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Estimates of sums of zero multiplicities for eigenfunctions of the Laplace-Beltrami operator. (English. Russian original) Zbl 1149.58011
J. Math. Sci., New York 146, No. 1, 5509-5512 (2007); translation from Fundam. Prikl. Mat. 11, No. 5, 85-90 (2005).
Summary: We obtain an upper estimate $$N-\chi(M)$$ for the sum $$\mathbb Q_N$$ 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 $$M$$. Stronger estimates, but equivalent asymptotically $$(N\to\infty )$$, are given for the cases of the sphere $$S ^{2}$$ and the projective plane $$\mathbb R^{2}$$. Asymptotically sharper estimates are shown for the case of a domain on the plane.
##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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