6
1. INTRODUCTION
In Chapter 5 we continue our stationary algebro-geometric approach to the
Toda hierarchy and provide a detailed derivation of the standard ^-function repre-
sentation of all stationary TL5 solutions (cf. Theorem 5.2).
In Chapter 6 we finally complete the algebro-geometric approach to the Toda
hierarchy. In addition to a detailed discussion of the time-dependent BA-function in
Theorem 6.2 we derive the well-known (time-dependent) ^-function representation
of the TLr equations with #-gap initial conditions in Theorem 6.3. Our detailed
account in Chapter 5 and 6 also leads to an alternative ^-function representation
of b in Corollaries 5.6 and 6.5 which, much to our surprise, seems to have escaped
notice in the literature thus far.
In Chapter 7 we turn to the KM hierarchy and its connection with the Toda
hierarchy. In addition to developing a recursive approach to the KM hierarchy
(which appears to be new) and in particular to a computation of Q22+2 in (1-14),
we describe at length the Miura-type transformation (1.31), (1.32) and especially
the transfer of solutions from the Toda to the KM hierarchy in Theorem 7.2.
In analogy to Chapter 4, Chapter 8 establishes spectral properties of the self-
adjoint realization D of M in
£2(Z)
®
C2
associated with finite-gap solutions p of
the KMP equations. Theorem 8.2, in particular, reduces the spectral analysis of the
Dirac-type difference operator D to that of the Jacobi operators Hk, the self-adjoint
realizations of Lk, k = 1, 2 in £2(Z) (cf. (1.27), (1.29)-(1.32)) by using factorization
(commutation) methods indicated in (1.28).
Finally, in Chapter 9 we complete the principal objective of this exposition
and derive all real-valued algebro-geometric quasi-periodic finite-gap solutions of
the KM hierarchy in Theorems 9.3 and 9.5. Isospectral manifolds of finite-gap
KM solutions are briefly considered in Remark 9.4 and a brief outlook on possible
applications of these completely integrable lattice models ends this exposition.
For convenience of the reader, and for the sake of being self-contained, we ad-
ded Appendix A which summarizes basic facts on hyperelliptic curves and their
^-functions and defines the notation used in the main body of this exposition. Ap-
pendix B records the principal results of Chapters 3-5 in the important special case
of periodic rather than quasi-periodic Jacobi operators by explicitly invoking Flo-
quet theory. Appendix C finally records the simplest explicit examples associated
with genus g = 0 and 1.
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