We define the random magnetic Laplacian with spatial white noise as magnetic field on the two-dimensional torus using paracontrolled calculus. It yields a random self-adjoint operator with pure point spectrum and domain a random subspace of nonsmooth functions in L 2 . We give sharp bounds on the eigenvalues which imply an almost sure Weyl-type law.

Certains droits réservés. Cet article est mis à disposition selon les termes de la Licence Internationale d'Attribution Creative Commons BY-NC-ND 4.0 https://creativecommons.org/licenses/by-nc-nd/4.0/ Confluentes Mathematici est membre du Centre Mersenne pour l'édition scientifique ouverte http:// www.centre-mersenne.org/ e-

This article is devoted to the spectral analysis of the electromagnetic Schrödinger operator on the Euclidean plane. In the semiclassical limit, we derive a pseudo-differential effective operator that allows us to describe the spectrum in various situations and appropriate regions of the complex plane. Not only results of the self-adjoint case are proved (or recovered) in the proposed unifying framework, but also new results are established when the electric potential is complex-valued. In such situations, when the non-self-adjointness comes with its specific issues (lack of a “spectral theorem”, resolvent estimates), the analogue of the “low-lying eigenvalues” of the self-adjoint case are still accurately described and the spectral gaps estimated.