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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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