Exponential decay for the Schr{？}dinger equation on a dissipative waveguide

We prove exponential decay for the solution of the Schr{\"o}dinger equation on a dissipative waveguide. The absorption is effective everywhere on the boundary but the geometric control condition is not satisfied. The proof relies on separation of variables and the Riesz basis property for the eigenfunctions of the transverse operator. The case where the absorption index takes negative values is also discussed.