On the negative spectrum of the hierarchical Schr\"{o}dinger operator

TL;DR

This paper investigates the spectral properties of the Schrödinger operator on Dyson's hierarchical lattice, providing explicit spectral descriptions and analyzing phase transitions from recurrence to transience.

Contribution

It offers an explicit spectral analysis of the hierarchical Schrödinger operator and explores the phase transition related to the spectral dimension.

Findings

Explicit spectrum and eigenfunctions for the unperturbed operator

Description of the resolvent and parabolic kernel

Identification of a continuous phase transition from recurrent to transient behavior

Abstract

This paper is devoted to the spectral theory of the Schr\"{o}dinger operator on the simplest fractal: Dyson's hierarchical lattice. An explicit description of the spectrum, eigenfunctions, resolvent and parabolic kernel are provided for the unperturbed operator, i.e., for the Dyson hierarchical Laplacian. Positive spectrum is studied for the perturbations of the hierarchical Laplacian. Since the spectral dimension of the operator under consideration can be an arbitrary positive number, the model allows a continuous phase transition from recurrent to transient underlying Markov process. This transition is also studied in the paper.