Open Collections

Description

A rank-one perturbation $A+K$ of an operator $A$ is one where the range of $K$ is just one-dimensional. Being rather restrictive, they form a small class of perturbations. Yet, rank-one perturbations are related to many deep questions. Here we focus on a relation with Anderson-type Hamiltonians. These are random perturbations which are obtained by taking a countable sum of rank-one perturbation, each weighted by a randomly chosen coupling constant. Such perturbations are non-compact almost surely. Under mild conditions, the essential parts of two realizations of an Anderson-type Hamiltonian are almost surely related by a rank one perturbation.