Vol. 7, No. 6, 2014

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
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 $\left(-{\Delta }_{g}+\nu \right)$, where $\nu$ 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 $\nu$, 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