On multiplicity bounds for Schrödinger eigenvalues on Riemannian surfaces

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 + υ), where υ 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 υ, where p > 1. We also discuss similar multiplicity bounds for Laplace eigenvalues on singular Riemannian surfaces.