The discrete wave equation with applications to scattering theory and quantum chaos
Carsten Peterson
TL;DR
<3-5 sentence high-level summary> The paper develops a unified algebraic and harmonic-analytic framework for the discrete wave equation, starting with the one-dimensional lattice and progressing to wave equations on regular and biregular trees. It reveals explicit, Chebyshev-polynomial–driven solutions, finite propagation speed, and energy conservation, while connecting these discrete models to continuous analogues via spectral transforms and Bernstein-Szegő theory. The work then applies Lax–Phillips scattering theory to regular trees, giving explicit outgoing/incoming representations and a scattering operator, and extends the analysis to biregular trees where spherical harmonic analysis yields explicit wave propagators and a delocalization theorem for eigenfunctions on biregular graphs. Together, the results illuminate the wave dynamics on non-Euclidean combinatorial geometries and pave the way for higher-rank generalizations to affine buildings.
Abstract
With a view towards studying the multitemporal wave equation on affine buildings recently introduced by Anker-Rémy-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.