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.