Spectral problem on graphs and L-functions

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,Qp). 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 ∆(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. E-mail: [email protected]