FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2005, VOLUME 11, NUMBER 5, PAGES 85-90

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

V. N. Karpushkin

Abstract

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We obtain an upper estimate $N$- c (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 $$c (M) is the Euler characteristic of $M$. There are given more strong estimates, but equivalent asymptotically ($N\; \to$¥), for the cases of the sphere $S2$ and the projective plane $$R². Asymptotically more sharp estimate are shown for the case of a domain on the plane.

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