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 para-controlled calculus. It is a self-adjoint operator with pure point spectrum. We prove estimates on its eigenvalues which imply a Weyl law for H. Finally, we give a version of Brezis-Gallouët inequality which implies existence and uniqueness for the cubic nonlinear Schrödinger equation with multiplicative noise.

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

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

Antoine Mouzard. The Anderson Hamiltonian on a two-dimensional manifold. 2020. ⟨hal-02931734v2⟩