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

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