Eigenvalue-counting correlation asymptotics for white-noise Schrödinger operators

Let $N(\lambda)$ be the eigenvalue counting function of the Schrödinger operator considered in the paper, and define

$$\chi(\lambda,\mu)=\operatorname{Corr}[N(\lambda),N(\mu)]=\frac{\operatorname{Cov}[N(\lambda),N(\mu)]}{\sqrt{\operatorname{Var}[N(\lambda)]\operatorname{Var}[N(\mu)]}}.$$

Here $\kappa$ is the parameter appearing in the model. The eigenvalue-counting correlation conjecture. For $d=1,2,3$, as $(\lambda,\mu)\to\infty$, the correlation $\chi(\lambda,\mu)$ is asymptotic to an explicit function of $\min\{\lambda,\mu\}$, $\max\{\lambda,\mu\}$, $d$, and $\kappa$. This is motivated by the paper's correlation-scaling result and an informal Abelian/Tauberian relation; determining the explicit asymptotic requires estimates for eigenvalue-counting variances and covariances, and the conjecture remains open.