On rotated CMV operators and orthogonal polynomials on the unit circle. (English) Zbl 07917828
Summary: 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) [M. J. Cantero et al., Linear Algebra Appl. 362, 29–56 (2003; Zbl 1022.42013)], 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.
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 47B36 | Jacobi (tridiagonal) operators (matrices) and generalizations |
| 81P45 | Quantum information, communication, networks (quantum-theoretic aspects) |
Citations:
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