2D random magnetic Laplacian with white noise magnetic field

Abstract

We define the random magnetic Laplacien 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.

Strichartz inequalities with white noise potential on compact surfaces

“…We proved that Strichartz inequalities are stable under suitable perturbation, that is lower-order perturbation in the sense of previous Proposition 2.1. One can show that the magnetic Laplacian with white noise magnetic field constructed in [23] is also a lower order perturbation of the Laplacian on the two-dimensional torus in this sense. Thus Theorem 2.3 also gives Strichartz inequalities for the associated Schrödinger group.…”

“…Note that the work [24] is self-contained and a gentle introduction to the paracontrolled calculus on manifold in the spatial framework. For another example of singular random operators, see [23] where Morin and Mouzard construct the magnetic Laplacian with white noise magnetic field on T 2 . With this work, we show that this approach is also well-suited for the study of dispersive PDEs.…”

“…While the Anderson Hamiltonian can be interpreted as the electric Laplacian −∆ + V with electric field V = ξ, one can consider as an analogy the magnetic Laplacian with magnetic field B = ξ space white noise. This this the content of [23] where Morin and Mouzard construct…”