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