Rank-one perturbations and Anderson-type Hamiltonians

Abstract

Motivated by applications of the discrete random Schrödinger operator, mathematical physicists and analysts began studying more general Anderson-type Hamiltonians; that is, the family of self-adjoint operators $$H_{\omega }=H+V_{\omega }$$ on a separable Hilbert space $\mathcal{H}$, where the perturbation is given by $$V_{\omega }={\sum }_n{\omega }_n(\cdot ,{\phi }_n){\phi }_n$$ with a sequence $\left\{{\phi }_n\right\}\subset \mathcal{H}$ and independent identically distributed random variables ${\omega }_n$. We show that the essential parts of Hamiltonians associated to any two realizations of the random variable are (almost surely) related by a rank-one perturbation. This result connects one of the least trackable perturbation problem (with almost surely noncompact perturbations) with one where the perturbation is “only” of rank-one perturbations. The latter presents a basic application of model theory. We also show that the intersection of the essential spectrum with open sets is almost surely either the empty set, or it has nonzero Lebesgue measure.