The discrete wave equation with applications to scattering theory and quantum chaos

TL;DR

This paper develops the theory of the discrete wave equation on graphs, computes scattering operators explicitly, and extends eigenfunction delocalization results from regular to biregular graphs, with applications to quantum chaos.

Contribution

It systematically analyzes the discrete wave equation on graphs, explicitly computes scattering operators, and extends eigenfunction delocalization results to biregular graphs.

Findings

Explicit scattering operators for regular and biregular trees.

Extension of eigenfunction delocalization to biregular graphs.

Foundational properties of the discrete wave equation on graphs.

Abstract

With a view towards studying the multitemporal wave equation on affine buildings recently introduced by Anker-R\'emy-Trojan [arXiv:2312.06860], we systematically develop the basic properties of the discrete wave equation on $\mathbb{Z}$ and use this to explain existing results about the wave equation on regular graphs. Furthermore, we explicitly compute the incoming and outgoing translation representations and the scattering operator, in the sense of Lax-Phillips, for regular and biregular trees. Finally, we use the wave equation on biregular graphs to extend a result of Brooks-Lindenstrauss about delocalization of eigenfunctions on regular graphs to the setting of biregular graphs.