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

Abstract

We consider a non-selfadjoint $h$-differential model operator $P_h$ in the semiclassical limit ($h\to 0$) 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.

46 pages, 4 figures, Commun. Math. Phys. (2016)

References in corpus (3)

• Equilibrium distribution of zeros of random polynomials
• Spectral properties of non-self-adjoint operators
• Almost sure Weyl asymptotics for non-self-adjoint elliptic operators on compact manifolds