Quantum strips in higher dimensions

Abstract

We consider the Dirichlet Laplacian in unbounded strips on ruled surfaces in any space dimension. We locate the essential spectrum under the condition that the strip is asymptotically flat. If the Gauss curvature of the strip equals zero, we establish the existence of discrete spectrum under the condition that the curve along which the strip is built is not a geodesic. On the other hand, if it is a geodesic and the Gauss curvature is not identically equal to zero, we prove the existence of Hardy-type inequalities. We also derive an effective operator for thin strips, which enables one to replace the spectral problem for the Laplace-Beltrami operator on the two-dimensional surface by a one-dimensional Schroedinger operator whose potential is expressed in terms of curvatures. In the appendix, we establish a purely geometric fact about the existence of relatively parallel adapted frames for any curve under minimal regularity hypotheses.