Absence of singular continuous spectra and embedded eigenvalues for one dimensional quantum walks with general long-range coins

TL;DR

This paper extends previous work on quantum walks by analyzing general long-range perturbations of the coin operator, proving the absence of singular continuous spectra and embedded eigenvalues using generalized eigenfunctions.

Contribution

It introduces a broader class of long-range perturbations in quantum walks and establishes spectral properties previously known only for short-range cases.

Findings

No singular continuous spectrum for general long-range perturbations

Embedded eigenvalues are absent in these quantum walks

Generalized eigenfunctions (Jost solutions) are constructed for analysis

Abstract

This paper is a continuation of the paper \cite{W} by the third author, which studied quantum walks with special long-range perturbations of the coin operator. In this paper, we consider general long-range perturbations of the coin operator and prove the non-existence of a singular continuous spectrum and embedded eigenvalues. The proof relies on the construction of generalized eigenfunctions (Jost solutions) which was studied in the short-range case in \cite{MSSSSdis}.