Quantum strips on surfaces

Motivated by the theory of quantum waveguides, we investigate the spectrum of the Laplacian, subject to Dirichlet boundary conditions, in a curved strip of constant width that is defined as a tubular neighbourhood of an infinite curve in a two-dimensional Riemannian manifold. Under the assumption that the strip is asymptotically straight in a suitable sense, we localise the essential spectrum and find sufficient conditions which guarantee the existence of geometrically induced bound states. In particular, the discrete spectrum exists for non-negatively curved strips which are studied in detail. The general results are used to recover and revisit the known facts about quantum strips in the plane. As an example of non-positively curved quantum strips, we consider strips on ruled surfaces.