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Homogenisation for the Robin eigenvalue problem on manifolds and flexibility of optimal Schrödinger potentials

Chia-Chun Lo

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 $M$. 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.

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 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 . 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 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 in some dual Sobolev space.

Paper Structure

This paper contains 15 sections, 29 theorems, 121 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Homogenisation for the Robin Problem
  3. Flexibility of Schrödinger potentials
  4. Background in homogenisation methods
  5. Plan of the paper
  6. Notation and conventions
  7. Measures on Riemannian manifolds
  8. Asymptotic notation
  9. Function spaces and embedding theorems
  10. Controlled homogenisation for the Robin problem
  11. Heuristics
  12. The homogenisation construction
  13. Analytic properties of perforated manifolds
  14. Approaching the Schrödinger problem in the limit
  15. Flexibility of Schrödinger potentials

Key Result

Theorem 1.4

Fix a constant $\alpha>0$. Let $M$ be a closed Riemannian manifold of dimension $d\ge2$. Let the potential $V$ on $M$ be admissible and non-negative. Then there exists a family $\{\Omega^{\varepsilon}\}_{\varepsilon>0}$ of domains $\Omega^{\varepsilon}\subseteq M$ such that for all $k\in\mathbb{N}$ as $\varepsilon\to0$.

Figures (1)

  • Figure 1: Example of the construction of a perforated domain in dimension $2$, for a potential $V$ which changes sign across a region lying in the middle of the domain. Left: the domain; within it a maximal $\varepsilon$-separated set $S^\varepsilon$ and the associated Voronoi tessellation, with the boundaries between cells indicated by dotted lines. Centre: the domain with perforations, partitioned into three sets: ---the cells associated with $S_\partial^\varepsilon$ , which intersect the boundary. ---the cells associated with $S_\ocircle^\varepsilon$, where $V$ can be close to zero. ---the cells associated with $S_\odot^\varepsilon$; these lie entirely in the interior of $\Omega$, where $V$ is bounded sufficiently far away from zero. From all such cells we remove a metric ball whose radius is given by \ref{['eqn:radius-dfn']}; the holes are proportionally larger on cells where the magnitude of $V$ is larger. Right: a single perforated cell $V^{\varepsilon,\alpha}_s$ highlighted.

Theorems & Definitions (55)

  • Definition 1.1: Admissible potentials
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Schrödinger as homogenised Robin; on a closed Riemannian manifold
  • Theorem 1.5: Schrödinger as homogenised Robin; on a Riemannian manifold with Lipschitz boundary
  • Theorem 1.6: Flexibility of Schrödinger potentials
  • Remark 1.7
  • Proposition 2.1: Sobolev multiplication theorem, I
  • Proposition 2.2: Sobolev multiplication theorem, II
  • Proposition 2.3: $\exp\mathrm{L}\cong\mathrm{L}\log\mathrm{L}^*$
  • ...and 45 more