Shifts of Random Energy Levels by a Local Perturbation

TL;DR

This paper analyzes how a local perturbation impacts the energy levels in systems modeled by random matrix theory, providing explicit formulas for joint distributions and correlations.

Contribution

It introduces an analytic expression for the joint distribution of energy levels before and after perturbation, including correlation functions for the unitary ensemble.

Findings

Derived the joint distribution function of energy levels

Obtained the two-point correlation function for the unitary ensemble

Provided insights into the spectral shifts caused by local perturbations

Abstract

We consider the effect of a local perturbation on the energy levels of a system described by random matrix theory. An analytic expression for the joint distribution function of initial and final energy levels is obtained. In the case of unitary ensemble we also find the two-point correlation function of initial and final densities of states.