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Russian Mathematical Surveys, 2022, Volume 77, Issue 1, Pages 81–97
DOI: https://doi.org/10.1070/RM9916 (Mi rm9916) 		 
This article is cited in 1 scientific paper (total in 1 paper)

Spectrum of the Laplace operator on closed surfaces

D. A. Popov

Lomonosov Moscow State University, Belozersky Research Institute of Physico-Chemical Biology
English version PDF (505 kB) Citations (1)
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DOI: https://doi.org/10.1070/RM9916
Abstract: A survey is given of classical and relatively recent results on the distribution of the eigenvalues of the Laplace operator on closed surfaces. For various classes of metrics the dependence of the behaviour of the second term in Weyl's formula on the geometry of the geodesic flow is considered. Various versions of trace formulae are presented, along with ensuing identities for the spectrum. The case of a compact Riemann surface with the Poincaré metric is considered separately, with the use of Selberg's formula. A number of results on the stochastic properties of the spectrum in connection with the theory of quantum chaos and the universality conjecture are presented.
Bibliography: 51 titles.
Keywords: spectrum, Laplace operator, Weyl's formula, geodesic flow, trace formulae, quantum chaos, universality conjecture.
Received: 10.01.2020
Russian version:
Uspekhi Matematicheskikh Nauk, 2022, Volume 77, Issue 1(463), Pages 91–108
DOI: https://doi.org/10.4213/rm9916
Bibliographic databases:
Document Type: Article
UDC: 517.984.5
MSC: Primary 58J50; Secondary 11F72
Language: English
Original paper language: Russian
Citation: D. A. Popov, “Spectrum of the Laplace operator on closed surfaces”, Uspekhi Mat. Nauk, 77:1(463) (2022), 91–108; Russian Math. Surveys, 77:1 (2022), 81–97
Citation in format AMSBIB
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    • This publication is cited in the following 1 articles:
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    References:78
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