Scattering in twisted waveguides

We consider a twisted quantum waveguide, i.e. a domain of the form Ωθ:=rθω×R where ω⊂R2 is a bounded domain, and rθ=rθ(x3) is a rotation by the angle θ(x3) depending on the longitudinal variable x3. We investigate the nature of the essential spectrum of the Dirichlet Laplacian Hθ, self-adjoint in L2(Ωθ), and consider related scattering problems. First, we show that if the derivative of the difference θ1−θ2 decays fast enough as |x3|→∞, then the wave operators for the operator pair (Hθ1,Hθ2) exist and are complete. Further, we concentrate 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 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 Hθ is empty.