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 = −∆x −∆y + V (x, y) with Dirichled boundary condition on an unbounded domain Ω, and we introduce the notion of a repulsive waveguide along the direction of the first group of variables x. If Ω is a repulsive waveguide, we prove a sharp estimate for the Helmholtz equation Hu − λu = 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.