Hardy Inequalities in Strips on Ruled Surfaces

Problems linking the geometry of two-dimensional manifolds and the spectrum of associated Laplacians have been considered for more than a century. While classical motivations come from theories of elasticity and electromagnetism, the same rather simple models can be also remarkably successful in describing even rather complicated phenomena in quantum heterostructures. Here, an enormous amount of recent research has been undertaken on both the theoretical and experimental aspects of binding in curved striplike waveguide systems. More specifically, as a result of theoretical studies, it is well known now that the Dirichlet Laplacian in an infinite planar strip of uniform width always possesses eigenvalues below its essential spectrum whenever the strip is curved and asymptotically straight. We refer to [13, 15] for initial proofs and to [8, 19, 21] for reviews with many references on the topic. The existence of the curvature-induced bound states is interesting from several respects. First of all, one deals with a purely quantum effect of geometrical origin, with negative consequences for the electronic transport in nanostructures. From the mathematical point of view, the strips represent a class of noncompact noncomplete manifolds for which the spectral results of this type are nontrivial, too.