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Multiplicities of singularities of eigenfunctions for the Laplace-Beltrami operator. (English. Russian original) Zbl 0843.58122
Funct. Anal. Appl. 29, No. 1, 62-64 (1995); translation from Funkts. Anal. Prilozh. 29, No. 1, 80-82 (1995).
The author deals with the multiplicities of singularities of eigenfunctions for the Laplace-Beltrami operator. Let $$M$$ be a smooth compact 2-dimensional Riemann manifold. Let $$\chi(M)$$ be the Euler characteristic of the manifold $$M$$. Let $$f_N$$ be the eigenfunction of the Laplace-Beltrami operator on $$M$$ with index $$N$$. Set $$\Gamma= \{x\in M/\partial M: f_N(x)= 0\}$$. Assume that all the eigenfunctions are smooth. This is the case for $$S^2$$, $$\mathbb{R} P^2$$, and $$T^2$$. Assume that there exists a neighborhood $$U$$ of the boundary such that the set $$\Gamma \cap U$$ is the union of some continuous curves. Denote by $$m_\ell$$ the number of points $$c$$ in the boundary of the manifold $$M$$ such that in a neighborhood of the point $$c$$ the set $$\Gamma$$ consists of $$\ell$$ branches entering $$c,\ell\in [1,+ \infty)$$.
The main theorem in this paper is:
Theorem. $$\sum_{a\in \Gamma} (k(a)- 1)+ \sum m_\ell \ell/2\leq N- \chi (M)$$.
Here $$k(a)$$ is the degree of the principal homogeneous part of the function $$f_N$$ at the point $$a\in \Gamma$$. If $$M$$ is $$\mathbb{R} P^2$$ or $$S^2$$, then a sharper estimate holds (see Theorem 2 in this paper).

##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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##### References:
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