Spectral statistics of random Schr\"{o}dinger operator with growing potential

TL;DR

This paper studies the spectral behavior of a class of random Schrödinger operators with potentials that grow with distance, revealing how the spectral statistics depend on the growth rate parameter.

Contribution

It introduces a model of random Schrödinger operators with growing potential and analyzes their spectral statistics, a novel approach in the study of disordered quantum systems.

Findings

Spectral statistics vary with the growth parameter α.

The model exhibits different spectral regimes depending on α.

New insights into localization and delocalization phenomena.

Abstract

In this work we investigate the spectral statistics of random Schr\"{o}dinger operators $H^\omega=-\Delta+\sum_{n\in\mathbb{Z}^d}(1+|n|^\alpha)q_n(\omega)|\delta_n\rangle\langle\delta_n|$, $\alpha>0$ acting on $\ell^2(\mathbb{Z}^d)$ where $\{q_n\}_{n\in\mathbb{Z}^d}$ are i.i.d random variables distributed uniformly on $[0,1]$.