Geometrically Induced Spectrum in Curved Leaky Wires
Abstract
Introduction The aim of the present paper is to elucidate some geometrically induced spectral properties for the Laplacian in L 2 (R 2 ) perturbed by a negative multiple of the Dirac measure of an infinite curve \\Gamma in the plane. This problem has at least two motivations. On the physics side we note that quantum mechanics of electrons confined to narrow tubelike regions has attracted a considerable interest, because such systems represent a natural model for semiconductor "quantum wires". In some examples the region in question is a strip or tube with hard walls -- see, e.g., [DE] and references therein -- while other treatments assume even stronger localization to a curve 1 or a graph -- a rich bibliography to such models can be found in [KS]. Various interesting spectral effects were found in such a setting related to the geometry and topology of the underlying restricted configuration space. One of th