Bounds on resonances for the Laplacian on perturbations of half-space

The resonances of the Laplacian on perturbations of half-spaces of dimensions greater than or equal to two, with either Dirichlet or Neumann boundary conditions, are studied. An upper bound for the resonance counting function is proven. If the domain has an elliptic, nondegenerate, nonglancing periodic billiard trajectory, it is shown that there exists a sequence of resonances that converge to the real axis.