Colloquium Publications. American Mathematical Society 51, Part 2. Providence, RI: American Mathematical Society (ISBN 0-8218-3675-7/hbk). xxi, 467-1044. $ 67.00 (2005).
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. Szegő 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].