Scattering on univalent graphs from $L$-function viewpoint

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.