Scattering in twisted waveguides

摘要

We consider a twisted quantum waveguide, i.e. a domain of the form Ω_θ:= rθω × ? where ω C ?~2 is a bounded domain, and r_θ = r_θ(x3) is a rotation by the angle θ(χ_3) depending on the longitudinal variable x3. We investigate the nature of the essential spectrum of the Dirichlet Laplacian Hθ, self-adjoint in L~2(Ω_θ), and consider related scattering problems. First, we show that if the derivative of the difference θ_1 - θ_2 decays fast enough as |x3| →∞, then the wave operators for the operator pair (H_(θ1), H_(θ2)) exist and are complete. Further, we concentrate on appropriate perturbations of constant twisting, i.e. θ' = β - ε with constant β ? ?, and ε which decays fast enough at infinity together with its first derivative. In that case the unperturbed operator corresponding to ε is an analytically fibered 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_θ is empty.