This two-part volume gives a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analog of the spectral theory of one-dimensional Schrodinger operators. Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szego's theorems), limit theorems for the density of the zeros of orthogonal polynomials, matrix representations for multiplication by $z$ (CMV matrices), periodic Verblunsky coefficients from the point of view of meromorphic functions on hyperelliptic surfaces, and connections between the theories of orthogonal polynomials on the unit circle and on the real line. The book is suitable for graduate students and researchers interested in analysis.
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This review is from: Orthogonal Polynomials on the Unit Circle (Colloquium Publications (Amer Mathematical Soc)) (Hardcover)
While my background is in theoretical physics, I have never delved into the theory of the discrete Schrodinger operator. But this book can be considered an intensive exploration of that operator. Simon summarises a lot of research, and certainly not just by him, that reveals an amazing depth and complexity.
This book is really a text for the graduate or research level in mathematical physics. It is rather abstruse. Which may be a stumbling block to the average physicist. While the Schrodinger equation is the basis of non-relativistic quantum mechanics, going from results in this book to broader applications might not be easy.
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
3
books
cite this book:
- Trace Ideals and Their Applications, Second Edition by Barry Simon in Back Matter
- Classical and Quantum Orthogonal Polynomials in One Variable (Encyclopedia of Mathematics and its Applications) by Mourad E. H. Ismail in Back Matter
- Spectral Properties of Banded Toeplitz Matrices by Albrecht Böttcher in Back Matter
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