L functions in scattering on p-adic multiloop surfaces

Abstract

The scattering processes on multiloop infinite graphs with one and the same valence p+1 of all vertices are studied herein. These graphs are discrete spaces of a constant negative curvature since they can be interpreted as quotients of the p-adic hyperbolic plane over free acting discrete subgroups of the projective group PGL(2,Qp). They are, in fact, identical to p-adic multiloop surfaces. Releasing from a graph its subgraph containing all loops, which is called the reduced graph Tred, we can get L functions corresponding to these finite closed graphs. For an infinite graph, the notion of spherical functions is introduced. These are the eigenfunctions of a discrete Laplace operator acting on the graph. s-wave scattering processes are considered and in this way they define the scattering amplitudes ci. In the discrete case, their number coincides with ‖Tred‖—the number of vertices of the reduced graph. Taking the product over all ci, we obtain the determinant of the scattering matrix which turns out to be presented as a ratio of two L functions: C∼L(α+)/L(α−). Here the L function is the Ihara–Selberg function depending only on the form of Tred. All the dependence on the initial value of p remains in arguments α± = t/2p ±√t2/4p2−1/p, t being the eigenvalue of the Laplacian. The Hashimoto–Bass theorem is used for expressing L function L(u) of any finite graph via the determinant of a local operator Δ(u) acting on this graph.