On the Joint Distribution of Energy Levels of Random Schroedinger Operators

TL;DR

This paper establishes a new probabilistic bound on the simultaneous occurrence of localized eigenvalues in random Schrödinger operators on graphs, extending existing estimates through a novel spectral averaging approach.

Contribution

It introduces a generalized bound on joint eigenvalue probabilities for localized states, expanding the theoretical understanding of spectral properties in random operators.

Findings

Derived a bound extending Wegner estimates

Introduced a new multiparameter spectral averaging principle

Provided insights into eigenvalue localization probabilities

Abstract

We consider operators with random potentials on graphs, such as the lattice version of the random Schroedinger operator. The main result is a general bound on the probabilities of simultaneous occurrence of eigenvalues in specified distinct intervals, with the corresponding eigenfunctions being separately localized within prescribed regions. The bound generalizes the Wegner estimate on the density of states. The analysis proceeds through a new multiparameter spectral averaging principle.