Poisson kernel expansions for Schrödinger operators on trees

Abstract

We study Schrödinger 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. R

Quantum ergodicity on graphs: From spectral to spatial delocalization

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“…A closely related result on Poisson transformations for general trees has recently been obtained by Anantharaman and Sabri [AS19]. Their perspective, however, is complementary to ours: They start with a given graph and a Schrödinger operator and construct a suitable Poisson kernel in terms of Green's functions.…”

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