Homogenisation for the Robin eigenvalue problem on manifolds and flexibility of optimal Schr\"odinger potentials

TL;DR

This paper demonstrates that the spectrum of Schr"odinger eigenvalue problems on manifolds can be approximated by Robin eigenvalue problems through homogenisation, and explores the flexibility of optimal potentials in this context.

Contribution

It introduces a homogenisation approach to approximate Schr"odinger spectra with Robin problems on manifolds and shows the flexibility of extremal potentials in eigenvalue optimization.

Findings

Robin eigenvalues approximate Schr"odinger eigenvalues via homogenisation.

Optimal potentials can be approached by smooth potentials in dual Sobolev spaces.

Results apply to manifolds with boundary and sign-indefinite potentials.

Abstract

We show that the spectrum of a Schr\"odinger 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 $M$. 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 Schr\"odinger potentials: for certain problems where it is known that there exists some potential $V$ which extremises some Schr\"odinger 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.