps H (Ω θ )bijectively onto H (Ω). Hence, for g = θ ′ we get Q [ w ] = Q θ ′ [ U θ w ] , w ∈ H (Ω θ ) , PHILIPPE BRIET, HYNEK KOVAˇR´IK, AND GEORGI RAIKOV which implies H θ ′ = U θ H θ U − θ . (2.4)Assume now that g ∈ C ( R ) with g, g ′ ∈ L ∞ ( R ). Set G ( x ) := R x g ( s ) ds , x ∈ R . Then U G maps bijectively H (Ω G ) onto H (Ω). Therefore, Proposition 2.1 and the unitarity of U G : L (Ω G ) → L (Ω) imply the following Corollary 2.2.
Assume that ω ⊂ R is a bounded domain with boundary ∂ω ∈ C , and g ∈ C ( R ) with g, g ′ ∈ L ∞ ( R ) . Then the domain of the operator H g coincides with H (Ω) ∩ H (Ω) . Furthermore, if g ∈ C ( R ) with g, g ′ ∈ L ∞ ( R ) we have H g u = (cid:0) − ∂ − ∂ − ( ∂ + g ∂ τ ) (cid:1) u, u ∈ H (Ω) ∩ H (Ω) , (2.5)since H G ϕ = − ∆ ϕ , ϕ ∈ H (Ω G ) ∩ H (Ω G ).2.3. Existence and completeness of the wave operators.
Next we show that underappropriate assumptions on the difference g − g , the wave operators for the operator pair( H g , H g ) exist and are complete, and hence the absolutely continuous spectra of the operators H g and H g coincide. Theorem 2.3.
Assume that ω ⊂ R is a bounded domain with C -boundary. Let g j ∈ C ( R ; R ) with g j , g ′ j ∈ L ∞ ( R ) , j = 1 , . Suppose that there exists α > such that k φ − α ( g − g ) k L ∞ ( R ) < ∞ , (2.6) the function φ α being defined in (2.1) . Then we have H − g − H − g ∈ S (L (Ω)) . (2.7)Theorem 2.3 is proven in Section 4. By a classical result from the stationary scattering theory(see the original work [4] or [22, Corollary 3, Section 3, Chapter XI], [27, Chapter 6, Section2, Theorem 6]), this theorem implies the following Corollary 2.4.
Under the assumptions of Theorem 2.3 the wave operators s − lim t →±∞ e itH g e − itH g P ac ( H g ) for the operator pair ( H g , H g ) exist and are complete. Therefore, the absolutely continuousparts of H g and H g are unitarily equivalent, and, in particular, σ ac ( H g ) = σ ac ( H g ) . (2.8)Corollary 2.4 admits an equivalent formulation in terms of the operator pair ( H θ , H θ ): Corollary 2.5.
Assume that ω ⊂ R is a bounded domain with C -boundary. Let θ j ∈ C ( R ; R ) with θ j , θ ′ j , θ ′′ j ∈ L ∞ ( R ) , j = 1 , . Suppose that there exists α > such that k φ − α ( θ ′ − θ ′ ) k L ∞ ( R ) < ∞ , Then the wave operators s − lim t →±∞ e it H θ J e − it H θ P ac ( H θ ) , J := U − θ U θ , for the operator pair ( H θ , H θ ) exist and are complete. Therefore, the absolutely continuousparts of H θ and H θ are unitarily equivalent, and, in particular, σ ac ( H θ ) = σ ac ( H θ ) . CATTERING IN TWISTED WAVEGUIDES 5
Constant twisting.
In our remaining results, we concentrate on the case of appropriateperturbations 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 , unitaryin L (Ω), by( F u )( x ω , k ) = (2 π ) − / Z R e − ikx u ( x ω , x ) dx , k ∈ R , x ω ∈ ω. Then we have ˆ H β = F H β F ∗ = Z ⊕ 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 ( ω ) ∩ H ( ω )as h β ( k ) = − ∆ ω + ( βi∂ τ − k ) , − ∆ ω being the self-adjoint operator generated in L ( ω ) by the closed quadratic form Z ω |∇ v | dx ω , v ∈ H ( ω ) . Note that for all k ∈ R the resolvent h β ( k ) − is compact, and h β ( k ) has a purely discretespectrum. Let 0 < E ( k ) ≤ E ( 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 orthogonalprojection onto Ker( h β ( k ) − E n ( k )), k ∈ R and n ∈ N . By [5, 10] we have σ ( H β ) = σ ac ( H β ) = [ E (0) , ∞ ) . (2.11)A detailed discussion of the properties of E n ( k ) is given in Section 5. It turns out that thefunctions E n ( k ) are piecewise analytic, and that for any given k ∈ R , the function E n ( k ) canbe analytically extended into an open neighborhood of k . We denote such an extension by˜ E n,k ( k ). If k is a point where E n ( k ) is analytic, then of course ˜ E n,k ( · ) = E n ( · ). With thisnotation at hand, we introduce the following subsets of R : E := (cid:8) E ∈ R : ∃ n ∈ N , ∃ k ∈ R : E n ( k ) = E ∧ ∂ k ˜ E n,k ( k ) = 0 (cid:9) , E := (cid:8) E ∈ R : ∃ k ∈ R , ∃ n, m ∈ N , n = m : E n ( k ) = E m ( k ) = E ∧∧ ∂ k ˜ E n,k ( k ) ∂ k ˜ E m,k ( k ) < (cid:9) . We then define the set E of critical levels as follows: E = E ∪ E . (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 5.4.2.5.
Absence of singular continuous spectrum of H β − ε .Theorem 2.7. Let θ ′ ( x ) = β − ε ( x ) , where ε ∈ C ( R , R ) is such that k ε φ − k ∞ + k ε ′ φ − k ∞ < ∞ , (2.13) 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; PHILIPPE BRIET, HYNEK KOVAˇR´IK, AND GEORGI RAIKOV (b)
The point spectrum of H θ ′ has no accumulation points in R \ E ; (c) The singular continuous spectrum of H θ ′ is empty. Theorem 2.7 is proven in Subsection 8.2.
Remark 2.8. If ε ∈ C ( R , R ) is such that ε ′ is bounded and k ε φ − α k ∞ < ∞ for some α > σ ac ( H θ ′ ) = σ ac ( H β ) = [ E (0) , ∞ ) . Note that in order to prove the absence of singular continuous spectrum of H θ ′ we need strongerhypothesis on ε and ε ′ , see equation (2.13).By [22, Section XI.3], Corollary 2.4 and Theorem 2.7 part (c) imply Corollary 2.9.
Under the assumptions of Theorem 2.7 the wave operators for the operatorpair ( H β , H θ ′ ) exist and are asymptotically complete. Proof of Proposition 2.1
Denote by C ∞ (Ω θ ) the class of functions u ∈ C ∞ (Ω θ ), compactly supported in Ω θ . Set˙ C ∞ (Ω θ ) := (cid:8) u ∈ C ∞ (Ω θ ) | u | ∂ Ω θ = 0 (cid:9) . Lemma 3.1.
Under the assumptions of Proposition 2.1 there exists a constant c ∈ (0 , ∞ ) such that k u k (Ω θ ) ≤ c Z Ω θ ( | ∆ u | + | u | ) dx (3.1) for any u ∈ ˙ C ∞ (Ω θ ) .Proof. Our argument will follow closely the proof of [18, Chapter 3, Lemma 8.1]. We have Z Ω θ ( | ∆ u | + c | u | ) dx = Z Ω θ X j,k =1 | ∂ j ∂ k u | + c | u | dx + 2 Z ∂ Ω θ K (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂ν (cid:12)(cid:12)(cid:12)(cid:12) dS, u ∈ ˙ C ∞ (Ω θ ) , (3.2)(see [18] or [25, Chapter 5, Section 5, Problem 6]) where c ∈ (0 , ∞ ) is an arbitrary constantwhich is to be specified later, K is the mean curvature, and ν is the exterior normal unit vectorat ∂ Ω θ . Our assumptions on ∂ω and θ imply that for any u ∈ ˙ C ∞ (Ω θ ) we have2 Z ∂ Ω θ K (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂ν (cid:12)(cid:12)(cid:12)(cid:12) dS ≥ − c Z ∂ Ω θ |∇ u | dS. (3.3)with c := 2 sup x ∈ ∂ Ω θ | K ( x ) | ≤ ( x ω ,x ) ∈ ∂ω × R (cid:8) ( | θ ′′ ( x ) | + θ ′ ( x ) ) | x ω | + (1 + θ ′ ( x ) | x ω | ) | κ ( x ω ) | (cid:9) , where κ ( x ω ) is the curvature of ∂ω at the point x ω ∈ ∂ω . Let us check that for any ε > c ( ε ) such that for any v ∈ C ∞ (Ω θ ) we have Z ∂ Ω θ | v | dS ≤ Z Ω θ (cid:0) ε |∇ ω v | + c ( ε ) | v | (cid:1) dx (3.4) CATTERING IN TWISTED WAVEGUIDES 7 where, as above, ∇ ω := ( ∂ , ∂ ) T . In order to prove this, we note the inequality Z ∂ Ω θ | v | dS ≤ c Z R Z ∂ω θ ( x | v | ds ! dx (3.5)where c := sup ( x ω ,x ) ∈ ∂ω × R (cid:0) θ ′ ( x ) | x ω | (cid:1) / , and ω θ ( a ) is the cross-section of Ω θ with the plane { x = a } , a ∈ R .Next, since ω is a bounded domain with sufficiently regular boundary, we find that for any ε > c ( ε ) such that for any x ∈ R and any w ∈ C ∞ ( ω θ ( x ) ) we have Z ∂ω θ ( x | w | ds ≤ Z ω θ ( x (cid:0) ε |∇ w | + c ( ε ) | w | (cid:1) dx ω (3.6)(see e.g. [18, Chapter 2, Eq. (2.25)]). Choosing w = v ( · , x ) in (3.6), integrating with respectto x , and bearing in mind (3.5), we get Z ∂ Ω θ | v | dS ≤ Z Ω θ (cid:0) c ε |∇ ω v | + c c ( ε ) | v | (cid:1) dx which implies (3.4) with c ( ε ) = c c ( ε/c ). Now the combination of (3.3) and (3.4) implies2 Z ∂ Ω θ K (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂ν (cid:12)(cid:12)(cid:12)(cid:12) dS ≥ − c Z Ω θ ε X j,k =1 | ∂ j ∂ k u | + c ( ε ) |∇ u | dx. (3.7)Further, we have Z Ω θ |∇ u | dx = − Re Z Ω θ ∆ uudx ≤ Z Ω θ (cid:0) | ∆ u | + | u | (cid:1) dx. (3.8)Combining (3.2), (3.7), and (3.8), we find that for any ε > Z Ω θ (cid:0) (1 + c c ( ε ) / | ∆ u | + c | u | (cid:1) dx ≥ Z Ω θ (1 − c ε ) X j,k =1 | ∂ j ∂ k u | + ( c − c c ( ε ) / | u | dx which yields (3.1) under appropriate choice of c , c and ε . (cid:3) Denote by ˜H (Ω θ ) the Hilbert space (cid:8) u ∈ H (Ω θ ) | ∆u ∈ L (Ω θ ) (cid:9) with scalar product gener-ated by the quadratic form R Ω θ ( | ∆ u | + | u | ) dx . Lemma 3.2.
Under the assumptions of Proposition 2.1 we have u ∈ ˜H (Ω θ ) if and only if u ∈ H (Ω θ ) ∩ H (Ω θ ) .Proof. By Z Ω θ ( | ∆ u | + | u | ) dx ≤ k u k (Ω θ ) , u ∈ H (Ω θ ) , (3.9)and (3.1), we have Z Ω θ ( | ∆ u | + | u | ) dx ≍ k u k (Ω θ ) , u ∈ ˙ C ∞ (Ω θ ) . (3.10) PHILIPPE BRIET, HYNEK KOVAˇR´IK, AND GEORGI RAIKOV
Evidently, the class ˙ C ∞ (Ω θ ) is dense in H (Ω θ ) ∩ H (Ω θ ). Then (3.9) easily implies that˙ C ∞ (Ω θ ) is dense in ˜H (Ω θ ) as well. Now the claim of the lemma follows from (3.10). (cid:3) Proof of Proposition 2.1.
Let L be the operator − ∆ with domain C ∞ (Ω θ ), and L ∗ be theadjoint of L in L (Ω θ ). If v ∈ D( L ∗ ), then a standard argument from the theory of distributionsover C ∞ (Ω θ ) shows that L ∗ v = − ∆ v ∈ L (Ω θ ). Since H θ is a restriction of L ∗ , we find that u ∈ D ( H θ ) implies that H θ u = − ∆ u ∈ L (Ω θ ). On the other hand, u ∈ D ( H θ ) implies u ∈ D ( Q θ ) = H (Ω θ ). By Lemma 3.2 we have u ∈ H (Ω θ ) ∩ H (Ω θ ), i.e.D ( H θ ) ⊆ H (Ω θ ) ∩ H (Ω θ ) . (3.11)If we now suppose that D ( H θ ) = H (Ω θ ) ∩ H (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 (Ω θ ) ∩ H (Ω θ ), which contradicts the self-adjointnessof H θ . Therefore, (2.2) holds true, and the proof of Proposition 2.1 is complete. (cid:3) Proof of Theorem 2.3
For the proof of Theorem 2.3 we need an auxiliary result, Lemma 4.1, preceded by somenecessary notation.Let { µ j } j ∈ N be the the non-decreasing sequence of the eigenvalues of the operator − ∆ ω . Since H g ≥ µ Ω , and µ >
0, the operator H g is invertible. Lemma 4.1.
Let g ∈ C ( R ; R ) with g, g ′ ∈ L ∞ ( R ) .(i) Assume f ∈ L ( R ) . Then we have f ( x ) H − g ∈ S (L (Ω)) . (4.1) (ii) Assume h ∈ L ( R ) . Then we have h ( x ) ∂ j H − g ∈ S (L (Ω)) , j = 1 , , . (4.2) Proof.
By Corollary 2.2 the operator H H − g is bounded, so that it suffices to prove (4.1) –(4.2) for g = 0. Evidently, k f H − k S (L (Ω)) = X j ∈ N k f ( − ∂ + µ j ) − k S (L ( R )) == (2 π ) − X j ∈ N Z R | f ( s ) | ds Z R dξ ( ξ + µ j ) = (2 π ) − X j ∈ N µ − / j Z R | f ( s ) | ds Z R dξ ( ξ + 1) . (4.3)Set N ( λ ) := { j ∈ N | µ j < λ } , λ >
0. By the celebrated Weyl law, we have N ( λ ) = | ω | π λ (1 + o (1)) as λ → ∞ where | ω | is the area of ω (see the original work [26] or [23, TheoremXIII.78]). Therefore, the series P j ∈ N µ − γj = γ R ∞ µ λ − γ − N ( λ ) dλ is convergent if and only if γ >
1. In particular, X j ∈ N µ − / 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 − = ∂ j (( − ∆ ω ) ⊗ R ) − / h (( − ∆ ω ) ⊗ R ) / H − . CATTERING IN TWISTED WAVEGUIDES 9
Since the operators ∂ j (( − ∆ ω ) ⊗ R ) − / , j = 1 ,
2, are bounded, it suffices to show that h (( − ∆ ω ) ⊗ R ) / H − ∈ S (L (Ω)) . (4.5)Applying a standard interpolation result (see e.g. [24, Theorem 4.1] or [3, Section 4.4]), andbearing in mind (4.4), we get k h (( − ∆ ω ) ⊗ R ) / H − k S (L (Ω)) = X j ∈ N µ j k h ( − ∂ + µ j ) − k S (L ( R )) ≤≤ (2 π ) − X j ∈ N µ j Z R | h ( s ) | ds Z R dξ ( ξ + µ j ) = (2 π ) − X j ∈ N µ − / j Z R | h ( s ) | ds Z R dξ ( ξ + 1) < ∞ which implies (4.5). Finally, we prove (4.2) with g = 0 and j = 3. To this end it suffices toapply again [24, Theorem 4.1] and (4.4), and get k h∂ H − k S (L (Ω)) = X j ∈ N k h∂ ( − ∂ + µ j ) − k S (L ( R )) ≤≤ (2 π ) − X j ∈ N Z R | h ( s ) | ds Z R ξ dξ ( ξ + µ j ) = (2 π ) − X j ∈ N µ − / j Z R | h ( s ) | ds Z R ξ dξ ( ξ + 1) < ∞ . (cid:3) Proof of Theorem 2.3.
For z ∈ ρ ( H g ) ∩ ρ ( H g ) we have( H g − z ) − − ( H g − z ) − = ∂∂z ( H g − z ) − W ( H g − z ) − =( H g − z ) − W ( H g − z ) − + ( H g − z ) − W ( H g − z ) − (4.6)with W := ∂ τ ( g − g ) ∂ τ + ∂ ( g − g ) ∂ τ + ∂ τ ( g − g ) ∂ . Choosing z = 0, we obtain H − g − H − g = − (cid:0) φ α/ ∂ τ H − g (cid:1) ∗ (cid:0) ( g − g ) φ − α φ α/ ∂ H − g + ( g − g ) φ − α φ α/ ∂ τ H − g (cid:1) − (cid:0) φ α/ ∂ H − g (cid:1) ∗ ( g − g ) φ − α φ α/ ∂ τ H − g − (cid:0) φ α/ ∂ τ H − g (cid:1) ∗ (cid:0) ( g − g ) φ − α φ α/ ∂ H − g + ( g − g ) φ − α φ α/ ∂ τ H − g (cid:1) − (cid:0) φ α/ ∂ H − g (cid:1) ∗ ( g − g ) φ − α φ α/ ∂ τ H − g . Since the multipliers by ( g − g ) φ − α and ( g − g ) φ − α are bounded operators by (2.6) and g j ∈ L ∞ ( R ), j = 1 ,
2, while φ α/ ∂ ℓ H − g j ∈ S (L (Ω)) , ℓ = τ, , j = 1 , , by Lemma 4.1 (ii), it suffices to show that φ α/ ∂ ℓ H − g j ∈ S / (L (Ω)) , ℓ = τ, , j = 1 , . (4.7)In what follows we write g instead of g j , j = 1 ,
2. Commuting multipliers by functions φ whichdepend only on x and belong to appropriate H¨ormander classes, with the resolvent H − g , andbearing in mind that [ φ, H − g ] = − H − g ( φ ′′ + 2 φ ′ ( ∂ + g∂ τ )) H − g , we obtain φ α/ ∂ τ H − g = φ α/ ∂ τ H − g φ α/ H − g − φ α/ ∂ τ H − g φ ′′ α/ H − g + 2 φ α/ ∂ τ H − g ( φ ′ α/ ) φ − α/ H − g − φ α/ ∂ τ H − g φ ′ α/ φ − α/ ( ∂ + g∂ τ ) H − g (cid:16) φ α/ H − g − (cid:16) φ ′′ α/ + φ ′ α/ ( ∂ + g∂ τ ) (cid:17) H − g (cid:17) , (4.8) φ α/ ∂ H − g = φ α/ ∂ H − g φ α/ H − g − φ α/ ∂ H − g φ ′′ α/ H − g + 2 φ α/ ∂ H − g ( φ ′ α/ ) φ − α/ H − g − φ α/ ∂ H − g φ ′ α/ φ − α/ ( ∂ + g∂ τ ) H − g (cid:16) φ α/ H − g − (cid:16) φ ′′ α/ + φ ′ α/ ( ∂ + g∂ τ ) (cid:17) H − g (cid:17) + φ ′ α/ φ − α/ H − g (cid:16) φ α/ H − g − (cid:16) φ ′′ α/ + 2 ( ∂ + g∂ τ ) (cid:17) H − g (cid:17) . (4.9)Bearing in mind that S p ⊂ S q if p < q , and that H − g is a bounded operator, we find thatLemma 4.1 implies that all the terms at the r.h.s. of (4.8) and (4.9) can be presented eitheras a product of an operator in S and an operator in S , or as a product of three operators in S , which yields (4.7), and the proof of Theorem 2.3 is complete. (cid:3) 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 unitarilyequivalent to R ⊕ R h β ( k ) dk with h β ( k ) = − ∆ ω + ( iβ∂ τ − k ) , k ∈ R . The goal of the section isto establish various properties of the fiber operator h β ( k ), which will be used later in Section 7for 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 ( ω ) ∩ H ( ω ) , form aself-adjoint holomorphic family of type (A) in the sense of Kato.Proof. Note that h β ( k ) = h β (0) − β i k∂ τ + k , and that h β (0) is self-adjoint on H ( ω ) ∩ H ( ω ). Let u ∈ H ( ω ). Then for any ε > k β i∂ τ u k ≤ ( u, h β (0) u ) L ( ω ) ≤ k u k k h β (0) u k ≤ ε − k u k + ε k h β (0) u k . Hence βi∂ τ is relatively bounded with respect to h β (0) with relative bound zero and theassertion follows from [16, Theorem VII.2.6]. (cid:3) From Lemma 5.1 and the Rellich Theorem, [16, Theorem VII.3.9], it follows that all theeigenvalues 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 ∈ R be given. Then (cid:12)(cid:12)p λ ℓ ( k ) − p λ ℓ ( k ) (cid:12)(cid:12) ≤ | k − k | , k ∈ R . (5.2) Proof.
By [16, Theorem VII.3.9] there exists an analytic normalized eigenvector ψ ℓ ( k ) associ-ated to λ ℓ ( k ). From the Feynman-Hellmann formula, see e.g. [16, Section VII.3.4], we obtain | ∂ k λ ℓ ( k ) | = 4 | (( k − iβ∂ τ ) ψ ℓ ( k ) , ψ ℓ ( k )) L ( ω ) | ≤ k ( k − iβ∂ τ ) ψ ℓ ( k ) k ( ω ) ≤ ψ ℓ ( k ) , h β ( k ) ψ ℓ ( k )) L ( ω ) = 4 λ ℓ ( k ) , k ∈ R . CATTERING IN TWISTED WAVEGUIDES 11
Hence | ∂ k λ ℓ ( k ) | ≤ p λ ℓ ( k ) k ∈ R . (5.3)By integrating this differential inequality we arrive at (5.2). (cid:3) 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 + k , ∀ k ∈ R .On the other hand, since every E n ( k ) coincides with one of the functions λ ℓ ( k ) locally onintervals 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 ( k ) ≥ E (0) + c k , k ∈ R , (5.4)for some c ∈ (0 , k R > E ( 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 ( k ) > R on R \ I R . Weclaim 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 thatsup j ∈L R max k ∈ I R | ∂ k λ j ( k ) | ≤ k R + 2 √ R. Let k ∞ ∈ I R be an accumulation point of the sequence { k j } . Hence, for any ε > J ε ⊂ L R such that | λ j ( k ∞ ) − R | ≤ ε for all j ∈ J ε . This means that R is anaccumulation point of the spectrum of h β ( k ∞ ) which contradicts the fact that σ ( h β ( k ∞ )) isdiscrete. 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 theinterval I R . Next, by (5.4) it follows that none of the eigenvalues λ ℓ ( k ) , ℓ ∈ L , is constant andtherefore, 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 (5.5), 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. (cid:3) 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 thefunctions (5.1), it follows that E ⊂ E c . Hence E ∩ ( a, b ) is finite too. (cid:3) Lemma 5.5.
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 p n ( k ) ∂ k E n ( k ) = 2 p n ( k ) ( k − iβ∂ τ ) 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 orthonormaleigenvectors φ jn ( k ) , j = 1 , . . . , q ( n, I ), analytic on I , associated with E n ( k ). Here q ( n, I ) de-notes the multiplicity of E n ( k ) on I . Since ( φ jn ( k ) , φ in ( k )) L ( ω ) = δ i,j for all k ∈ I , where δ i,j is the Kronecker symbol, we have( h β ( k ) φ jn ( k ) , φ in ( k )) L ( ω ) = E n ( k ) δ i,j k ∈ I. By differentiating this identity with respect to k , we easily obtain2 (( k − iβ∂ τ ) φ jn ( k ) , φ in ( k )) L ( ω ) = ∂ k E n ( k ) δ i,j k ∈ R , (5.7)Hence for any u ∈ L ( ω )2 p n ( k ) ( k − iβ∂ τ ) p n ( k ) u = 2 q ( n,I ) X i,j =1 φ in ( k ) ( φ in ( k ) , ( k − iβ∂ τ ) φ jn ( k )) L ( ω ) ( φ jn ( k ) , u ) L ( ω ) = ∂ k E n ( k ) q ( n,I ) X j =1 φ jn ( k ) ( φ jn ( k ) , u ) L ( ω ) = ∂ k E n ( k ) p n ( k ) u. (cid:3) For the next lemma we need the following definition. Let
I ⊂ R be an open interval. Fix0 < η < |I| / 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 . (5.9) Lemma 5.6.
Suppose that I ⊂ R is an open interval. Let λ ( k ) and µ ( k ) be two analyticfunctions from the family (5.1) and assume that there is exactly one point k ∈ I such that λ ( k ) = µ ( k ) , and λ ( k ) = µ ( k ) for k = k ∈ I . Let π λ ( k ) and π µ ( k ) be the eigenprojectionsassociated with λ ( k ) and µ ( k ) . Then in the sense of quadratic forms on L ( ω ) we have χ I ( λ ( k )) π λ ( k ) ( k − iβ∂ τ ) π µ ( k ) χ I ( µ ( k )) ≤ (5.10) ≤ b λ,µ | λ ( k ) − µ ( k ) | ( χ I ( λ ( k )) π λ ( k ) + χ I ( µ ( k )) π µ ( k )) for all k ∈ I, k = k , where b λ,µ > 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 iλ ( k ) , i = 1 , . . . , q ( λ )and ψ jµ ( 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 iλ ( k ) , ψ jµ ( k )) L ( ω ) = 0 k ∈ I, k = k with respect to k we arrive at2(( k − iβ∂ τ ) ψ iλ ( k ) , ψ jµ ( k )) L ( ω ) = ( λ ( k ) − µ ( k ))( ∂ k ψ iλ ( k ) , ψ jµ ( k )) L ( ω ) k ∈ I, k = k . (5.11) CATTERING IN TWISTED WAVEGUIDES 13
Note that for all k = k , k ∈ I we have π λ ( k ) = q ( λ ) X j =1 ψ jλ ( k ) ( ψ jλ ( k ) , · ) L ( ω ) , π µ ( k ) = q ( µ ) X i =1 ψ iµ ( k ) ( ψ iµ ( k ) , · ) L ( ω ) . Let u ∈ L ( ω ) and let max ≤ j ≤ q ( µ ) max ≤ i ≤ q ( λ ) sup k ∈ I | ( ∂ k ψ iλ ( k ) , ψ jµ ( k )) L ( ω ) | =: ˜ b λ,µ . (5.12)From (5.11) we obtain( u, χ I ( λ ( k )) π λ ( k ) ( k − iβ∂ τ ) π µ ( k ) χ I ( µ ( k )) u ) L ( ω ) == 12 q ( λ ) X j =1 q ( µ ) X i =1 χ I ( λ ( k )) χ I ( µ ( k ))( u, ψ iλ ( k )) ( ψ jµ ( k ) , u ) ( λ ( k ) − µ ( k ))( ∂ k ψ jλ ( k ) , ψ iµ ( k )) ≤ ˜ b λ,µ | λ ( k ) − µ ( k ) | q ( λ ) X j =1 q ( µ ) X i =1 (cid:0) χ I ( λ ( k )) | ( u, ψ jλ ( k )) | + χ I ( µ ( k )) | ( ψ iµ ( k ) , u ) | (cid:1) ≤ b λ,µ | λ ( k ) − µ ( k ) | (cid:0) χ I ( λ ( k ))( u, π λ ( k ) u ) + χ I ( µ ( k )) ( u, π µ ( k ) u ) (cid:1) , for all k = k , k ∈ I , where b λ,µ = ˜ b λ,µ max { q ( λ ) , q ( µ ) } . (cid:3) The conjugate operator
This section is devoted to the construction of the conjugate operator A occurring in the Mourreestimates obtained in the subsequent two sections.Pick γ ∈ C ∞ ( R ; R ), and introduce the operatorˆ A = i γ ∂ k + ∂ k γ ) , D( ˆ A ) = S ( R ) , (6.1)with S ( R ) being the Schwartz class on R . Proposition 6.1.
Let γ ∈ C ∞ ( R ; R ) . Then the operator ˆ A defined in (6.1) is essentiallyself-adjoint in L ( R ) .Proof. Without loss of generality we may assume that there exist a < b such that γ ( a ) = γ ( b ) = 0 and γ ( k ) > k ∈ ( a, b ). Consider solutions u ± to the equations( ˆ A ∗ u )( k ) = i γ ( k ) ∂ k + ∂ k γ ( k )) u ( k ) = ± i u ( k ) . (6.2)A direct calculation gives u ± ( k ) = exp (cid:16) Z kk ± − γ ′ ( r )2 γ ( r ) dr (cid:17) , k ∈ ( a, b ) (6.3)for some k ∈ ( a, b ). The positivity of γ in ( a, b ) implies that γ ′ ( a ) ≥ γ ′ ( b ) ≤
0. Henceby the Taylor expansion there exists an ε > d a , d b such that γ ( r ) ≤ d a ( r − a ) for r ∈ ( a, a + ε ) , γ ( r ) ≤ d b ( b − r ) for r ∈ ( b − ε, b ) . This, combined with (6.3), yields u + ( k ) = (cid:16) γ ( k ) γ ( k ) (cid:17) / exp (cid:16) Z kk drγ ( r ) (cid:17) ≥ (cid:16) γ ( k ) γ ( k ) (cid:17) / exp (cid:16) Z kb − ε drd b ( b − r ) (cid:17) ≥ c ε ( b − k ) − − db , k ∈ ( b − ε, b ) , for some c ε >
0. Hence u + L ( R ). The same argument shows that u − ( k ) ≥ ˜ c ε ( k − a ) − − da , ∀ k ∈ ( a, a + ε ) , ˜ c ε > , which implies u − L ( R ). We thus conclude that ˆ A has deficiency indices (0 ,
0) and thereforeis essentially self-adjoint. (cid:3)
We define the self-adjoint operator ˆ A as the closure of ˆ A in L ( R ).Further, we describe explicitly the action of the unitary group generated by ˆ A .Given a k ∈ R and a function γ ∈ C ∞ ( R ), we consider the initial value problem ddt 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 ) | / f ( ϕ ( t, k )) (6.5) defines a strongly continuous one-parameter unitary group on L ( R ) . Moreover, ˆ A is thegenerator 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.4.1], it follows that ϕ ∈ C ∞ ( R ). Moreover, ∂ k ϕ ( t, k ) = exp (cid:16) − Z t γ ′ ( ϕ ( s, k )) ds (cid:17) ∀ t ≥ , ∀ 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. Inorder to verify that W ( t ) is strongly continuous on L ( R ), let that f ∈ L ( R ). We then have k W ( t ) f − f k ( R ) ≤ k ∂ k ϕ ( t, k ) / ( f ◦ ϕ ( t, k ) − f ) k ( R ) + 2 k ( ∂ k ϕ ( t, k ) / − f k ( R ) (6.7) ≤ c Z supp γ (cid:0) | f ( ϕ ( t, k )) − f ( k ) | + | ∂ k ϕ ( t, k ) / − | | f ( k ) | (cid:1) 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 ) → t → k on compact subsets of R . Since supp γ is compact, (6.7) implies that k W ( t ) f − f k L ( R ) → , t → . Moreover, using (6.4), a direct calculation gives ddt ( W ( t ) f )( k ) (cid:12)(cid:12) t =0 = − γ ′ ( k ) f ( k ) − γ ( k ) f ′ ( k ) = ( i ˆ A f )( k ) , f ∈ S ( R ) . Hence by [21, Theorem VIII.10] it follows that ˆ A generates the unitary group W ( t ). (cid:3) CATTERING IN TWISTED WAVEGUIDES 15
Let γ be as in Theorem 7.2. By Proposition 6.1 and [21, Theorem VIII.33] it follows that theoperator ω ⊗ ˆ A is essentially self-adjoint on C ∞ ( ω ) ⊗ S ( R ). The same is true for the operator F ∗ ( ω ⊗ ˆ A ) F . We define the conjugate operator A in L (Ω) as its closure: A = ¯ A , A = F ∗ ( ω ⊗ ˆ A ) F , D( A ) = C ∞ ( ω ) ⊗ S ( R ) . (6.8)Let Γ be the operator in L ( R ) acting as(Γ ψ )( x ) := (2 π ) − / Z R ˆ γ ( x − t ) ψ ( t ) dt, ˆ γ := F γ, (6.9)where F denotes the Fourier transform from L ( R ) onto L ( R ):( F f )( k ) = (2 π ) − / Z R e − iks f ( s ) ds, f ∈ L ( R ) . A direct calculation then shows that A = − ω ⊗ (Γ x + x Γ) . Mourre estimates for a constant twisting
In this section we establish a Mourre estimate for the commutator [ H β , iA ] with β constantand A defined in (6.8).In the sequel, we use the following notation. Given a self-adjoint positive operator S , invertiblein L (Ω), we denote by D( S ν ) ∗ , ν >
0, the completion of L (Ω) with respect to the norm k S − ν u k L (Ω) . Lemma 7.1.
The commutator [ ˆ H β , i ( ω ⊗ ˆ A )] defined as a quadratic form on C ∞ (Ω) extendsto a bounded operator from D( ˆ H β ) into D( ˆ H β ) ∗ . Moreover, [ ˆ H β , i ( ω ⊗ ˆ A )] = 2 γ ( k )( k − iβ∂ τ ) . (7.1) Proof.
Let u ∈ C ∞ (Ω). A simple calculation then gives( ˆ H β u, i ( ω ⊗ ˆ A ) u ) L (Ω) − ( i ( ω ⊗ ˆ A ) u, ˆ H β u ) L (Ω) = 2( u, γ ( k − iβ∂ τ ) u ) L (Ω) . Hence (7.1) follows. Moreover, from the above equation we easily obtain | ([ ˆ H β , i ( ω ⊗ ˆ A )] u, u ) L (Ω) | ≤ C (cid:0) k ˆ H / β u k (Ω) + k u k (Ω) (cid:1) . (7.2)So [ ˆ H β , i ( ω ⊗ ˆ A )] is a bounded operator from D( ˆ H / β ) into D( ˆ H / β ) ∗ , and hence it is alsobounded from D( ˆ H β ) into D( ˆ H β ) ∗ . (cid:3) Theorem 7.2.
Let E ∈ R \ E . Then there exist δ > , a function γ ∈ C ∞ ( R ) and a positiveconstant c = c ( E, δ ) such that in the form sense on L (Ω) we have χ I ( ˆ H β ) [ ˆ H β , i ( ω ⊗ ˆ A )] χ I ( ˆ H β ) ≥ c χ I ( ˆ H β ) , (7.3) where I = ( E − δ, E + δ ) , χ I is given by (5.9) and the commutator [ ˆ H β , i ( ω ⊗ ˆ A )] is understoodas a bounded operator from D( ˆ H β ) into D( ˆ H β ) ∗ . Proof.
First of all we chose δ small enough such thatdist( E, ( E c \ E )) > δ, (7.4)which is possible in view of Lemma 5.4. Recall that E ⊂ E c . Next we define K ( n, E ) = (cid:8) k ∈ R : E n ( k ) = E (cid:9) . Note that by Lemma 5.4 K ( n, E ) is finite for every n and K ( n, E ) = ∅ for all n > N E + δ . Inthe rest of the proof we use the notation N = N E + δ . Let K ( E ) = ∪ ∞ n =1 K ( n, E ) = ∪ Nn =1 K ( n, E ) , and define K ( E ) = { k ∈ R : there exists a unique n such that E n ( k ) = E } , K ( E ) = K ( E ) \ K ( 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 ) isgiven by a union of finitely many non-degenerate disjoint open intervals: B ( n, E ) = ∪ G n 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 ( j, n ) := Q ( j, n ) if Q ( j, n ) ∩ K ( n, E ) ⊂ K ( E ) Q ( j, n ) := Q ( j, n ) if Q ( j, n ) ∩ K ( n, E ) ⊂ K ( E ) . By the hypothesis on E we can take δ small enough so that Q ( j, n ) ∩ Q ( j, m ) = ∅ n = m, and at the same time Q ( j, n ) ∩ K ( E ) = Q ( i, m ) ∩ K ( E )implies Q ( j, n ) ∩ Q ( i, m ) = ∅ . Hence, for δ sufficiently small, we can construct intervals J ,l with l = 1 , . . . , L ( E ), and J ,p with p = 1 , . . . , P ( E ), such that J ,l ∩ J ,l ′ = ∅ l = l ′ , J ,p ∩ J ,p ′ = ∅ p = p ′ , J ,l ∩ J ,p = ∅ ∀ l, p, (7.5)and such that M ( E ) := N [ n =1 (cid:16) ∪ j Q ( j, n ) (cid:17) = L ( E ) [ l =1 J ,l , M ( E ) := N [ n =1 (cid:16) ∪ j Q ( j, n ) (cid:17) = P ( E ) [ p =1 J ,p . Moreover, each J ,l contains exactly one element k ,l of K ( E ) and each J ,p contains exactlyone element k ,p of K ( E ). By construction, we have M ( E ) := M ( E ) ∪ M ( E ) = ∪ Nn =1 B ( n, E ) , M ( E ) ∩ M ( E ) = ∅ . CATTERING IN TWISTED WAVEGUIDES 17
With these preliminaries we can proceed with the estimation of the commutator. From Lemma7.1 we find that χ I ( ˆ H β ) [ ˆ H β , i ( ω ⊗ ˆ A )] χ I ( ˆ H β ) = (7.6)= 2 ∞ X n,m =1 Z ⊕ 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 X n,m =1 Z ⊕M ( 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 X n,m =1 Z ⊕M ( 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 ( E ), for each l = 1 , . . . , L ( E ) there exists exactly one n ( l ) ≤ N such that N X n,m =1 Z ⊕ J ,l χ I ( E n ( k )) p n ( k ) γ ( k )( k − iβ∂ τ ) χ I ( E m ( k )) p m ( k ) dk == Z ⊕ J ,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 ,l , it is analytic on J ,l . Hence by Lemma 5.5 we obtain Z ⊕ J ,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 == Z ⊕ J ,l χ 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 ,l , ∀ l = 1 , . . . , L ( E ) . (7.7)Note that | ∂ k E n ( l ) ( k ) | is strictly positive on J ,l . Therefore we have d := min ≤ l ≤ L ( E ) inf k ∈ J ,l | ∂ k E n ( l ) ( k ) | > . Hence, N X n,m =1 Z ⊕M ( 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 N X n =1 Z ⊕M ( E ) χ 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 ,p we have N X n,m =1 Z ⊕ J ,p χ I ( E n ( k )) p n ( k ) γ ( k )( k − iβ∂ τ ) χ I ( E m ( k )) p m ( k ) dk == Z ⊕ J ,p X 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 ) ⊂ { , . . . , N } . Moreover, from the construction of the intervals J ,p it followsthat there exists a family of analytic eigenfunctions λ s ( k ) , s ∈ S ( p ), with S ( p ) being a finitesubset of N , such that each E r ( k ) coincides with some λ s ( k ) on J ,p ∩ ( −∞ , k ,p ) and withsome λ s ′ ( k ) on J ,p ∩ ( k ,p , ∞ ), where k ,p is the only element of K ( E ) contained in J ,p . Let π s ( k ) be the eigenprojection associated with λ s ( k ). With the help of Lemma 5.6 we obtain Z ⊕ J ,p X 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 == Z ⊕ J ,p X 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 == Z ⊕ J ,p X s ∈ S ( p ) χ I ( λ s ( k )) γ ( k ) ∂ k λ s ( k ) π s ( k ) dk ++ Z ⊕ J ,p X 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 ,p are mutually disjoint and also disjoint from the set M ( E ), see (7.5),and since the functions ∂ k λ s ( k ) are either all strictly negative or all strictly positive on everyinterval J ,p , by the construction of J ,p , we can choose γ such that, in addition to (7.7), itholds γ ( k ) ∂ k λ s ( k ) = | ∂ k λ s ( k ) | ∀ k ∈ J ,p , ∀ s ∈ S ( p ) , ∀ p = 1 , . . . , P ( E ) . (7.10)Moreover, d := min ≤ p ≤ P ( E ) min s ∈ S ( p ) inf k ∈ J ,p | ∂ k λ s ( k ) | > . Now, to control the last term in (7.9) assume that s = s ′ and let b λ s ,λ s ′ be the constant givenin Lemma 5.6 with I = J ,p . Note that | J ,p | decreases as δ →
0. From the explicit expressionfor b λ s ,λ s ′ , see (5.12), it is then easily seen that there exists b p >
0, independent of δ , such thatmax s,s ′ ∈ S ( p ) ,s = s ′ b λ s ,λ s ′ ≤ b p . CATTERING IN TWISTED WAVEGUIDES 19
Hence, (7.9), in combination with Lemmata 5.5 and 5.6, yields Z ⊕ J ,p X 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 − c p b p δ ) Z ⊕ J ,p X s ∈ S ( p ) χ I ( λ s ( k )) π s ( k ) dk == ( d − c p b p δ ) Z ⊕ J ,p X r ∈ R ( p ) χ I ( E r ( k )) p r ( k ) dk, where c p > p . Therefore we obtain N X n,m =1 Z ⊕M ( 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 − C E δ ) N X n =1 Z ⊕M ( E ) χ I ( E n ( k )) p n ( k ) dk, with C E = max ≤ p ≤ P ( E ) c p b p . Taking into account (7.8), we thus conclude that for δ smallenough there exists some c > N X n,m =1 Z ⊕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 X n =1 Z ⊕M ( E ) χ I ( E n ( k )) p n ( k ) dk = c ∞ X n =1 Z ⊕ R χ I ( E n ( k )) p n ( k ) dk = c χ I ( ˆ H β ) . In view of (7.6) this proves the theorem. (cid:3)
Corollary 7.3.
Let E ∈ R \ E and I = ( E − δ, E + δ ) be given as in Theorem 7.2. Then χ I ( H β ) [ H β , iA ] χ I ( H β ) ≥ c χ 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 conjugateoperator is defined by (6.1) and (6.8) .Proof. This follows from (2.9), (6.8) and Theorem 7.2. (cid:3) Perturbation of the constant twisting
Mourre estimate for [ H θ ′ , iA ] . In the sequel we will suppose that θ ′ ( x ) = β − ε ( x ) . In this section we will prove a Mourre estimate for the commutator [ H θ ′ , iA ], see below Theorem8.2. Notice that H θ ′ acts as H θ ′ = H β + W, W = (2 εβ − ε ) ∂ τ + 2 ε ∂ τ ∂ + + ε ′ ∂ τ (8.1)on H (Ω) ∩ H (Ω), cf. Corollary 2.2. Together with (8.1) we will also use the decomposition H θ ′ = H + U, U = W − β ∂ τ − β ∂ τ ∂ . (8.2) Lemma 8.1.
Let χ I ∈ C ∞ ( R ) be given by (5.9) . Then the operator χ I ( H θ ′ ) − χ I ( H β ) iscompact in L (Ω) .Proof. The Helffer-Sj¨ostrand formula, [7, 8], gives χ I ( H θ ′ ) − χ I ( H β ) = − π Z R ∂ ˜ χ∂ ¯ z ( H θ ′ − z ) − W ( H β − z ) − dx dy, (8.3)where z = x + iy , and ˜ χ is a compactly supported quasi-analytic extension of χ I I in R whichsatisfies sup x ∈ R (cid:12)(cid:12)(cid:12) ∂ ˜ χ∂z ( x + iy ) (cid:12)(cid:12)(cid:12) ≤ const y , | y | ≤ . (8.4)Since ( H θ ′ − z ) − W ( H − z ) − is compact whenever y = 0, see [5], it follows that ∂ ˜ χ∂ ¯ z ( H θ ′ − z ) − W ( H β − z ) − is compact for all ( x, y ) ∈ R with y = 0. Moreover, by the resolventequation the norm of ( H θ ′ − z ) − W ( H − z ) − is bounded by a constant times y − . In viewof (8.4) the integrand on the right hand side of (8.3) is then uniformly norm-bounded in R and hence χ I ( H θ ′ ) − χ I ( H β ) is compact. (cid:3) Theorem 8.2.
Let E ∈ R \ E and let ε satisfy (2.13) . Choose δ > and γ ∈ C ∞ ( R ) as inTheorem 7.2. Then there exists a positive constant c and a compact operator K in L (Ω) suchthat P I ( E,δ ) [ H θ ′ , iA ] P I ( E,δ ) ≥ c P I ( E,δ ) + P I ( E,δ ) K P I ( E,δ ) , (8.5) where P I ( E,δ ) is the spectral projection for the interval I ( E, δ ) := ( E − δ/ , E + δ/ , associated to H θ ′ .Proof. Let I = ( E − δ, E + δ ). We proceed in several steps. First we show that there exists c > K in L (Ω) such that χ I ( H θ ′ )[ H β , iA ] χ I ( H θ ′ ) ≥ c χ I ( H β ) + K . (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 > χ I ( H β )[ H β , iA ] χ I ( H β ) ≥ cχ 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 )) = 2 χ I ( H β )( i∂ + 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) − ( i∂ + βi∂ τ ) and χ I ( H θ ′ )( H θ ′ + 1) are bounded so χ I ( H β )( i∂ + βi∂ τ ) is bounded too and K := χ I ( H β )( i∂ + βi∂ τ )Γ( χ I ( H θ ′ ) − χ I ( H β )) is compact. The same arguments show that K := ( 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∂ − βi∂ τ ) χ I ( H θ ′ )is compact. Putting K = K + K concludes the first step of the proof. CATTERING IN TWISTED WAVEGUIDES 21
Next we consider χ I ( H θ ′ )[ W, iA ] χ I ( H θ ′ ). For the sake of simplicity we now write s insteadof x . Defining η ( s ) := 2 ε ( s ) β − ε ( s ) we get [ W, iA ] = [ η , iA ] ∂ τ + [ ε ∂ s , iA ] ∂ τ + [ ∂ s ε, iA ] ∂ τ . (8.7)We first deal with the term [ η ∂ τ , iA ] = i [ η, A ] ∂ τ = − i [ η, Γ s + s Γ] ∂ τ . For an appropriate testfunction φ we obtain √ π ([ η, Γ s + s Γ] φ )( s ) =: ( T φ )( s ) = η ( s ) Z R ˆ γ ( s − s ′ ) s ′ φ ( s ′ ) ds ′ + η ( s ) Z R s ˆ γ ( s − s ′ ) φ ( s ′ ) ds ′ − Z R ˆ γ ( s − s ′ ) η ( s ′ ) s ′ φ ( s ′ ) ds ′ − Z R s ˆ γ ( s − s ′ ) η ( s ′ ) φ ( s ′ ) ds ′ . Hence T is an integral operator on L ( R ) with the kernel T ( s, s ′ ) = η ( s )ˆ γ ( s − s ′ ) s ′ + η ( s ) s ˆ γ ( s − s ′ ) − ˆ γ ( s − s ′ ) η ( s ′ ) s ′ − s ˆ γ ( s − s ′ ) η ( s ′ ) . To control the s -dependence we rewrite the kernel as T ( s, s ′ ) = η ( s )ˆ γ ( s − s ′ )( s ′ − s ) + 2 η ( s ) s ˆ γ ( s − s ′ ) (8.8) − γ ( s − s ′ ) η ( s ′ ) s ′ − ( s − s ′ )ˆ 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 k ˆ g k L q ( R ) ≤ (2 π ) − p k g k L p ( R ) and the interpolation result which we alreadyused in the proof of Lemma 4.1 (see [24, Theorem 4.1] or [3, Section 4.4]), imply that theintegral operator with a kernel of the form f ( s ) g ( s − s ′ ), s, s ′ ∈ R , belongs to the class S q , andhence is compact on L ( R ). By (2.13), both functions η ( s ) and sη ( s ) are in L q ( R ) for q largeenough. Since γ ∈ C ∞ ( R ), its Fourier transform ˆ γ is in the Schwartz class on R and thereforein any L p ( R ) with p ≥
1. Therefore, the operator [ η, Γ s + s Γ] is compact on L ( R ). In order toensure the compactness of χ I ( H θ ′ ) T ∂ τ χ I ( H θ ′ ) on L (Ω), we note that by Corollary 2.2 andthe closed graph theorem the operators H − β H θ ′ and H − θ ′ H β are bounded on L (Ω). Since H θ ′ χ I ( H θ ′ ) is bounded too, it suffices to prove that H − β T ∂ τ H − β (8.9)is compact on L (Ω). To this end we point out that H β ≥ ω ⊗ ( − ∆ ω ) and that the operators ∂ τ ( − ∆ ω ) − and ( − ∆ ω ) − are respectively bounded and compact on L ( ω ). Hence ( ω ⊗ ( − ∆ ω )) − T ∂ τ ( ω ⊗ ( − ∆ ω )) − is a product of a bounded and a compact operator and henceis compact on L (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 forthe the operator [ ∂ τ ε∂ s , iA ] = − i ∂ τ [ ε∂ s , Γ s + s Γ] , with the help of the integration by parts we find that([ ε∂ s , Γ s + s Γ] φ )( s ) = (2 π ) − / ( R φ )( s ) + (2 π ) − / ( R φ )( s )= (2 π ) − / Z R R ( s, s ′ ) φ ( s ′ ) ds ′ + (2 π ) − / Z R R ( s, s ′ ) φ ( s ′ ) ds ′ , where the integral kernels R ( s, s ′ ) and R ( s, s ′ ) of the operators R and R are given by R ( s, s ′ ) = ε ( s ) (cid:0) ˆ γ ′ ( s − s ′ )( s ′ − s ) + ˆ γ ( s − s ′ ) + 2 s ˆ γ ′ ( s − s ′ ) (cid:1) (8.10) R ( s, s ′ ) = ε ( s ′ ) (cid:0) − ˆ γ ′ ( s − s ′ ) s ′ + ˆ γ ( s − s ′ ) − s ˆ γ ( s − s ′ ) (cid:1) + ε ′ ( s ′ ) (cid:0) ˆ γ ( s − s ′ ) s ′ + s ˆ γ ( s − s ′ ) (cid:1) , and ˆ γ ′ denotes the derivative of ˆ γ . As above we need to write also the kernel T ( s, s ′ ) as asum of the terms of the form f ( s ) g ( s − s ′ ) and f ( s ′ ) g ( s − s ′ ): R ( s, s ′ ) = ε ( s ′ ) (cid:0) − γ ′ ( s − s ′ ) s ′ + ˆ γ ( s − s ′ ) − ( s − s ′ ) ˆ γ ( s − s ′ ) (cid:1) + ε ′ ( s ′ ) (cid:0) γ ( s − s ′ ) s ′ + ( s − s ′ )ˆ γ ( s − s ′ ) (cid:1) . (8.11)Using the assumptions of Theorem 2.7 and the fact that ˆ γ ′ is the Schwartz class on R , weconclude as before that R and R are compact on L ( R ) and therefore χ I ( H θ ′ ) ∂ τ [ ε ∂ s , Γ s + s Γ] χ I ( H θ ′ )is compact on L (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 I ( H θ ′ ) + K (8.12)where I = ( E − δ, E + δ ) and K is compact. Now we fix η = δ/ P I ( E,δ ) . (cid:3) Proof of Theorem 2.7.
In order to prove Theorem 2.7, we will need, in addition to theMourre estimate established in Theorem 8.2, a couple of technical results. We introduce thenorm k u k +2 ,θ := (cid:0) k H θ ′ u k (Ω) + k u k (Ω) (cid:1) / , u ∈ H (Ω) ∩ H (Ω) , and recall that if ε satisfies (2.13), then in view of Corollary 2.2 k u k +2 ,θ ≍ k u k +2 , ≍ k u k H (Ω) , u ∈ H (Ω) ∩ H (Ω) . (8.13) Proposition 8.3.
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 |≤ k e itA u k +2 ,θ < ∞ . (b) The operator B = [ H , iA ] defined as a quadratic form on D( A ) ∩ D( H ) is boundedon L (Ω) . (c) The operator B = [ H θ ′ , iA ] defined as a quadratic form on D( A ) ∩ D( H θ ′ ) is boundedfrom D( H θ ′ ) into D( H / θ ′ ) ∗ . (d) There is a common core C for A and H so that A maps C into H (Ω) .Proof. Note that H = − ∆ and that D( H θ ′ ) = D( H ) = H (Ω) ∩ H (Ω), in view of Corollary2.2. To prove assertion ( a ), pick f ∈ D( H θ ′ ) and denote g = F f . By Lemma 6.2,( e itA f )( x ) = F ∗ (cid:2) ( ∂ k ϕ ( t, k )) / g ( x ω , ϕ ( t, k )) (cid:3) . (8.14)Hence, k ∆ ( e itA f ) k (Ω) = k e itA (∆ ω f ) + ∂ ( e itA f ) k (Ω) (8.15) ≤ k ∆ ω f k (Ω) + k k ( ∂ k ϕ ( t, k )) / g ( x ω , ϕ ( t, k )) k (Ω) , CATTERING IN TWISTED WAVEGUIDES 23 where we have used the fact that e itA : L (Ω) → L (Ω) and F ∗ : L (Ω) → L (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 k ∂ k ϕ ( t, k ) g ( x ω , ϕ ( t, k )) k (Ω) ≤ k c Z ω × [ − k c ,k c ] ∂ k ϕ ( t, k ) | g ( x ω , ϕ ( t, k )) | dk dx ω + Z Ω k | g ( x ω , k ) | dk dx ω ≤ k c Z Ω | g ( x ω , z ) | dz dx ω + Z Ω k | g ( x ω , k ) | dk dx ω = k c k f k (Ω) + k ∂ f k (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 k e itA f k , = k ∆ ( e itA f ) k (Ω) + k f k (Ω) ≤ const k f k (Ω) . This implies that sup | t |≤ k e itA f k +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 , iA ] = ω ⊗ F ∗ (2 γ ( k ) k ) F which is a bounded operator on L (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) − / [ H β , iA ]( H β + 1) − is bounded on L (Ω). On the other hand, from the proof ofTheorem 8.2 it follows that the same is true for the operator ( H β + 1) − / [ W, iA ]( H β + 1) − .Since ( H β +1) α ( H θ ′ +1) − α , α = 1 / ,
1, is bounded, we conclude that ( H θ ′ +1) − / B ( H θ ′ +1) − is bounded on L (Ω) and (c) follows. To prove (d) we define C := D( − ∆ ω ) ⊗ S ( R ). Bydefinition of A , C is a core for A . On the other hand, C is also a core for H . Since F : C → C is a bijection and since ˆ A : S → S , it follows that A : C → C ⊂ H (Ω). (cid:3) Lemma 8.4.
Let ε satisfy assumptions of Theorem 2.7. Then ( H θ ′ + 1) − [ B, A ] ( H θ ′ + 1) − is bounded as an operator on L (Ω) .Proof. Recall that B = [ H θ ′ , iA ]. We write B = B + B , where B = [[ H β , iA ] , iA ] , B = [[ W, iA ] , iA ] . As for the term B , a direct calculation gives B = F ∗ [[ ˆ H β , i ( ω ⊗ ˆ A )] , i ( ω ⊗ ˆ A )] F = F ∗ (cid:0) γ ( k ) + γ ( k ) γ ′ ( k )( k − iβ ∂ τ ) (cid:1) F . (8.16)Let u ∈ L (Ω). Similarly as in (7.2) we find out that | ( u, ( k − iβ ∂ τ ) u ) L (Ω) | ≤ ( u, ( ˆ H β + 1) u ) L (Ω) . Since γ and γ ′ are bounded, the last inequality implies that also ( H β + 1) − B ( H β + 1) − isbounded. From Proposition 2.1 it then follows that( H θ ′ + 1) − B ( H θ ′ + 1) − is bounded too. As for the remaining part of the double commutator, we first note that inview of (8.7) and of the fact that the operators ∂ τ ( H θ ′ + 1) − and ∂ τ ( H θ ′ + 1) − are bounded,it suffices to show that h [ η , F ∗ ˆ A F ] + [ ε ′ , F ∗ ˆ A F ] + [ ε ∂ s , F ∗ ˆ A F ] , F ∗ ˆ A F i (8.17) is a bounded operator on L ( R ). Let u ∈ L ( R ) and recall that( F ∗ ˆ A F u )( s ) = − √ π (cid:16) Z R ˆ γ ( s − s ′ ) s ′ u ( s ′ ) ds ′ + s Z R ˆ γ ( s − s ′ ) u ( s ′ ) ds ′ (cid:17) . It will be useful to introduce the shorthandsˆ γ j ( r ) = r j ˆ γ ( r ) . Note that ˆ γ j ∈ S ( R ) for all j ∈ N . We have − √ π [ η , F ∗ ˆ A F ] u = 2 η ( s ) Z R ˆ γ ( s − s ′ ) s ′ u ( s ′ ) ds ′ + η ( s ) Z R ˆ γ ( s − s ′ ) u ( s ′ ) ds ′ (8.18) − Z R ˆ γ ( s − s ′ ) s ′ η ( s ′ ) u ( s ′ ) ds ′ − s Z R ˆ γ ( s − s ′ ) η ( s ′ ) u ( s ′ ) ds ′ =: X j =1 T j u. Accordingly, −√ π [ T , F ∗ ˆ A F ] u = η ( s ) Z R Z R ˆ γ ( s − s ′ ) s ′ ˆ γ ( s ′ − s ′′ ) u ( s ′′ ) ds ′′ ds ′ + η ( s ) Z R Z R ˆ γ ( s − s ′ ) s ′ s ′′ ˆ γ ( s ′ − s ′′ ) u ( s ′′ ) ds ′′ ds ′ − Z R Z R ˆ γ ( s − s ′ ) η ( s ′ ) ˆ γ ( s ′ − s ′′ ) s ′ s ′′ u ( s ′′ ) ds ′′ ds ′ − s Z R Z R ˆ γ ( s − s ′ ) η ( s ′ ) ˆ γ ( s ′ − s ′′ ) u ( s ′′ ) ds ′′ ds ′ =: X j =1 T ,j u. Note that s ′ ˆ γ ( s − s ′ )ˆ γ ( s ′ − s ′′ ) = s ˆ γ ( s − s ′ )ˆ γ ( s ′ − s ′′ ) + ˆ γ ( s − s ′ )ˆ γ ( s ′ − s ′′ ) − s ˆ γ ( s − s ′ )ˆ γ ( s ′ − s ′′ ) , which implies T , u ( s ) = s η ( s ) ˆ γ ∗ (ˆ γ ∗ u ) + η ( s ) ˆ γ ∗ (ˆ γ ∗ u ) − s η ( s ) ˆ γ ∗ (ˆ γ ∗ u ) . Hence, by a repeated use of the Young inequality k g ∗ h k p ≤ C k g k q k h k r , q + 1 r = 1 + 1 p , (8.19)with p = q = 2 and r = 1, we get k T , u k ≤ C , (cid:0) k s η k ∞ k ˆ γ k + k η k ∞ k ˆ γ k k ˆ γ k + k s η k ∞ k ˆ γ k k ˆ γ k (cid:1) k u k , for some C , < ∞ . Moreover, since s ′ s ′′ ˆ γ ( s − s ′ )ˆ γ ( s ′ − s ′′ ) = s ′ ˆ γ ( s − s ′ )ˆ γ ( s ′ − s ′′ ) − s ˆ γ ( s − s ′ )ˆ γ ( s ′ − s ′′ ) + ˆ γ ( s − s ′ )ˆ γ ( s ′ − s ′′ ) , with the help of (8.19) we obtain k T , u k ≤ k T , u k + C , (cid:0) k η k ∞ k ˆ γ k + k s η k ∞ k ˆ γ k k ˆ γ k (cid:1) k u k . CATTERING IN TWISTED WAVEGUIDES 25
As for T , , we note that T , u = ˆ γ ∗ ( sη (ˆ γ ∗ u )) − ˆ γ ∗ ( s η (ˆ γ ∗ u )) , which, in combination with (8.19), implies k T , u k ≤ C , (cid:0) k sη k ∞ k ˆ γ k k ˆ γ k + k s η k ∞ k ˆ γ k k ˆ γ k (cid:1) k u k . Next, for T , u we find T , u = − ˆ γ ∗ ( η (ˆ γ ∗ u )) + ˆ γ ∗ ( sη (ˆ γ ∗ u )) − ˆ γ ∗ ( sη (ˆ γ ∗ u )) + ˆ γ ∗ ( s η (ˆ γ ∗ u )) . By using again (8.19) we get k T , u k ≤ C , (cid:0) k η k ∞ k ˆ γ k + 2 k sη k ∞ k ˆ γ k k ˆ γ k + k s η k ∞ k ˆ γ k (cid:1) k u k . This implies that k [ T , F ∗ ˆ A F ] u k = 12 X j =1 k T ,j u k ≤ C k u k , for some C < ∞ . As for the term [ T , F ∗ ˆ A F ] u , we find out that − √ π [ T , F ∗ ˆ A F ] u = − η ˆ γ ∗ (ˆ γ ∗ u ) − η ˆ γ ∗ (ˆ γ ∗ u ) + 2 s η ˆ γ ∗ (ˆ γ ∗ u ) − γ ∗ ( sη (ˆ γ ∗ u )) − ˆ γ ∗ ( η (ˆ γ ∗ u )) . Hence by (8.19) k [ T , F ∗ ˆ A F ] u k ≤ C (cid:0) k η k ∞ ( k ˆ γ k + k ˆ γ k k ˆ γ k ) + k sη k ∞ k ˆ γ k k ˆ γ k (cid:1) k u k . Next we consider the last term on the right hand side of (8.18). A direct calculation gives − √ π [ T , F ∗ ˆ A F ] u = 2 ˆ γ ∗ ( sη (ˆ γ ∗ u )) − γ ∗ ( η (ˆ γ ∗ u ))+ 2 ˆ γ ∗ ( s η (ˆ γ ∗ u )) − ˆ γ ∗ ( sη (ˆ γ ∗ u )) . By the Young inequality, k [ T , F ∗ ˆ A F ] u k ≤ C (cid:0) k ˆ γ k k ˆ γ k k sη k ∞ + k ˆ γ k k η k ∞ + k ˆ γ k k s η k ∞ (cid:1) k u k , with some C < ∞ . The same argument applies to [ T , F ∗ ˆ A F ]. We thus conclude that thefirst term in (8.17) defines a bounded operator in L ( R ). The same arguments apply to thesecond term in (8.17) replacing η by ε ′ . As for the last term in (8.17), integration by partsshows that − √ π [ ε∂ s , F ∗ ˆ A F ] u = ε ( s ) Z R ˆ γ ′ ( s − s ′ ) s ′ u ( s ′ ) ds ′ + ε ( s ) Z R ˆ γ ( s − s ′ ) u ( s ′ ) ds ′ + ε ( s ) s Z R ˆ γ ′ ( s − s ′ ) u ( s ′ ) ds ′ + Z R (cid:2) ˆ γ ( s − s ′ )( ε ( s ′ ) + s ′ ε ′ ( s ′ ) − ˆ γ ′ ( s − s ′ ) s ′ ε ( s ′ )) (cid:3) u ( s ′ ) ds ′ + s Z R (cid:2) ˆ γ ( s − s ′ ) ε ′ ( s ′ ) − ˆ γ ′ ( s − s ′ ) ε ( s ′ ) (cid:3) u ( s ′ ) ds ′ . Note that each term on the right hand side of the above equation is of the same type as oneof the terms that we have already treated above, with ˆ γ replaced by ˆ γ ′ when necessary. Since r j ˆ γ ′ ∈ S ( R ) for all j ∈ N , by following the same line of arguments as above we obtain (cid:13)(cid:13)(cid:2) [ ε ∂ s , F ∗ ˆ A F ] , F ∗ ˆ A F (cid:3) u (cid:13)(cid:13) ≤ ˜ C (cid:0) k ε (1 + s ) k ∞ + k ε ′ (1 + s ) k ∞ (cid:1) k u k . for some constant ˜ C < ∞ . This together with the previous estimates implies that (8.17)defines a bounded operator in L ( R ). (cid:3) With these prerequisites, we can finally state the result about the nature of the essentialspectrum of H θ ′ : Corollary 8.5.
Let ε satisfy the assumptions of Theorem 2.7. Let E ∈ R \ E be given anddefine the interval I ( E, δ ) = ( E − δ/ , E + δ/ as in Theorem 8.2. Then: (a) I ( E, δ ) contains at most finitely many eigenvalues of H θ ′ , each having finite multiplic-ity; (b) The point spectrum of H θ ′ has no accumulation point in I ( E, δ ) ; (c) H θ ′ has no singular continuous spectrum in I ( E, δ ) .Proof. Since A is self-adjoint, the statement follows from Proposition 8.3, Lemma 8.4, Theorem8.2 and [20, Theorem 1.2]. (cid:3) 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 finitesubcovering: J ⊂ ∪ Nn =1 I ( E n , δ n ) . (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 immediatelyfrom (a). To prove (c) assume that σ sc ( H θ ′ ) ∩ ( R \ E ) = ∅ . Since the set E is locally finite, seeLemma 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 ′ ) = ∅ . (cid:3) Acknowledgements.
The authors were partially supported by the Bernoulli Center, EPFL,Lausanne, within the framework of the Program “
Spectral and Dynamical Properties of Quan-tum Hamiltonians ”, January – June 2010. H. Kovaˇr´ık gratefully acknowledges also partialsupport of the MIUR-PRIN08 grant for the project “
Trasporto ottimo di massa, disuguaglianzegeometriche e funzionali e applicazioni ”, and would like to thank as well the Faculty of Math-ematics of Pontificia Universidad Cat´olica de Chile for the warm hospitality extended to him.G. Raikov was partially supported by by the Chilean Science Foundation
Fondecyt under Grant1130591, and by
N´ucleo Cient´ıfico ICM
P07-027-F.
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Ph. Briet
Universit´e du Sud Toulon VarCentre de Physique Th´eoriqueCNRS-Luminy, Case 90713288 Marseille, FranceE-mail: [email protected]
H. Kovaˇr´ık
Dipartimento di MatematicaUniversit`a degli studi di BresciaVia Branze, 3825123 Brescia, ItalyE-mail: [email protected]