2. TH E TODA HIERARCHY 11
using (2.11), (2.21), and (2.22). Since obviously
(2.32) (5n, [(L*0+1)+ - (L*0+1)_]£n) = 0
by (2.6), this settles the case m = n in (2.29). The remaining cases are settled one
by one in a similar fashion.
Before we turn to a discussion of the stationary Toda hierarchy we briefly sketch
the main steps leading to (2.10)-(2.13). If Ker(L(t) - z), z G C denotes the two-
dimensional nullspace of L(t) z (in the algebraic sense as opposed to the functional
analytic one), we seek a representation of P20+2W m Ker(L(t) z) of the form
(2.33) P 2
S + 2
(i)|
K e r ( L ( t )
_
z )
= 2a(t)Fg(z,t)S+ +Gg+1(z,t),
where Fg and Gg+i are polynomials in z of the type
g g
(2.34) Fg(z,t) = ^ V / ^ * ) , Gg+1(z,t) = - ^
+ 1
+ £ ^ 5
f l
- i ( * ) + /
9 +
i(*).
j = 0 j = 0
with ft(t) = {fe(n,t)}neZ t°°(Z), gt(t) = {ge(n,t)}neZ e t°°(Z). The Lax
equation (2.12) restricted to Ker(L(i) z) then yields
0 = {L - [P2g+2,L}}\ ={L + (L~ z)P2g+2}\K,e
(2.35)
Ker(L-z) ' ^ ' " ^ 9 + z J i K r ( L - z )
{a[^-^z +
2(b+
+
z)Fg+-2(b
+ z)Fg + G++1-G;+1]S^
+ [-b+(b +
z)—+2(a-)2Fg-
- 2 a 2 F / + (b + z)(Gg+l - Gg+l)}} \Ket{L_z).
Hence one obtains
(2.36) -a - ^ = 2(6+ + z)Fg+ - 2(6 + z)Fg + G;+1 - G++1,
(2.37) b=(b + z)^=+ 2{a-fF; -
2a2Fg+
+ (6+ z)(Gj
+ 1
- Gg+1).
Upon summing (2.36) (adding G^+i G^+i and neglecting a trivial summation
constant) one infers
(2.38) d = -a[2(b+ + *)F+ + G++1 + Gp+i], p G N0.
Insertion of (2.38) into (2.37) then implies
(2.39) b = -2[(b +
z)2F9
+ (b + z)Gs+i + a
2
F+ - (a")
2
^"] , ? G N0.
Insertion of (2.34) into (2.38) and (2.39) then produces the recursion relation (2.11)
(except for the relation involving fg+\ which serves as a definition) and the result
(2.13). Relation (2.33) then yields (2.10). We omit further details and just record
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