Work

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

Year: 2025

Type: article

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 $$M$$ 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 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.

Source: Journal of Geometric Analysis

DOI: 10.1007/s12220-025-02188-w

Author: Chia-Chun Lo

Institution: King's College London

Language: English

Topic: Spectral Theory in Mathematical Physics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,290

Funder: King's College London