Preface The name of the game Jacobi operators appear in a variety of applications. They can be viewed as the discrete analogue of Sturm-Liouville operators and their investigation has many similarities with Sturm-Liouville theory. Spectral and inverse spectral theory for Jacobi operators plays a fundamental role in the investigation of completely inte- grable nonlinear lattices, in particular the Toda lattice and its modified counterpart, the Kac-van Moerbeke lattice. Why I have written this book Whereas numerous books about Sturm-Liouville operators have been written, only few on Jacobi operators exist. In particular, there is currently no monograph available which covers all basic topics (like spectral and inverse spectral theory, scattering theory, oscillation theory and positive solutions, (quasi-)periodic opera- tors, spectral deformations, etc.) typically found in textbooks on Sturm-Liouville operators. In the case of the Toda lattice a textbook by M. Toda [230] exists, but none of the recent advances in the theory of nonlinear lattices are covered there. Audience and prerequisites As audience I had researchers in mind. This book can be used to get acquainted with selected topics as well as to look up specific results. Nevertheless, no previous knowledge on difference equations is assumed and all results are derived in a self- contained manner. Hence the present book is accessible to graduate students as well. Previous experience with Sturm-Liouville operators might be helpful but is not necessary. Still, a solid working knowledge from other branches of mathematics is needed. In particular, I have assumed that the reader is familiar with the theory of (linear) self-adjoint operators in Hilbert spaces which can be found in (e.g.) x m
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