On Rotated CMV Operators and Orthogonal Polynomials on the Unit Circle

TL;DR

This paper introduces rotated CMV operators, a generalized form of CMV operators derived from orthogonal polynomials on the unit circle, with applications to split-step quantum walks and transfer matrix computations.

Contribution

It extends the theory of CMV operators by defining rotated versions via rotated OPUCs, including their LM-factorisation and second kind polynomials, and applies these to transfer matrix calculations.

Findings

Rotated CMV operators can be constructed similarly to original CMV operators.

The paper develops rotated second kind polynomials for these operators.

LM-factorisation of rotated CMV operators enables transfer matrix computations.

Abstract

Split-step quantum walk operators can be expressed as a generalised version of CMV operators with complex transmission coefficients, which we call rotated CMV operators. Following the idea of Cantero, Moral and Velazquez's original construction of the original CMV operators from the theory of orthogonal polynomials on the unit circle (OPUC), we show that rotated CMV operators can be constructed similarly via a rotated version of OPUCs with respect to the same measure, and admit an analogous LM-factorisation as the original CMV operators. We also develop the rotated second kind polynomials corresponding to the rotated OPUCs. We then use the LM-factorisation of rotated alternate CMV operators to compute the Gesztesy-Zinchenko transfer matrices for rotated CMV operators.