Stationary scattering theory for unitary operators with an application to quantum walks

TL;DR

This paper develops a comprehensive stationary scattering theory framework for unitary operators in a two-Hilbert spaces setting, with applications to anisotropic quantum walks, providing new formulas for wave operators and scattering matrices.

Contribution

It introduces a general stationary scattering theory for unitary operators with explicit conditions and formulas, applied to quantum walks, advancing the mathematical understanding of quantum scattering.

Findings

Established conditions for existence and coincidence of wave operators.

Derived explicit representation formulas for scattering matrices.

Applied the theory to a class of anisotropic quantum walks.

Abstract

We present a general account on the stationary scattering theory for unitary operators in a two-Hilbert spaces setting. For unitary operators $U_0,U$ in Hilbert spaces ${\cal H}_0,{\cal H}$ and for an identification operator $J:{\cal H}_0\to{\cal H}$, we give the definitions and collect properties of the stationary wave operators, the strong wave operators, the scattering operator and the scattering matrix for the triple $(U,U_0,J)$. In particular, we exhibit conditions under which the stationary wave operators and the strong wave operators exist and coincide, and we derive representation formulas for the stationary wave operators and the scattering matrix. As an application, we show that these representation formulas are satisfied for a class of anisotropic quantum walks recently introduced in the literature.