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

Abstract

This paper is a continuation of work 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 gauge transformation and construction of generalized eigenfunctions (Jost solutions) which was studied in the short-range case.