A simple construction of the Anderson operator via its quadratic form in dimensions two and three

Antoine Mouzard (1) , El Maati Ouhabaz (2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 IMB - Institut de Mathématiques de Bordeaux

We provide a simple construction of the Anderson operator in dimensions two and three. This is done through its quadratic form. We rely on an exponential transform instead of the regularity structures or paracontrolled calculus which are usually used for the construction of the operator. The knowledge of the form is robust enough to deduce important properties such as positivity and irreducibility of the corresponding semigroup. The latter property gives existence of a spectral gap.

Mots clés: Anderson form; singular stochastic operator; Schrödinger operator; renormalization; positivity; spectral gap

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

Antoine Mouzard, El Maati Ouhabaz. A simple construction of the Anderson operator via its quadratic form in dimensions two and three. 2023. ⟨hal-04197062⟩