On the resonances of the Laplacian on waveguides

Abstract

The resonances for the Dirichlet and Neumann Laplacian are studied on compactly perturbed waveguides. An upper bound on the number of resonances near the physical plane is proven. In the absence of resonances, an upper bound is proven for the localised resolvent. This is then used to prove that the existence of a quasimode whose asymptotics is bounded away from the thresholds implies the existence of resonances converging to the real axis.