On multiplicity bounds for Schrödinger eigenvalues on Riemannian surfaces

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.