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

This article is cited in 9 scientific papers (total in 9 papers)

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:
\begin{equation} \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} \end{equation}
($\alpha(\lambda)$ 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
Received: 26.01.1977

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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    Citing articles on Google Scholar: Russian citations, English citations
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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  mathnet  crossref  mathscinet  zmath
    2. N.L. Balazs, A. Voros, “Chaos on the pseudosphere”, Physics Reports, 143:3 (1986), 109  crossref
    3. C. Matthies, F. Steiner, “Selberg’s ζ function and the quantization of chaos”, Phys Rev A, 44:12 (1991), R7877  crossref  mathscinet  adsnasa  isi
    4. Robert Graham, Ralph Hübner, Péter Szépfalusy, “Level statistics of a noncompact integrable billiard”, Phys Rev A, 44:11 (1991), 7002  crossref  mathscinet  isi
    5. R. Aurich, C. Matthies, M. Sieber, F. Steiner, “Novel rule for quantizing chaos”, Phys Rev Letters, 68:11 (1992), 1629  crossref  mathscinet  zmath  adsnasa
    6. J. Bolte, F. Steiner, “The Selberg trace formula for bordered Riemann surfaces”, Comm Math Phys, 156:1 (1993), 1  crossref  mathscinet  zmath  adsnasa
    7. E. B. Bogomolny, B. Georgeot, M.-J. Giannoni, C. Schmit, “Trace formulas for arithmetical systems”, Phys Rev E, 47:4 (1993), R2217  crossref  mathscinet  zmath  adsnasa  isi
    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  crossref
    9. D. A. Popov, “On the Selberg Trace Formula for Strictly Hyperbolic Groups”, Funct. Anal. Appl., 47:4 (2013), 290–301  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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