Cyclicity in rank-one perturbation problems

TL;DR

This paper investigates the cyclicity property of vectors in rank-one perturbation problems for self-adjoint and unitary operators, showing that certain vectors are generically cyclic across families of such perturbations, with applications to Anderson-type Hamiltonians.

Contribution

It demonstrates that for fixed non-zero vectors, cyclicity is a common property in rank-one perturbation families, extending understanding of spectral characteristics in operator theory.

Findings

Cyclicity is prevalent for fixed vectors in rank-one perturbations.

The set of exceptions where vectors are not cyclic is small.

Results have implications for Anderson-type Hamiltonians.

Abstract

The property of cyclicity of a linear operator, or equivalently the property of simplicity of its spectrum, is an important spectral characteristic that appears in many problems of functional analysis and applications to mathematical physics. In this paper we study cyclicity in the context of rank-one perturbation problems for self-adjoint and unitary operators. We show that for a fixed non-zero vector the property of being a cyclic vector is not rare, in the sense that for any family of rank-one perturbations of self-adjoint or unitary operators acting on the space, that vector will be cyclic for every operator from the family, with a possible exception of a small set with respect to the parameter. We discuss applications of our results to Anderson-type Hamiltonians.