  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
Schr\"odinger and wave equations associated to $H$, and Strichartz estimates
for the Schr\"odinger equation. Additionally, we deduce that the operator $H$
does not admit eigenvalues.
