Dispersion for Schr\"odinger operators on regular trees

TL;DR

This paper establishes sharp dispersive decay estimates for Schr"odinger operators on regular trees, extending known results to new models and providing precise asymptotics.

Contribution

It proves sharp $t^{-3/2}$ decay estimates for the adjacency matrix and Schr"odinger equation on regular trees, including first-order asymptotics, extending previous work on periodic operators.

Findings

Proves $t^{-3/2}$ decay for adjacency matrix on regular trees

Establishes $t^{-3/2}$ decay for Schr"odinger equation on metric regular trees

Provides first-order asymptotics for the decay estimates

Abstract

We prove dispersive estimates for two models~: the adjacency matrix on a discrete regular tree, and the Schr\"odinger equation on a metric regular tree with the same potential on each edge/vertex. The latter model can be thought of as an extension of the case of periodic Schr\"odinger operators on the real line. We establish a $t^{-3/2}$-decay for both models which is sharp, as we give the first-order asymptotics.