Evolution equations on non flat waveguides

Abstract

We investigate the dispersive properties of evolution equations on waveguides with a non flat shape. More precisely we consider an operatorH=-Delta_{x}-Delta_{y}+V(x,y)with Dirichled boundary condition on an unbounded domain Omega, and we introduce the notion of a repulsive waveguide along the direction of the first group of variables x. If Omega is a repulsive waveguide, we prove a sharp estimate for the Helmholtz equation Hu-lambda u=f. As consequences we prove smoothing estimates for the Schrödinger and wave equations associated to H, and Strichartz estimates for the Schrödinger equation. Additionally, we deduce that the operator H does not admit eigenvalues.