Bound states in curved quantum waveguides

Abstract

A free quantum particle living on a curved planar strip Ω of a fixed width d with Dirichlet boundary conditions is studied. It can serve as a model for electrons in thin films on a cylinder‐type substrate, or in a curved quantum wire. Assuming that the boundary of Ω is infinitely smooth and its curvature decays fast enough at infinity, it is proved that a bound state with energy below the first transversal mode exists for all sufficiently small d. A lower bound on the critical width is obtained using the Birman–Schwinger technique.