On rotated CMV operators and orthogonal polynomials on the unit circle

Abstract

Split-step quantum walk operators can be expressed as a generalized 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 $\mathcal{LM}$-factorization as the original CMV operators. We also develop the rotated second kind polynomials corresponding to the rotated OPUCs. We then use the $\mathcal{LM}$-factorization of rotated alternate CMV operators to compute the Gesztesy–Zinchenko transfer matrices for rotated CMV operators.

Keywords:

• CMV matrix
• orthogonal polynomials on the unit circle
• unitary operator
• quantum walk

2020 Mathematics Subject Classifications:

• Primary 42C05, 47ACA and 47B36
• Secondary81P45

Acknowledgments

The author would like to thank his advisor Darren C. Ong for his guidance throughout the project; the anonymous reviewers for useful suggestions and improvements to the manuscript; and Maxim Zinchenko for helpful discussions regarding the initial conditions of the Gesztesy–Zinchenko difference equation and second kind polynomials.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The author is supported in part by Xiamen University Malaysia Research Fund (grant numbers XMUMRF/2020-C5/IMAT/0011 and XMUMRF/2023C11/IMAT/0024).