Two point eigenvalue correlation for a class of non-selfadjoint operators under random perturbations

TL;DR

This paper analyzes the eigenvalue correlations of a class of non-selfadjoint semiclassical operators under small random perturbations, revealing eigenvalue repulsion at close range and decoupling at long range.

Contribution

It provides an asymptotic formula for the two-point eigenvalue density, demonstrating eigenvalue repulsion and decoupling phenomena in the semiclassical limit.

Findings

Eigenvalues exhibit close range repulsion.

Eigenvalues show long range decoupling.

Derived an asymptotic formula for two-point eigenvalue density.

Abstract

We consider a non-selfadjoint $h$-differential model operator $P_h$ in the semiclassical limit ($h\rightarrow 0$) subject to random perturbations with a small coupling constant $\delta$. Assume that $\exp(-\frac{1}{Ch}) < \delta \ll h^{\kappa}$ for constants $C,\kappa>0$ suitably large. Let $\Sigma$ be the closure of the range of the principal symbol. We study the $2$-point intensity measure of the random point process of eigenvalues of the randomly perturbed operator $P_h^{\delta}$ and prove an $h$-asymptotic formula for the average $2$-point density of eigenvalues. With this we show that two eigenvalues of $P_h^{\delta}$ in the interior of $\Sigma$ exhibit close range repulsion and long range decoupling.