On estimates for the discrete eigenvalues of two-dimensional quantum waveguides

Abstract

In this paper, we give upper estimates for the number and sum of discrete eigenvalues below the bottom of the essential spectrum counting multiplicities of quantum waveguides in two dimensions. We consider both straight and curved waveguides of constant width, and the estimates are presented in terms of norms of the potential. For the curved quantum waveguide, we assume that the waveguide is not self-intersecting and its curvature is a continuous and bounded function on R. The estimates are new, particularly for the case of curved quantum waveguides.