#### Vol. 7, No. 6, 2014

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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