Scattering in twisted waveguides

Abstract We consider a twisted quantum waveguide, i.e. a domain of the form Ω θ : = r θ ω × R where ω ⊂ R 2 is a bounded domain, and r θ = r θ ( x 3 ) is a rotation by the angle θ ( x 3 ) depending on the longitudinal variable x 3 . We investigate the nature of the essential spectrum of the Dirichlet Laplacian H θ , 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 | x 3 | → ∞ , then the wave operators for the operator pair ( H θ 1 , H θ 2 ) exist and are complete. Further, we concentrate on appropriate perturbations of constant twisting, i.e. θ ′ = β − e with constant β ∈ R , and e which decays fast enough at infinity together with its first derivative. In that case the unperturbed operator corresponding to e 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.