# One-dimensional quantum walks with a general perturbation of the radius 1

## TL;DR

This paper provides a comprehensive analysis of the discrete spectrum of one-dimensional quantum Hamiltonians with a general perturbation of radius 1, advancing understanding of localized states in quantum walks.

## Contribution

It offers a complete characterization of the discrete spectrum for a broad class of one-dimensional Hamiltonians with radius-1 perturbations, extending previous specific cases.

## Key findings

- Complete description of the discrete spectrum for the model.
- Identification of conditions for spectral localization.
- Extension of spectral analysis to general radius-1 perturbations.

## Abstract

We give a complete description of the discrete spectrum of one-dimensional Hamiltonian with a general perturbation of the radius $1$.