A spectral problem on graphs and $L$-functions

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.