A spectral problem on graphs and L-functions
Abstract
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.
- Publication:
-
Russian Mathematical Surveys
- Pub Date:
- December 1999
- DOI:
- 10.1070/RM1999v054n06ABEH000231
- Bibcode:
- 1999RuMaS..54.1197C