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

We obtain an upper estimate $N-\chi(M)$ for the sum $Q_N$ of singular zero multiplicities of the $N$th eigenfunction of the Laplace–Beltrami operator on the two-dimensional, compact, connected Riemann manifold $M$, where $\chi(M)$ is the Euler characteristic of $M$. There are given more strong estimates, but equivalent asymptotically ($N\to\infty$), for the cases of the sphere $S^2$ and the projective plane $\mathbb R^2$. Asymptotically more sharp estimate are shown for the case of a domain on the plane.