An eigenfunction expansion formula for one-dimensional two-state quantum walks

TL;DR

This paper provides a direct proof of an eigenfunction expansion formula for one-dimensional two-state quantum walks, enhancing understanding of their spectral properties and Green functions, especially in the two-phase model.

Contribution

It introduces a new direct proof method for the eigenfunction expansion formula specific to one-dimensional two-state quantum walks, extending prior results from CMV matrix theory.

Findings

Derived a concrete formula for the spectral measure in the two-phase model

Established properties of Green functions for these quantum walks

Connected eigenfunction expansion to spectral analysis in quantum walks

Abstract

The purpose of this paper is to give a direct proof of an eigenfunction expansion formula for one-dimensional 2-state quantum walks, which is an analog of that for Sturm-Liouville operators due to Weyl, Stone, Titchmarsh and Kodaira. In the context of the theory of CMV matrix it had been already established by Gesztesy-Zinchenko. Our approach is restricted to the class of quantum walks mentioned above whereas it is direct and it gives some important properties of Green functions. The properties given here enable us to give a concrete formula for a positive-matrix-valued measure, which gives directly the spectral measure, in a simplest case of the so-called two-phase model.