Resonances on geometrically finite graphs

TL;DR

This paper studies resonances on geometrically finite regular graphs, proving meromorphic continuation of the resolvent, characterizing resonant states, and highlighting their similarities to hyperbolic manifolds with explicit computations.

Contribution

It introduces the spectral theory of resonances on geometrically finite graphs, including meromorphic continuation and geometric characterization, with explicit examples.

Findings

Resonances are finitely many and explicitly computable.

Resonances exhibit phenomena similar to hyperbolic manifolds.

Examples include algebraic curves over finite fields.

Abstract

In analogy with the spectral theory of geometrically finite hyperbolic manifolds, we initiate the study of resonances on geometrically finite (q+1)-regular graphs of groups. We prove the meromorphic continuation of the resolvent of the adjacency operator on such spaces and give a geometric characterization of the resonant states. In contrast to the hyperbolic surfaces setting, geometrically finite graphs have only finitely many resonances and may be computed explicitly, yet exhibit many of the same qualitative phenomena as in the hyperbolic manifolds setting. Particularly interesting examples arise from algebraic curves over finite fields.