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 Izv. Akad. Nauk SSSR Ser. Mat., 1978, Volume 42, Issue 3, Pages 484–499 (Mi izv1776)

Selberg's trace formula for the Hecke operator generated by an involution, and the eigenvalues of the Laplace–Beltrami operator on the fundamental domain of the modular group $PSL(2,\mathbf Z)$

A. B. Venkov

Abstract: In this paper a derivation is given of a generalized Selberg trace formula corresponding to the odd eigenfunctions of the Laplace–Beltrami operator in the space $L_2(\Gamma\setminus H)$, where the discrete group $\Gamma$ is $\Gamma=PSL(2,\mathbf Z)$ and $H$ is the upper halfplane (the Dirichlet problem on half of the fundamental domain). As an application a generalization is obtained of Minakshisundaram's formula:
$$\int_0^\infty e^{-t\lambda} d\alpha(\lambda)=\frac1t\cdot\frac1{24}+\frac{\ln t}{\sqrt t}\cdot\frac1{8\sqrt\pi}+\frac1{\sqrt t}\cdot\frac1{8\sqrt\pi}(\mathbf C-\ln2)+O_{t\to0,t>0}$$
($\alpha(\lambda)$ is the corresponding spectral density; $\mathbf C$ is Euler's constant) and also an asymptotic formula characterizing the irregularity of the distribution of the eigenvalues. Similar results are also obtained for all the eigenvalues of the discrete spectrum of the Laplace–Beltrami operator in the space $L_2(\Gamma\setminus H)$ when $\Gamma$ is the indicated group.
Bibliography: 18 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1978, 12:3, 448–462

Bibliographic databases:

UDC: 517.43+519.4+511.3
MSC: Primary 10D05; Secondary 35J05

Citation: A. B. Venkov, “Selberg's trace formula for the Hecke operator generated by an involution, and the eigenvalues of the Laplace–Beltrami operator on the fundamental domain of the modular group $PSL(2,\mathbf Z)$”, Izv. Akad. Nauk SSSR Ser. Mat., 42:3 (1978), 484–499; Math. USSR-Izv., 12:3 (1978), 448–462

Citation in format AMSBIB
\Bibitem{Ven78} \by A.~B.~Venkov \paper Selberg's trace formula for the Hecke operator generated by an involution, and the eigenvalues of the Laplace--Beltrami operator on the fundamental domain of the modular group~$PSL(2,\mathbf Z)$ \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1978 \vol 42 \issue 3 \pages 484--499 \mathnet{http://mi.mathnet.ru/izv1776} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=480349} \zmath{https://zbmath.org/?q=an:0392.43015|0416.43010} \transl \jour Math. USSR-Izv. \yr 1978 \vol 12 \issue 3 \pages 448--462 \crossref{https://doi.org/10.1070/IM1978v012n03ABEH001991} 

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This publication is cited in the following articles:
1. A. B. Venkov, “Spectral theory of automorphic functions, the Selberg zeta-function, and some problems of analytic number theory and mathematical physics”, Russian Math. Surveys, 34:3 (1979), 79–153
2. N.L. Balazs, A. Voros, “Chaos on the pseudosphere”, Physics Reports, 143:3 (1986), 109
3. C. Matthies, F. Steiner, “Selberg’s ζ function and the quantization of chaos”, Phys Rev A, 44:12 (1991), R7877
4. Robert Graham, Ralph Hübner, Péter Szépfalusy, “Level statistics of a noncompact integrable billiard”, Phys Rev A, 44:11 (1991), 7002
5. R. Aurich, C. Matthies, M. Sieber, F. Steiner, “Novel rule for quantizing chaos”, Phys Rev Letters, 68:11 (1992), 1629
6. J. Bolte, F. Steiner, “The Selberg trace formula for bordered Riemann surfaces”, Comm Math Phys, 156:1 (1993), 1
7. E. B. Bogomolny, B. Georgeot, M.-J. Giannoni, C. Schmit, “Trace formulas for arithmetical systems”, Phys Rev E, 47:4 (1993), R2217
8. R. Aurich, J. Bolte, C. Matthies, M. Sieber, F. Steiner, “Crossing the entropy barrier of dynamical zeta functions”, Physica D: Nonlinear Phenomena, 63:1-2 (1993), 71
9. D. A. Popov, “On the Selberg Trace Formula for Strictly Hyperbolic Groups”, Funct. Anal. Appl., 47:4 (2013), 290–301
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