Page 317 - 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
2
2
∗
∗
(ξξ − 1)∇ ξ =2ξ (∇ξ) ,
is satisfied by each of the functions
pU n (x) − ωqU n (y) 2
ξ n = (ω = −1), n =1, 2, 3,...,
U n (1)
where U n (x) is a determinant of order (n + 1) obtained by bordering an
nth-order Hankelian as follows:
x

x /3
3

x /5
5
,
[a ij ] n
U n (x)=

···

x
2n−1
/(2n − 1)
111 ··· 1 •

n+1
where
1 2 2(i+j−1) 2 2(i+j−1)
a ij = [p x + q y − 1],
i + j − 1
2
2
p + q =1,
and x and y are prolate spheriodal coordinates. The argument x in U n (x)
refers to the elements in the last column, so that U n (1) is the determinant
obtained from U n (x) by replacing the x in the last column only by 1.
A note on this solution is given in Section 6.2 on brief historical notes.
Some properties of U n (x) and a similar determinant V n (x) are proved in
Section 4.10.3.
6.11 The Relativistic Toda Equation — A Brief
Note
The relativistic Toda equation in a function R n and a substitution for R n
in terms of U n−1 and U n are given in Section 6.2.9. The resulting equation
can be obtained by eliminating V n and W n from the equations
2
H (2) (U n ,U n )=2(V n W n − U ), (6.11.1)
x n
aH (2) (U n ,U n−1 )= aU n U n−1 + V n W n−1 , (6.11.2)
x
2
2
2
V n+1 W n−1 − U = a (U n+1 U n−1 − U ), (6.11.3)
n n
(2)
is the one-variable Hirota operator (Section 5.7),
where H x
1
a = √ ,
1+ c 2