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 Uspekhi Mat. Nauk, 1999, Volume 54, Issue 6(330), Pages 109–148 (Mi umn231)

A spectral problem on graphs and $L$-functions

L. O. Chekhov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: This paper is concerned with a scattering process on multiloop infinite $(p+1)$-valent graphs (generalized trees). These graphs are one-dimensional connected simplicial complexes that are quotients of a regular tree with respect to free actions of discrete subgroups of the projective group $PGL(2,\mathbb Q_p)$. As homogeneous spaces, they are identical to $p$-adic multiloop surfaces. The Ihara–Selberg $L$-function is associated with a finite subgraph, namely, the reduced graph containing all loops of the generalized tree. We study a spectral problem and introduce spherical functions as the eigenfunctions of a discrete Laplace operator acting on the corresponding graph. We define the $S$-matrix and prove that it is unitary. We present a proof of the Hashimoto–Bass theorem expressing the $L$-function of any finite (reduced) graph in terms of the determinant of a local operator $\Delta (u)$ acting on this graph and express the determinant of the $S$-matrix as a ratio of $L$-functions, thus obtaining an analogue of the Selberg trace formula. The points of the discrete spectrum are also determined and classified using the $L$-function. We give a number of examples of calculations of $L$-functions.

DOI: https://doi.org/10.4213/rm231

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English version:
Russian Mathematical Surveys, 1999, 54:6, 1197–1232

Bibliographic databases:

Document Type: Article
MSC: Primary 11F72, 11M06, 11M41, 20E08, 05C05, 11R42, 11S40; Secondary 58G25, 33C55, 35J05, 81U20

Citation: L. O. Chekhov, “A spectral problem on graphs and $L$-functions”, Uspekhi Mat. Nauk, 54:6(330) (1999), 109–148; Russian Math. Surveys, 54:6 (1999), 1197–1232

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/umn231
• https://doi.org/10.4213/rm231
• http://mi.mathnet.ru/eng/umn/v54/i6/p109

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This publication is cited in the following articles:
1. L. O. Chekhov, N. V. Puzyrnikova, “Integrable systems on graphs”, Russian Math. Surveys, 55:5 (2000), 992–994
2. Phys. Usp., 44:4 (2001), 424–427
3. L. O. Chekhov, “Integrable deformations of systems on graphs with loops”, Russian Math. Surveys, 57:3 (2002), 587–588
4. Alain Comtet, Jean Desbois, Christophe Texier, “Functionals of Brownian motion, localization and metric graphs”, J Phys A Math Gen, 38:37 (2005), R341
5. Christophe Texier, “ζ-regularized spectral determinants on metric graphs”, J Phys A Math Theor, 43:42 (2010), 425203
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