Weyl law for the Anderson Hamiltonian on a two-dimensional manifold

Antoine Mouzard (1, 2)

1 UR - Université de Rennes
2 IRMAR - Institut de Recherche Mathématique de Rennes

We define the Anderson Hamiltonian H on a two-dimensional manifold using high order paracontrolled calculus. It is a self-adjoint operator with pure point spectrum. We get lower and upper bounds on its eigenvalues which imply an almost sure Weyl-type law for H.

Mots clés: Schrödinger operator; Anderson Hamiltonian; Paracontrolled calculus; White noise

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Probabilités [math.PR]; Théorie spectrale [math.SP]

Antoine Mouzard. Weyl law for the Anderson Hamiltonian on a two-dimensional manifold. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2022, 58 (3), pp.1385-1425. ⟨10.1214/21-AIHP1216⟩. ⟨hal-02931734v4⟩