On the resonances of the Laplacian on waveguides

The resonances for the Dirichlet and Neumann Laplacian are studied on compactly perturbed waveguides. 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. The following upper bound to the number of resonances is also proven: # kj ∈ Res(Δ), dist(kj , physical plane) < 1 + |kj |/2, |kj | < r