Exponential splitting of bound states in a waveguide with a pair of distant windows

We consider Laplacian in a straight planar strip with Dirichlet boundary which has two Neumann “windows” of the same length the centers of which are 2l apart, and study the asymptotic behaviour of the discrete spectrum as l → ∞. It is shown that there are pairs of eigenvalues around each isolated eigenvalue of a single-window strip and their distances vanish exponentially in the limit l → ∞. We derive an asymptotic expansion also in the case where a single window gives rise to a threshold resonance which the presence of the other window turns into a single isolated eigenvalue.