SCATTERING IN TWISTED WAVEGUIDES

Philippe Briet (1, 2, 3, 4) , Hynek Kovarik (1, 2, 3, 4) , Georgi Raikov (1, 2, 3, 4)

1 CPT - Centre de Physique Théorique - UMR 6207
2 UniBs - Università degli Studi di Brescia = University of Brescia
3 Facultad de Matemáticas [Santiago de Chile]
4 CPT - Centre de Physique Théorique - UMR 7332

We consider a twisted quantum waveguide, i.e., a domain of the form := r ! R where ! R2 is a bounded domain, and r = r (x3) is a rotation by the angle (x3) depending on the longitudinal variable x3. We investigate the nature of the essential spectrum of the Dirichlet Laplacian H , self-adjoint in L2( ), and consider related scattering problems. First, we show that if the derivative of the di erence 1 2 decays fast enough as jx3j ! 1, then the wave operators for the operator pair (H 1 ;H 2 ) exist and are complete. Further, we concentrate on appropriate perturbations of constant twisting, i.e. 0 = " with constant 2 R, and " which decays fast enough at in nity together with its rst derivative. In that case the unperturbed operator corresponding to " is an analytically bered Hamiltonian with purely absolutely continuous spectrum. Obtaining Mourre estimates with a suitable conjugate operator, we prove, in particular, that the singular continuous spectrum of H is empty.

Mots clés: Twisted waveguides; wave operators; Mourre estimates

Domaines: Théorie spectrale [math.SP]

Philippe Briet, Hynek Kovarik, Georgi Raikov. SCATTERING IN TWISTED WAVEGUIDES. Journal of Functional Analysis, 2014, 266 (1), pp.1-35. ⟨10.1016/j.jfa.2013.09.026⟩. ⟨hal-00916807⟩