Communications in Mathematical Physics

, Volume 223, Issue 2, pp 223-259

First online:

Szegő Difference Equations, Transfer Matrices¶and Orthogonal Polynomials on the Unit Circle

  • Leonid  GolinskiiAffiliated withMathematics Division, Institute for Low Temperature Physics and Engineering, 47 Lenin Avenue,¶Kharkov 61103, Ukraine. E-mail:
  • , Paul NevaiAffiliated withDepartment of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210-1174, USA. E-mail:

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We develop the theory of orthogonal polynomials on the unit circle based on the Szegő recurrence relations written in matrix form. The orthogonality measure and C-function arise in exactly the same way as Weyl's function in the Weyl approach to second order linear differential equations on the half-line. The main object under consideration is the transfer matrix which is a key ingredient in the modern theory of one-dimensional Schrödinger operators (discrete and continuous), and the notion of subordinacy from the Gilbert–Pearson theory. We study the relations between transfer matrices and the structure of orthogonality measures. The theory is illustrated by the Szegő equations with reflection coefficients having bounded variation.