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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