2D random magnetic Laplacian with white noise magnetic field

Abstract

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.

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“…This is similar in spirit to Theorem 6 in [12]. One can show that the magnetic Laplacian with white noise magnetic field constructed in [26] 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.…”

“…jT 1 T 2 j C jx 0 y 0 j (7-26) for all T 1 ; T 2 2 1 2 ; 3 2 and x 0 ; y 0 2 ‫ނ‬ 9 1=2 . Now the inequalities (7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25) and (7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26) imply the first part of the statement. The same inequalities imply k ‡ .v; T; x 0 /k .…”

“…Note that [27] is self-contained and is a gentle introduction to the paracontrolled calculus on manifolds in the spatial framework. For another example of singular random operators, see [26], where Morin and Mouzard construct the magnetic Laplacian with white noise magnetic field on ‫ޔ‬ 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 is the content of [26], where Morin and Mouzard construct…”