Local eigenvalue statistics of one-dimensional random nonselfadjoint pseudodifferential operators

We study a class of one-dimensional non-selfadjoint semiclassical elliptic pseudo-differential operators subject to small random perturbations. We compare two types of random perturbation: random potential and random matrix. It is known by recent works of Sj\"ostrand and Hager that, under suitable conditions on the law of the perturbation, the eigenvalues of the perturbed operator contained in the interior of the pseudospectrum will follow Weyl-asymptotics with probability close to one. We show that in the limit of the semiclassical parameter $h\to 0$, the local statistics of the eigenvalues of the perturbed operator in the interior of the pseudospectrum is universal in the sense that it only depends on the type of random perturbation and the principal symbol of the unperturbed operator. It is, however, independent of the law of the perturbation.

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