Abstract: A simple mathematical model for curved quantum waveguides is given by the Dirichlet Laplacian in an infinite strip-like region. The latter is usually defined as a tubular neighbourhood of constant width constructed around a reference curve in the plane. Making a thin-width limit in the Hamiltonian, one reveals an effective potential which is proportional to the minus square of the curvature of the reference curve. Such a potential represents an attractive interaction as soon as the curvature vanishes at infinity and, indeed, intensive studies inspired with this observation have proved the existence of nonempty discrete spectrum in all such quantum waveguides. There has succeeded an extensive investigation of other spectral and scattering properties.
In our talk we introduce a natural generalisation of these systems. Assuming that the quantum waveguide is produced on a curved surface, we define the quantum strip as a tubular neighbourhood of the reference curve in the surface. We introduce the corresponding Dirichlet Laplacian and derive the effective potential which now depends also on the surface curvature. We state some hypotheses by means of which we localise the essential spectrum and guarantee the existence of bound states. As an application, we establish these two spectral results for strips on surfaces of positive curvature. As a consequence, we reproduce and revise the known results for planar quantum strips, and study in more details strips on the paraboloid of revolution.