The Prime Geodesic Theorem and Quantum Mechanics on Finite Volume Graphs: A Review

Abstract

The prime geodesic theorem is reviewed for compact and finite volume Riemann surfaces and for finite and finite volume graphs. The methodology of how these results follow from the theory of the Selberg zeta function and the Selberg trace formula is outlined. Relationships to work on quantum graphs are surveyed. Extensions to compact Riemannian manifolds, in particular to three-dimensional hyperbolic spaces, are noted. Interconnections to the Selberg eigenvalue conjecture, the Ramanujan conjecture and Ramanujan graphs are developed.

Quantum chaos on discrete graphs

“…where |C(n)| is the cardinality of the set C(n), and u ∈ C with |u| sufficiently small to ensure the convergence of the infinite product. Following Ihara's original work, several authors (see e.g., [11] for a survey of the methods) have proved that…”

A Generalization of the prime geodesic theorem to counting conjugacy classes of free subgroups

“…This was first proven by [Huber, 1961] (in the n = 2 case and using Selberg's trace formula) and subsequently generalized by Margulis in his thesis, [Parry and Pollicott, 1983], [Knieper, 1997], [Gunesch, 2005] and others. [Hurt, 2001] and Sharp (see [Margulis, 2004]) have written two recent surveys.…”