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Zbl 1082.42021
Simon, Barry
Orthogonal polynomials on the unit circle. Part 2: Spectral theory. (English)
[B] Colloquium Publications. American Mathematical Society 51, Part 2. Providence, RI: American Mathematical Society. xxi, 467-1044. \$~67.00 (2005). ISBN 0-8218-3675-7/hbk

The two-part treatise by Barry Simon, the world renowned expert in mathematical physics, come out in the same AMS Colloquium Publications series as the celebrated book by G. Szegö on orthogonal polynomials 75 years earlier. The main subject is the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. Part 2 develops some more advanced topics of the OPUC theory. The author adopts the technique from the spectral theory of Schrödinger operators and Jacobi matrices to study the fine structure (absolutely continuous and singular components) of orthogonality measures on the unit circle based on the behavior of their Verblunsky coefficients. Chapter 9 deals with one of the top points of the modern OPUC theory -- Rakhmanov's theorem -- and its extensions due to Máté-Nevai-Totik, Khrushchev and Barrios-Lopéz. Various techniques of the spectral analysis are exhibited in Chapter 10. The bulk of Chapter 11 concerns an extremely beautiful theory of periodic Verblunsky coefficients and is very close to results for one-dimensional periodic Schrödinger operators. The key player here is meromorphic functions on hyperelliptic surfaces. Other topics addressed in this volume are the spectral analysis of specific classes of Verblunsky coefficients (sparse, random, subshifts etc.) as well as connections to Jacobi matrices and orthogonal polynomials on the real line. This completes with a reader's guide (topics and formulae) and a list of conjectures and open questions. Detailed historic and bibliographic notes are appended to each chapter. A reader is furnished with an extensive notation list and an exhaustive bibliography. The book will be of interest to a wide range of mathematicians. [See also the review of Part 1: Classical theory in Zbl 1082.42020].
[Leonid Golinskii (Kharkov)]

MSC 2000:: *42C05 General theory of orthogonal functions and polynomials
47B35 Toeplitz operators, etc.
30C85 Capacity and harmonic measure in the complex plane
30D55 H (sup p)-classes
42A10 Trigonometric approximation
05E35 Orthogonal polynomials (combinatorics)
34L99 Ordinary differential operators
42-02 Research monographs (Fourier analysis)
33-02 Research monographs (special functions)

Keywords: measures on the unit circle, Verblunsky coefficients; Rakhmanov's theorem; ratio asymptotics; spectral averaging; Lyapunov exponents; transfer matrices; periodic Verblunsky coefficients; meromorphic functions on hyperelliptic surfaces; sparse and random Verblunsky coefficients; Jacobi matrices

Citations: Zbl 1082.42020

Cited in: Zbl 1239.42026 Zbl 1185.34131 Zbl 1174.42032 Zbl 1131.39019 Zbl 1130.82017 Zbl 1108.37002 Zbl 1134.42017 Zbl 1127.47037 Zbl 1118.47023 Zbl 1082.42016 Zbl 1082.42020 Zbl 1086.42501

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