By introducing progressively finer perturbations to a spectral problem in a controlled manner — an example of a procedure known as homogenisation — one hopes to uncover properties of the problem by exhibiting it as the limit of a family of other ones. In this work, we employ this strategy to show that the spectrum of a Schrödinger eigenvalue problem posed on a Riemannian manifold M can be approached by that of a family of Robin eigenvalue problems posed on domains in M with many small perforations.
As an application, we identify the range of Sobolev spaces in whose dual we have a flexibility result for optimal Schrödinger potentials, in which a sequence of potentials which come close to optimising some eigenvalue may nevertheless remain bounded away from an optimiser.