We consider the Laplacian in a strip R×(0,d) with the boundary condition which is Dirichlet except at the segment of a length 2a of one of the boundaries where it is switched to Neumann. This operator is known to have a non-empty and simple discrete spectrum for any a>0. There is a sequence 0<a1<a2<⋯ of critical values at which new eigenvalues emerge from the continuum when the Neumann window expands. We find the asymptotic behavior of these eigenvalues around the thresholds showing that the gap is in the leading order proportional to (a−an)2 with an explicit coefficient expressed in terms of the corresponding threshold-energy resonance eigenfunction.

We consider the Laplace operator with Dirichlet boundary conditions on a planar domain and study the effect that performing a scaling in one direction has on the spectrum. We derive the asymptotic expansion for the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This method allows us, for instance, to obtain an approximation for the first Dirichlet eigenvalue for a large class of planar domains, under very mild assumptions.

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 $l\to\infty$. 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 $l\to\infty$. 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.