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A discreteness criterion for the spectrum of the Laplace–Beltrami operator on quasimodel manifolds. (Russian, English) Zbl 1018.58022
Sib. Mat. Zh. 43, No. 6, 1362-1371 (2002); translation in Sib. Math. J. 43, No. 6, 1103-1111 (2002).
The article is devoted to studying the spectrum of the Laplace-Beltrami operator on noncompact Riemannian manifolds of a special form, in particular, on model manifolds.
The author investigates the dependence of the spectrum of the Laplace-Beltrami operator on the metric of a manifold. A complete noncompact Riemannian manifold $$M$$ without boundary is considered which is representable as $$B\cup D$$, where $$B$$ is a compact set and $$D$$ is isometric to the product $$\mathbb R_+\times S_1\times S_2\times\dots\times S_k$$ ($$S_i$$ are compact Riemannian manifolds without boundary) with the metric $$ds^2$$. The author is interested in the case when $$M$$ is endowed with some Borel measure $$\mu$$ not necessarily coincident with the Riemannian volume. In this case the pair $$(M,\mu)$$ is called a weighted manifold.
The main result of the article is a discreteness criterion for the spectrum of the Laplace-Beltrami operator on a weighted manifold $$M$$ in terms of the volume and capacity of some domains on a weighted manifold $$M$$.

MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 58J05 Elliptic equations on manifolds, general theory 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 35P05 General topics in linear spectral theory for PDEs
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