The Kunz-Souillard approach to localization for Jacobi operators

TL;DR

This paper investigates spectral properties of Jacobi operators, demonstrating that certain perturbations lead to exponential localization and pure point spectra with exponentially decaying eigenfunctions, including specific examples with decaying potentials.

Contribution

It provides new results on how perturbations and decaying potentials induce localization and pure point spectra in Jacobi operators.

Findings

Perturbing diagonal coefficients causes exponential localization.

Examples of decaying potentials yield pure point spectrum.

Eigenfunctions decay exponentially in these cases.

Abstract

In this paper we study spectral properties of Jacobi operators. In particular, we prove two main results: (1) that perturbing the diagonal coefficients of Jacobi operator, in an appropriate sense, results in exponential localization, and purely pure point spectrum with exponentially decaying eigenfunctions; and (2) we present examples of decaying potentials $b_n$ such that the corresponding Jacobi operators have purely pure point spectrum.