Evolution Equations on Non-Flat Waveguides

We investigate the dispersive properties of evolution equations on waveguides with a non-flat shape. More precisely, we consider an operator H=-Delta(x)-Delta(y)+V(x,y) with Dirichlet boundary conditions 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 Schrodinger and wave equations associated to H, and Strichartz estimates for the Schrodinger equation. Additionally, we deduce that the operator H does not admit eigenvalues.