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

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