Vol. 7, No. 6, 2014

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
On multiplicity bounds for Schrödinger eigenvalues on Riemannian surfaces

Gerasim Kokarev

Vol. 7 (2014), No. 6, 1397–1420
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 (Δg + ν), where ν is C-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 Lp-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