Absence of the singular spectrum in a twisted Dirichlet-Neumann - PowerPoint PPT Presentation

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate Absence of the singular spectrum in a twisted Dirichlet-Neumann waveguide Ph. Briet 1 J.Dittrich 2 rík 3 D. Krejˇ ciˇ 1 Centre de Physique Théorique,CNRS, Marseille-France 2 Nuclear Physics Institute ASCR, ˇ Rež, Czech Republic 3 Czech Technical University in Prague, Czech Republic Differential Operators on Graphs and Waveguides TU-GRAZ 2019

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate Outline The Dittrich-Kˇ ríž problem 1 References 2 3 Questions The spectrum 4 Mourre estimate 5

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate The Dittrich-Kˇ ríž twisted problem Consider the following straight domain Ω in R 2 : In this talk we consider the case δ = 0 Consider the Laplace operator defined on H = L 2 (Ω) with DBC on D and NBC on N .

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate The Dittrich-Kˇ ríž twisted problem Twisted system free system Main problem → study of the scattering theory

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate The Hamiltonian Let D ( h ) = { ψ ∈ H 1 (Ω) | ψ ⌈ D = 0 } and � |∇ ψ | 2 dx h [ ψ ] := Ω Then D ( H ) = { ψ ∈ H 1 (Ω) , ∆ ψ ∈ H | ψ ⌈ D = 0 , ∂ y ψ ⌈ N = 0 } H ψ = − ∆ ψ

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate The Hamiltonian In fact we can prove that If ψ ∈ D ( H ) then ψ ∈ H 2 (Ω 0 ) for every open set Ω 0 ⊂ Ω such that { ( 0 , 0 ) , ( 0 , d ) } ∩ Ω 0 = ∅

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate References J.Dittrich, J.Kˇ ríž, Jour. Math. Phys 2002 Ph Briet, J. Dittrich, E. Soccorsi, Jour. Math Phys, 2014 D. Krejˇ ciˇ rík, E. Zuazua Jour. Diff. Equat. 2011 D. Krejˇ ciˇ rík, H. Kovarik Math Nachr. 2008 ...

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate Questions Point spectrum Absence of singular continuous spectrum Completeness of the wave operators

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate Wave operators Let Ω 1 = ( 0 , d ) × R − and Ω 2 = ( 0 , d ) × R + and χ j the characteristic function of Ω j . Let H 1 = − ∆ on Ω with BDC on { 0 } × R and NBC on { d } × R , H 2 = − ∆ on Ω with BDC on { d } × R and NBC on { 0 } × R Prove the existence of the TWO wave operators : j = 1 , 2 Ω ∓ t →±∞ e itH χ j e itH j j = s − lim and W ± t →±∞ e itH j χ j e itH P ac ( H ) = s − lim j Here P ac ( H j ) = I H

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate Completeness Prove that j ) ∗ = W ± (Ω ± j and the completeness relation P ac ( H ) = Ω ± 1 W ± 1 + Ω ± 2 W ± 2 If P ac ( H ) = I H i.e. σ sing = ∅ → asymptotic completeness

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate The spectrum Theorem: σ ess ( H ) = [ E , + ∞ ); E = π 2 4 d 2 Proof : → J.Dittrich, J.Kˇ ríž (See also Ph. Briet, H. Abdou Soimadou, D. Krejˇ ciˇ rík, to appear in ZAMP)

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate The spectrum Denote by σ pp ( H ) the set of all eigenvalues of H and σ d ( H ) the set of discrete eigenvalues of H . We know from J.Dittrich, J.Kˇ ríž that if δ > 0 then σ d ( H ) � = ∅ . We show that Theorem: If δ = 0 then σ pp ( H ) = ∅

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate strategy of proof Suppose that there exist an eigenvalue λ ∈ R and ψ ∈ H s.t. H ψ = λψ Then we construct a sequence ( ϕ n ) n ∈ N of D ( h ) s.t. 2 � ∂ x ψ � 2 = lim � h ( ψ, ϕ n ) − λ ( ψ, ϕ n ) � = 0 n →∞ then regularity properties of ψ and the fact ψ ∈ H 1 ⇒ ψ = 0. In fact ϕ n = ( ψ + 2 x ∂ x ψ ) χ n where χ n is an approximation of the identity function

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate Mourre estimate Let T := { E k } k ∈ N ∗ and let E ∈ R \ T and η > 0, s.t. ( E − η, E + η ) ∩ T = ∅ , P η := P ( E − η, E + η ) The conjugate operator : A = 1 2 ( F ( x ) i ∂ x + i ∂ x F ( x )) F ∈ C 2 ( R ) , F ( x ) ∼ x in a neighbourhoud of ±∞ Theorem: There exists a positive number α and a compact operator K on such that P η i [ H , A ] P η ≥ α P η + P η KP η

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate It also holds D ( A ) ∩ D ( H ) is a core of H . e itA leaves D ( H ) invariant and sup | t | < 1 � e itA ψ � < ∞ , ψ ∈ D ( H ) , the form i (( H ψ, A ψ ) − ( A ψ, H ψ )) on D ( A ) ∩ D ( H ) is bounded below. The associate operator B is s.t. D ( B ) ⊃ D ( H ) The operator associated to i (( B ψ, A ψ ) − ( A ψ, B ψ )) , is bounded from D ( H ) to D ( H ) ∗ See Georgescu-Gerard, JFA (2004) for a details about these conditions.

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate Mourre estimate Corollary: σ sc ( H ) = ∅ So H is purely absolutely continuous and the asymptotic completeness holds.

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate elements of proof Let E ∈ ( E 1 , E 2 ) , η as above and P η the spectral projector of H 1 . First we consider the operator H 1 ( H 2 ), then choose A as the generator of dilation group, A = 1 2 ( xi ∂ x + i ∂ x x ) So a simple calculation shows that in a suitable sense P η i [ H 1 , A ] P η = − 2 P η ∂ 2 x P η = 2 EP η + 2 ( H 1 − E ) P η + 2 P η ∂ 2 y P η � � ≥ 2 E − E 1 + o ( η ) P η → a strict Mourre estimate for H 1 ( H 2 ).

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate elements of proof For the twisting model let F ∈ C ∞ ( R ) st. F ( x ) = x if | x | > 1 and F ( x ) = 0 elsewhere, in particular near { ( 0 , 0 ) , ( 0 , d ) } . Let A = 1 2 ( F ( x ) i ∂ x + i ∂ x F ( x )) So P η [ H , A ] P η = − P η ( F ′ ∂ 2 x + ∂ 2 x F ′ ) P η = 2 EP η + P η F ′ ( H 1 − E )+( H 1 − E ) F ′ ) � y P η + P η 2 E ( F ′ − 1 ) + 1 + 2 P η F ′ ∂ 2 2 F ′′′ P η � � ≥ 2 E − E 1 + o ( η ) P η + P η KP η for some compact operator K → a Mourre estimate for H .

The Dittrich-Kˇ ríž problem References Questions The spectrum Mourre estimate Thanks for your attention