Homogenisation for the Robin Eigenvalue Problem on Manifolds and Flexibility of Optimal Schrödinger Potentials

Abstract We show that the spectrum of a Schrödinger eigenvalue problem posed on a closed Riemannian manifold M with non-negative potential can be approached by that of Robin eigenvalue problems with constant positive boundary parameter posed on a sequence of domains in $$M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> . We construct these Robin problems by means of a homogenisation procedure. We show a similar result for compact manifolds with non-empty boundary and sign-indefinite potential; in this case the Robin boundary parameter can be taken to be constant on each boundary component and to have constant magnitude. As an application, we prove a flexibility result for optimal Schrödinger potentials: for certain problems where it is known that there exists some potential V which extremises some Schrödinger eigenvalue, we show that this extremal eigenvalue is also approached by the corresponding eigenvalues for a sequence of smooth potentials which remain bounded away from V in some dual Sobolev space.