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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 $T_{\scriptsize red}$, $\alpha _\pm =t/2p\pm \sqrt {t^2/4p^21/p}$, $tp1$ 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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Theoretical and Mathematical Physics, 1995, 103:3, 723–737
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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
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\by L.~O.~Chekhov
\paper Scattering on univalent graphs from $L$function viewpoint
\jour TMF
\yr 1995
\vol 103
\issue 3
\pages 489506
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\mathscinet{http://www.ams.org/mathscinetgetitem?mr=1472314}
\zmath{https://zbmath.org/?q=an:0978.11047}
\transl
\jour Theoret. and Math. Phys.
\yr 1995
\vol 103
\issue 3
\pages 723737
\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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This publication is cited in the following articles:

L. O. Chekhov, “A spectral problem on graphs and $L$functions”, Russian Math. Surveys, 54:6 (1999), 1197–1232

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