Determinants and Their Applications in Mathematical Physics

302 6. Applications of Determinants in Mathematical Physics

6.10.6 The Ernst Equation

The Ernst equation, namely

(ξξ

∗

−1)∇

2

ξ=2ξ

∗

(∇ξ)

2

,

is satisfied by each of the functions

ξn=

pUn(x)−ωqUn(y)

Un(1)

(ω

2

=−1),n=1, 2 , 3 ,...,

whereUn(x) is a determinant of order (n+ 1) obtained by bordering an

nth-order Hankelian as follows:

Un(x)=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

x

x

3

/ 3

[aij]n x

5

/ 5

···

x

2 n− 1

/(2n−1)

111 ··· 1 •

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

,

where

aij=

1

i+j− 1

[p

2

x

2(i+j−1)

+q

2

y

2(i+j−1)

−1],

p

2

+q

2

=1,

andxandyare prolate spheriodal coordinates. The argumentxinUn(x)

refers to the elements in the last column, so thatUn(1) is the determinant

obtained fromUn(x) by replacing thexin the last columnonlyby 1.

A note on this solution is given in Section 6.2 on brief historical notes.

Some properties ofUn(x) and a similar determinantVn(x) are proved in

Section 4.10.3.

6.11 The Relativistic Toda Equation — A Brief

Note

The relativistic Toda equation in a functionRnand a substitution forRn

in terms ofUn− 1 andUnare given in Section 6.2.9. The resulting equation

can be obtained by eliminatingVnandWnfrom the equations

H

(2)

x (Un,Un)=2(VnWn−U

2

n), (6.11.1)

aH

(2)

x (Un,Un−^1 )=aUnUn−^1 +VnWn−^1 , (6.11.2)

Vn+1Wn− 1 −U

2

n=a

2

(Un+1Un− 1 −U

2

n), (6.11.3)

whereH

(2)

x is the one-variable Hirota operator (Section 5.7),

a=

1

√

1+c

2

,