A system of Schrödinger equations in a wave guide

We consider a system of Schrödinger equations in a wave guide, with coupling and damping at the boundary. It is related to the same problem on the one-dimensional cross-section. We prove in particular that we have a spectral gap and exponential decay of the energies for these problems. For the transverse problem we also study the localization of the eigenvalues and prove that the corresponding generalized eigenfunctions form a Riesz basis.