Construction and spectrum of the Anderson Hamiltonian with white noise potential on (\mathbf{R}^2) and (\mathbf{R}^3)

We propose a simple construction of the Anderson Hamiltonian with white noise potential on \(\mathbf{R}^2\) and \(\mathbf{R}^3\) 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 \(\mathbf{R}\). To prove this result, we extend the method of Kotani (1985) to our setting of singular random operators.

Recommended citation: Y.-S. Hsu and C. Labbé, Construction and spectrum of the Anderson Hamiltonian with white noise potential on \(\mathbb{R}^2\) and \(\mathbb{R}^3\), Probab. Math. Phys. 7 (2026), no.~1, 1--35; MR4998450