On multiplicity bounds for Schrödinger eigenvalues on Riemannian surfaces

Kokarev, Gerasim

doi:10.2140/apde.2014.7.1397

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On multiplicity bounds for Schrödinger eigenvalues on Riemannian surfaces

Gerasim Kokarev

Vol. 7 (2014), No. 6, 1397–1420

DOI: 10.2140/apde.2014.7.1397

Abstract

A classical result by Cheng in 1976, improved later by Besson and Nadirashvili, says that the multiplicities of the eigenvalues of the Schrödinger operator $(- Δ_{g} + ν)$ , where $ν$ is $C^{\infty}$ -smooth, on a compact Riemannian surface $M$ are bounded in terms of the eigenvalue index and the genus of $M$ . We prove that these multiplicity bounds hold for an $L^{p}$ -potential $ν$ , where $p > 1$ . We also discuss similar multiplicity bounds for Laplace eigenvalues on singular Riemannian surfaces.

Keywords

Schrödinger equation, eigenvalue multiplicity, nodal set, Riemannian surface

Mathematical Subject Classification 2010

Primary: 58J50, 35P99, 35B05

Milestones

Received: 19 November 2013

Revised: 5 May 2014

Accepted: 12 July 2014

Published: 18 October 2014

Authors

Gerasim Kokarev
Mathematisches Institut Universität München Theresienstraße 39 D-80333 München Germany

Vol. 7, No. 6, 2014