Page 310 - Determinants and Their Applications in Mathematical Physics

6.10 The Einstein and Ernst Equations 295

Exercise. The one-variable Hirota operators H x and H xx are defined in
Section 5.7 and the determinants A n and E n , each of which is a function of
ρ and z, are defined in (6.10.8) and (6.10.10). Apply Lemma 6.20 to prove
that
n − 1

H ρ (A n−1 ,E n ) − ωH z (A n ,E n−1 )= A n−1 E n ,
ρ
n − 2 2

H ρ (A n ,E n−1 ) − ωH z (A n−1 ,E n )= − A n E n−1 (ω = −1).
ρ
Using the notation
1

2
K (f, g)= H ρρ + H ρ + H zz (f, g),
ρ
where f = f(ρ, z) and g = g(ρ, z), prove also that
n(n − 2)
2
K (E n ,A n )= E n A n ,
ρ 2

2n − 4 1
2
K + (A n ,A n−1 )= − A n A n−1 ,
ρ ρ 2
/ 0
K 2 ρ n(n−2)/2 E n ,ρ n(n−2)/2 A n =0,
/ 2 0
K 2 ρ (n −4n+2)/2 A n−1 ,ρ n(n−2)/2 A n =0,
/ 2 0
K 2 ρ (n −2)/2 A n+1 ,ρ n(n−2)/2 A n =0.
(Sasa and Satsuma)

6.10.4 Preparatory Theorems

Define a Vandermondian (Section 4.1.2) V 2n (x) as follows:
j−1
V 2n (x)= x

i
2n
= V (x 1 ,x 2 ,...,x 2n ), (6.10.24)
(2n)
and let the (unsigned) minors of V 2n (c) be denoted by M (c). Also, let
ij
(2n)
M i (c)= M (c)= V (c 1 ,c 2 ,...,c i−1 ,c i+1 ,...,c 2n ),
i,2n
(2n)
M 2n (c)= M (c)= V 2n−1 (c). (6.10.25)
2n,2n
x j = z + c j ,
ρ
3
2
ε j = e ωθ j 1+ x 2 (ω = −1)
j
= τ j , (6.10.26)
ρ