On the spectrum of the hierarchical Schr\"{o}dinger type operators

TL;DR

This paper conducts a spectral analysis of a Schr"{o}dinger type operator on the Dyson hierarchical lattice, revealing the structure of its spectrum, effects of sparse and random potentials, and providing asymptotic and localization results.

Contribution

It introduces a detailed spectral analysis of Schr"{o}dinger operators with hierarchical structure, including sparse and random potentials, and derives new asymptotic and localization results.

Findings

Discrete spectrum can contain negative energies.

Spectral gaps of the base operator can host eigenvalues.

Precise asymptotics for sparse potentials with rapidly increasing distances.

Abstract

The goal of this paper is the spectral analysis of the Schr\"{o}dinger type operator $H=L+V$, the perturbation of the Taibleson-Vladimirov multiplier $L=\mathfrak{D}^{\alpha}$ by a potential $V$. Assuming that $V$ belongs to a certain class of potentials we show that the discrete part of the spectrum of $H$ may contain negative energies, it also appears in the spectral gaps of $L$. We will split the spectrum of $H$ in two parts: high energy part containing eigenvalues which correspond to the eigenfunctions located on the support of the potential $V,$ and low energy part which lies in the spectrum of certain bounded Schr\"{o}dinger-type operator acting on the Dyson hierarchical lattice. We pay special attention to the class of sparse potentials. In this case we obtain precise spectral asymptotics for $H$ provided the sequence of distances between locations tends to infinity fast enough. We also obtain certain results concerning localization theory for $H$ subject to (non-ergodic) random potential $V$. Examples illustrate our approach.