2011/09/09 by Briet, Philippe, Kovarik, Hynek, Raikov, Georgi
#35P05 #35P25 #47A10 #81Q10 #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1109.2111
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| goes to infinity, 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 εwhich decays fast enough at infinity together with its first derivative. In this case the unperturbed operator corresponding to εis 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.