Spectral problem on graphs and L-functions

TL;DR

This paper explores the spectral properties of generalized p+1-valent graphs, linking L-functions, spherical functions, and the Selberg trace formula, with applications to p-adic multiloop surfaces.

Contribution

It introduces a spectral framework for graphs related to p-adic surfaces, including the S-matrix, unitarity, and a trace formula connecting L-functions and graph spectra.

Findings

Established the unitarity of the S-matrix.

Proved the Hashimoto-Bass theorem relating L-functions to local operators.

Classified the discrete spectrum points via L-functions.

Abstract

The scattering process on multiloop infinite p+1-valent graphs (generalized trees) is studied. These graphs are discrete spaces being quotients of the uniform tree over free acting discrete subgroups of the projective group $PGL(2, {\bf Q}_p)$. As the homogeneous spaces, they are, in fact, identical to p-adic multiloop surfaces. The Ihara-Selberg L-function is associated with the finite subgraph-the reduced graph containing all loops of the generalized tree. We study the spectral problem on these graphs, for which we introduce the notion of spherical functions-eigenfunctions of a discrete Laplace operator acting on the graph. We define the S-matrix and prove its unitarity. We present a proof of the Hashimoto-Bass theorem expressing L-function of any finite (reduced) graph via determinant of a local operator $\Delta(u)$ acting on this graph and relate the S-matrix determinant to this L-function thus obtaining the analogue of the Selberg trace formula. The discrete spectrum points are also determined and classified by the L-function. Numerous examples of L-function calculations are presented.