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 \(M\).

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 \(M\).

Reviewer: V.Grebenev (Novosibirsk)

##### 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 |