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 operator $H=-{\Delta }_x-{\Delta }_y+V(x,y)$ with Dirichled boundary condition on an unbounded domain $\Omega$, and we introduce the notion of a \emph{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.

22 pages, 4 figures