# Spectral Analysis of Iterated Rank-One Perturbations

## TL;DR

This paper develops spectral analysis tools for self-adjoint operators with iterative rank-one perturbations, revealing spectrum localization and bounds, especially applied to Rademacher potentials.

## Contribution

It introduces new spectral theoretic methods for analyzing infinite sequences of random rank-one perturbations, including localization results and spectrum bounds.

## Key findings

- Localization of the Rademacher potential operator
- Establishment of bounds on spectrum types under rank-one perturbations
- Development of spectral tools for iterative random perturbations

## Abstract

The authors study the spectral theory of self-adjoint operators that are subject to certain types of perturbations. An iterative introduction of infinitely many randomly coupled rank-one perturbations is one of our settings. Spectral theoretic tools are developed to estimate the remaining absolutely continuous spectrum of the resulting random operators. Curious choices of the perturbation directions that depend on the previous realizations of the coupling parameters are assumed, and unitary intertwining operators are used. An application of our analysis shows localization of the random operator associated to the Rademacher potential. Obtaining fundamental bounds on the types of spectrum under rank-one perturbation, without restriction on its direction, is another main objective. This is accomplished by analyzing Borel/Cauchy transforms centrally associated with rank-one perturbation problems.