Correlations of Nearby Levels Induced by a Random Potential

TL;DR

This paper develops a new method to analyze level correlations in Hamiltonians with a deterministic part plus a random potential, especially for finite matrices where traditional methods fail.

Contribution

It introduces a contour integral representation for correlation functions in finite matrices, enabling analysis of level correlations beyond standard orthogonal polynomial techniques.

Findings

Recovered oscillating behavior of two-level correlations at short distances

Applicable to arbitrary deterministic Hamiltonians

Provides a new analytical tool for finite matrix random potentials

Abstract

We consider a Hamiltonian $H$ which is the sum of a deterministic part $H_0$ and of a random potential $V$. For finite $N \times N$ matrices, following a method introduced by Kazakov, we derive a representation of the correlation functions in terms of contour integrals over a finite number of variables. This allows one to analyse the level correlations, whereas the standard methods of random matrix theory, such as the method of orthogonal polynomials, are not available for such cases. At short distance we recover, for an arbitrary $H_0$, an oscillating behavior for the connected two-level correlation.