Poisson kernel expansions for Schr\"odinger operators on trees

TL;DR

This paper develops Poisson kernel expansions for Schrödinger operators on trees, enabling a Fourier transform framework that relates eigenfunctions to the boundary and spectrum of the operator.

Contribution

It introduces a novel Poisson kernel construction for Schrödinger operators on trees and establishes a Fourier analysis framework analogous to classical harmonic analysis.

Findings

Eigenfunctions generated by the Poisson kernel in the absolutely continuous spectrum

A Fourier inversion formula for Schrödinger operators on trees

A Plancherel formula linking spectrum and boundary measures

Abstract

We study Schr\"odinger operators on trees and construct associated Poisson kernels, in analogy to the laplacian on the unit disc. We show that in the absolutely continuous spectrum, the generalized eigenfunctions of the operator are generated by the Poisson kernel. We use this to define a "Fourier transform", giving a Fourier inversion formula and a Plancherel formula, where the domain of integration runs over the energy parameter and the geometric boundary of the tree.