The effective Hamiltonian in curved quantum waveguides and when it does not work
by David Krejcirík (Department of Theoretical Physics, Nuclear Physics Institute, Prague, Czech Republic)
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 diminishes. Both deformations due to bending and twisting 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 reference curves to have non-continuous and possibly vanishing curvature. For such curves, the distinguished Frenet frame need not exist and, moreover, the known approaches to establish the result do not work. We ask the question under which minimal regularity assumptions the effective one-dimensional approximation holds.
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. On the negative side, we construct an explicit waveguide for which the usefulness of the spectral information provided by the effective Hamiltonian is rather doubtful.