XVI Preface equations are considered. Again this applies to impurities in one-dimensional crys- tals. Furthermore, this chapter forms the main ingredient of the inverse scattering transform for the Toda equations. Chapter 11 tries to deform the spectra of Jacobi operators in certain ways. We compute transformations which are isospectral and such which insert a finite num- ber of eigenvalues. The standard transformations like single, double, or Dirichlet commutation methods are developed. These transformations can be used as pow- erful tools in inverse spectral theory and they allow us to compute new solutions from old solutions of the Toda and Kac-van Moerbeke equations in Chapter 14. Part II Chapter 12 is the first chapter on integrable lattices and introduces the Toda system as hierarchy of evolution equations associated with the Jacobi operator via the standard Lax approach. Moreover, the basic (global) existence and uniqueness theorem for solutions of the initial value problem is proven. Finally, the stationary hierarchy is investigated and the Burchnall-Chaundy polynomial computed. Chapter 13 studies various aspects of the initial value problem. Explicit for- mulas in case of reflectionless (e.g., (quasi-)periodic) initial conditions are given in terms of polynomials and Riemann theta functions. Moreover, the inverse scatter- ing transform is established. The final Chapter 14 introduces the Kac van-Moerbeke hierarchy as modified counterpart of the Toda hierarchy. Again the Lax approach is used to establish the basic (global) existence and uniqueness theorem for solutions of the initial value problem. Finally, its connection with the Toda hierarchy via a Miura-type transformation is studied and used to compute A^-soliton solutions on arbitrary background. Appendix Appendix A reviews the theory of Riemann surfaces as needed in this mono- graph. While most of us will know Riemann surfaces from a basic course on complex analysis or algebraic geometry, this will be mainly from an abstract viewpoint like in [86] or [129], respectively. Here we will need a more "computational" approach and I hope that the reader can extract this knowledge from Appendix A. Appendix B compiles some relevant results from the theory of Herglotz func- tions and Borel measures. Since not everybody is familiar with them, they are included for easy reference. Appendix C shows how a program for symbolic computation, Mathematic a®, can be used to do some of the computations encountered during the main bulk. While I don't believe that programs for symbolic computations are an indispens- able tool for doing research on Jacobi operators (or completely integrable lattices), they are at least useful for checking formulas. Further information and Mathe- matica® notebooks can be found at http://www.mat.univie.ac.at/~gerald/ftp/book-jac/ respectively ftp ://ftp.mat.univie.ac.at/pub/teschl/book-jac /

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