Jacobi Operators and Completely Integrable Nonlinear Lattices
Gerald Teschl
Starting from second order difference equations we move on to selfadjoint operators and develop discrete WeylTitchmarshKodaira theory, covering all classical aspects like Weyl mfunctions, spectral functions, the moment problem, inverse spectral theory, and uniqueness results. Next, we investigate some more advanced topics like locating the essential, absolutely continuous, and discrete spectrum, oscillation theory, trace formulas, random operators, almost periodic operators, (quasi)periodic operators, scattering theory, and spectral deformations.
Then, the Lax approach is used to introduce the Toda hierarchy and its modified counterpart, the Kacvan Moerbeke hierarchy. Uniqueness and existence theorems for the initial value problem, solutions in terms of Riemann theta functions, the inverse scattering transform, Bäcklund transformations, and soliton solutions are discussed.
Three appendices review the theory of compact Riemann surfaces, Herglotz functions, and show how one can use a program for symbolic computation, Mathematica, to do some of the work (Jacobi equations with Mathematica).
MSC91: 39Axx, 47B39, 37Kxx
Keywords: Jacobi operators, Toda hierarchy, Kacvan Moerbeke hierarchy.
Publisher:  American Mathematical Society  
Series:  Mathematical Surveys and Monographs, ISSN: 00765376  
Volume:  72  
Publication Year:  2000  
ISBN:  0821819402 / 9780821819401  
Paging:  351 pp; hardcover  
List Price:  $68  
Member Price:  $54  
Itemcode:  SURV/72 
Permission is granted to retrieve, print, and store a single copy for personal use only. If you like this book and want to support the idea of online versions, please consider buying this book. Note that all known mistakes are corrected in the online version, other than that it is identical to the printed version.

Preface
 Jacobi operators
 General properties
 Jacobi operators
 A simple example
 General second order difference expressions
 The infinite harmonic crystal in one dimension
 Foundations of spectral theory
 Weyl mfunctions
 Properties of solutions
 Positive solutions
 Weyl circles
 Canonical forms of Jacobi operators and the moment problem
 Some remarks on unbounded operators
 Inverse spectral theory
 Qualitative theory of spectra
 Locating the spectrum and spectral multiplicities
 Locating the essential spectrum
 Locating the absolutely continuous spectrum
 A bound on the number of eigenvalues
 Oscillation theory
 Prüfer variables and Sturm's theorem
 Classical Oscillation theory
 Renormalized oscillation theory
 Random Jacobi operator
 Random Jacobi operators
 The Lyaponov exponent and the density of states
 Almost periodic Jacobi operators
 Trace formulas
 Asymptotic expansions
 General trace formulas and xi functions
 Periodic Jacobi operators
 Floquet theory
 Connections with the spectra of finite Jacobi operators
 Polynomial identities
 A simple example  continued
 Perturbations of periodic operators
 Reflectionless Jacobi operators
 Spectral analysis and trace formulas
 Isospectral operators
 The finitegap case
 Further spectral interpretation
 Quasiperiodic Jacobi operators
 Riemann surfaces
 Solutions in terms of theta functions
 The elliptic case, genus one
 Illustrations of the RiemannRoch theorem
 Scattering theory
 Transformation operators
 The scattering matrix
 The Gel'fandLevitanMarchenko equations
 Inverse scattering theory
 Spectral deformations
 Commuting first order difference expressions
 The single commutation method
 Iteration of the single commutation method
 Application of the single commutation method
 A formal second commutation
 The double commutation method
 Iteration of the double commutation method
 The Dirichlet deformation method
Notes on literature
 The Toda system
 The Toda lattice
 Lax pairs, the Toda hierarchy, and hyperelliptic curves
 Stationary solutions
 Time evolution of associated quantities
 The initial value problem
 Finitegap solutions of the Toda hierarchy
 Quasiperiodic finitegap solutions and the timedependent BakerAkhiezer function
 A simple example  continued
 Inverse scattering transform
 Some additions in case of the Toda lattice
 The elliptic case  continued
 The Kacvan Moerbeke system
 The Kacvan Moerbeke hierarchy and its relation to the Toda hierarchy
 Kac and van Moerbeke's original equations
 Spectral theory for supersymmetric Diractype difference operators
 Associated solutions
 Nsoliton solutions on an arbitrary background
Notes on literature
Part 1: Jacobi Operators
Part 2: Completely Integrable Nonlinear Lattices
 Compact Riemann surfaces  a review
 Basic notation
 Abelian differentials
 Divisors and the RiemannRoch theorem
 Jacobian variety and Abel's map
 Riemann's theta function
 The zeros of the Riemann theta function
 Hyperelliptic Riemann surfaces
 Herglotz functions
 Jacobi equations with Mathematica
 The package DiffEqs and first order difference equations
 The package JacDEqs and Jacobi difference equation
 Simple properties of Jacobi difference equations
 Orthogonal Polynomials
 Recursions
 Commutation methods
 Toda lattice
 Kacvan Moerbeke lattice
Appendix
Bibliography
Glossary of notations
Index