Construction and spectrum of the AndersonHamiltonian with white noise potential on ℝ2 and ℝ3

Abstract

We propose a simple construction of the Anderson Hamiltonian with white noise potential on R² and R³ based on the solution theory of the parabolic Anderson model. It relies on a theorem of Klein and Landau (1981) that associates a unique self-adjoint generator to a symmetric semigroup satisfying some mild assumptions. Then, we show that almost surely the spectrum of this random Schrödinger operator is R. To prove this result, we extend the method of Kotani (1985) to our setting of singular random operators.