CHAPTER 2
The Toda Hierarchy, Recursion Relations, and
Hyperelliptic Curves
In this chapter we review the construction of the Toda hierarchy by using a
recursive approach first advocated by Al'ber [6] and derive the Burchnall-Chaundy
polynomials in connection with the stationary Toda hierarchy. Our recursive ap-
proach to the Toda hierarchy, though equivalent to the conventional one (see, e.g.,
[60], [70],[72], [77],[81],[82],[85]), markedly differs from the standard treatment.
We have chosen to present the formalism below since it most naturally yields
the Burchnall-Chaundy polynomials associated with the stationary Toda hierarchy
and hence the underlying hyperelliptic curves for algebro-geometric quasi-periodic
finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies to be conside-
red in Chapters 6 and 9. Moreover, as shown in Chapter 4 (cf. (4.32)-(4.36)),
this recursive approach provides a fundamental link to the spectral matrix of the
underlying Lax operator.
We start by introducing some notations. In the following we denote by £P(M),
where 1 p oo, M = N, N0 = NU {0}, Z, etc., the usual space of p-summable
respectively bounded (if p oo) complex-valued sequences / = {/(m)}mGM and by
f^(M) the corresponding restriction to real-valued sequences. The scalar product
in the Hilbert space
£2(M)
will be denoted by
(2.1) (/,?)= £7Rs(rc),
f,gee2(M).
nEM
Since £°°{Z) C
£P(Z)
in a natural way it suffices to make all further definitions for
p = oo. In £°°(Z) we introduce the shift operators
(2.2) (5
±
/)(n) = / ( n ± l ) , n Z, / G £°°{Z)
and in order to simplify notations we agree to use the short cuts
; ± = 5 ± / t h a t i s
-
/ ± ( n ) = / ( n ± 1)f
(f + 9)(n) = f(n) + g(n), (fg)(n) = f(n)g(n), n Z, f,g e t°°(Z)
whenever convenient. Moreover, if R : £°°(Z) £°°(Z) denotes a difference expres-
sion, let
(2.4) R = {R(m, n)}
m
,
n G Z
, R(m, n) = (5m, R6n)
denote its corresponding matrix representation with respect to the standard basis
I 1 Tfl
z==- Tl
(2.5) em = {6m(n)}nez, m G Z, 6m(n) = '
10, m ^ n
7
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