Determinants and Their Applications in Mathematical Physics

6.10 The Einstein and Ernst Equations 295

Exercise.The one-variable Hirota operatorsHxandHxxare defined in

Section 5.7 and the determinantsAnandEn, each of which is a function of

ρandz, are defined in (6.10.8) and (6.10.10). Apply Lemma 6.20 to prove

that

Hρ(An− 1 ,En)−ωHz(An,En− 1 )=

(

n− 1

ρ

)

An− 1 En,

Hρ(An,En− 1 )−ωHz(An− 1 ,En)=−

(

n− 2

ρ

)

AnEn− 1 (ω

2

=−1).

Using the notation

K

2

(f, g)=

(

Hρρ+

1

ρ

Hρ+Hzz

)

(f, g),

wheref=f(ρ, z) andg=g(ρ, z), prove also that

K

2

(En,An)=

n(n−2)

ρ

2

EnAn,

{

K

2

+

2 n− 4

ρ

}

(An,An− 1 )=−

1

ρ

2

AnAn− 1 ,

K

2

{

ρ

n(n−2)/ 2

En,ρ

n(n−2)/ 2

An

}

=0,

K

2

{

ρ

(n

2

− 4 n+2)/ 2

An− 1 ,ρ

n(n−2)/ 2

An

}

=0,

K

2

{

ρ

(n

2

−2)/ 2

An+1,ρ

n(n−2)/ 2

An

}

=0.

(Sasa and Satsuma)

6.10.4 Preparatory Theorems

Define a Vandermondian (Section 4.1.2)V 2 n(x) as follows:

V 2 n(x)=

∣

∣

x

j− 1

i

∣

∣

2 n

=V(x 1 ,x 2 ,...,x 2 n), (6.10.24)

and let the (unsigned) minors ofV 2 n(c) be denoted byM

(2n)

ij

(c). Also, let

Mi(c)=M

(2n)

i, 2 n

(c)=V(c 1 ,c 2 ,...,ci− 1 ,ci+1,...,c 2 n),

M 2 n(c)=M

(2n)

2 n, 2 n(c)=V^2 n−^1 (c). (6.10.25)

xj=

z+cj

ρ

,

εj=e

ωθj

√

1+x

2

j

(ω

2

=−1)

=

τj

ρ

, (6.10.26)