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 TMF, 1995, Volume 103, Number 3, Pages 489–506 (Mi tmf1318)

Scattering on univalent graphs from $L$-function viewpoint

L. O. Chekhov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The scattering process on multiloop infinite $p+1$-valent graphs is studied. These graphs are discrete spaces of a constant negative curvature being quotients of $p$-adic hyperbolic plane over free acting discrete subgroups of the projective group $PGL(2, {\mathbf Q}_p)$. They are, in fact, identical to $p$-adic multiloop surfaces. A finite subgraph containing all loops is called the reduced graph $T_{\scriptsize red}$, $L$-function is associated with this finite subgraph. For an infinite graph, we introduce the notion of spherical functions. They are eigenfunctions of a discrete Laplace operator acting on the graph. In scattering processes we define $s$-matrix and the scattering amplitudes $c_i$ imposing the restriction $c_i=A_{ret}(u)/A_{adv}(u)=\hbox {const}$ for all vertices $u\in T_{\scriptsize supp}$. $A_{ret}$ and $A_{adv}$ are retarded and advanced branches of a solution to eigenfucntion problem and $T_{\scriptsize supp}$ is a support domain for scattering centers. Taking the product over all $c_i$, we obtain the determinant of the scattering matrix which is expressed as a ratio of two $L$–functions: $C\sim L(\alpha _+)/L(\alpha _-)$. Here $L$–function is the Ihara–Selberg function depending only on the form of $T_{\scriptsize red}$, $\alpha _\pm =t/2p\pm \sqrt {t^2/4p^2-1/p}$, $t-p-1$ being the eigenvalue of the Laplacian. We present a proof of the Hashimoto–Bass theorem expressing $L$–function $L(u)$ of any finite graph via determinant of a local operator $\Delta (u)$ acting on this graph. Numerous examples of $L$-function calculations are presented.

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English version:
Theoretical and Mathematical Physics, 1995, 103:3, 723–737

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Document Type: Article
Language: English

Citation: L. O. Chekhov, “Scattering on univalent graphs from $L$-function viewpoint”, TMF, 103:3 (1995), 489–506; Theoret. and Math. Phys., 103:3 (1995), 723–737

Citation in format AMSBIB
\Bibitem{Che95} \by L.~O.~Chekhov \paper Scattering on univalent graphs from $L$-function viewpoint \jour TMF \yr 1995 \vol 103 \issue 3 \pages 489--506 \mathnet{http://mi.mathnet.ru/tmf1318} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1472314} \zmath{https://zbmath.org/?q=an:0978.11047} \transl \jour Theoret. and Math. Phys. \yr 1995 \vol 103 \issue 3 \pages 723--737 \crossref{https://doi.org/10.1007/BF02065871} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995TP54200011} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. L. O. Chekhov, “A spectral problem on graphs and $L$-functions”, Russian Math. Surveys, 54:6 (1999), 1197–1232
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