Scattering in twisted waveguides

Abstract: We consider a twisted quantum waveguide i.e. a domain of the form \Omega_{\theta} : = r_\theta \omega \times R, where \omega \subset R^2 is a bounded domain, and r_\theta = r_\theta(x_3) is a rotation by the angle \theta(x_3) depending on the longitudinal variable x_3. We investigate the nature of the essential spectrum of the Dirichlet Laplacian H_\theta, self-adjoint in L^2 (\Omega_\theta), and consider related scattering problems. First, we show that if the derivative of the difference \theta_1 - \theta_2 decays fast enough as |x_3| goes to infinity, then the wave operators for the operator pair (H_{\theta_1}, H_{\theta_2}) exist and are complete. Further, we concentrate on appropriate perturbations of constant twisting, i.e. \theta' = \beta - \epsilon, with constant \beta \in R, and \epsilon which decays fast enough at infinity together with its first derivative. In this case the unperturbed operator corresponding to \epsilon 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_\theta, is empty.