The precise shape of the eigenvalue intensity for a class of non-selfadjoint operators under random perturbations

We consider a non-selfadjoint $h$-differential model operator $P_h$ in the semiclassical limit ($h\rightarrow 0$) subject to small random perturbations. Furthermore, we let the coupling constant $\delta$ be $\exp\{-\frac{1}{Ch}\}\leq \delta \ll h^{\kappa}$ for constants $C,\kappa>0$ suitably large. Let $\Sigma$ be the closure of the range of the principal symbol. Previous results on the same model by Hager, Bordeaux-Montrieux and Sj\"ostrand show that if $\delta \gg\exp\{-\frac{1}{Ch}\}$ there is, with a probability close to $1$, a Weyl law for the eigenvalues in the interior of the of the pseudospectrum up to a distance $\gg\left(-h\ln{\delta h}\right)^{\frac{2}{3}}$ to the boundary of $\Sigma$. We study the intensity measure of the random point process of eigenvalues and prove an $h$-asymptotic formula for the average density of eigenvalues. With this we show that there are three distinct regions of different spectral behavior in $\Sigma$: The interior of the of the pseudospectrum is solely governed by a Weyl law, close to its boundary there is a strong spectral accumulation given by a tunneling effect followed by a region where the density decays rapidly.