A Semiclassical Weyl Law for Quantized Operators on the Torus Perturbed by Random Potentials

In this thesis we study the Weyl quantization of periodic symbols on the torus in a semiclassical regime where the parameter $N\to \infty$. This framework is natural for describing quantum observables that are periodic both in position and momentum. Since the torus is a compact phase space, the corresponding quantizations, when restricted to the space of periodic distributions, yield finite-dimensional operators, i.e., square matrices, whose size grows with $N$. Characterizing the spectrum of these operators is challenging, since they are not necessarily self-adjoint. To regularize the spectrum, we introduce small random perturbations in the form of diagonal matrices with compactly supported probability distributions. From a physical perspective, such perturbations can model random potentials in a quantum system. The main contribution of this work is the establishment of a probabilistic Weyl law for the spectrum of the perturbed operators. Using semiclassical analysis tools, Grushin problems, and techniques from complex analysis and probability theory, we prove that with high probability the eigenvalues of the perturbed operators roughly equidistribute in the range of their principal symbols. Specifically, for relatively compact sets with uniformly Lipschitz boundary, the eigenvalue counting function admits an asymptotic estimate governed by the measure of the set's preimage under the principal symbol. Our results demonstrate that small random potential perturbations can regularize the spectrum and recover Weyl-type asymptotics in this setting.