ESTIMATES OF SUMS OF ZERO MULTIPLICITIES FOR EIGENFUNCTIONS OF THE LAPLACE-BELTRAMI OPERATOR

摘要

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