Estimates of sums of zero multiplicities for eigenfunctions of the Laplace-Beltrami operator

Abstract

We obtain an upper estimate N−χ(M) for the sum Q N of singular zero multiplicities of the Nth eigenfunction of the Laplace-Beltrami operator on the two-dimensional, compact, connected Riemann manifold M, where χ M is the Euler characteristic ofM. Stronger estimates, but equivalent asymptotically (N a ∞), are given for the cases of the sphere S 2 and the projective plane ℝ2. Asymptotically sharper estimates are shown for the case of a domain on the plane.