The effective Hamiltonian in curved quantum waveguides under mild regularity assumptions

Abstract

The Dirichlet Laplacian in a curved three-dimensional tube built along a spatial (bounded or unbounded) curve is investigated in the limit when the uniform cross-section of the tube diminishes. Both deformations due to bending and twisting of the tube are considered. We show that the Laplacian converges in a norm-resolvent sense to the well known one-dimensional Schroedinger operator whose potential is expressed in terms of the curvature of the reference curve, the twisting angle and a constant measuring the asymmetry of the cross-section. Contrary to previous results, we allow the reference curves to have non-continuous and possibly vanishing curvature. For such curves, the distinguished Frenet frame standardly used to define the tube need not exist and, moreover, the known approaches to prove the result for unbounded tubes do not work. Our main ideas how to establish the norm-resolvent convergence under the minimal regularity assumptions are to use an alternative frame defined by a parallel transport along the curve and a refined smoothing of the curvature via the Steklov approximation.