Page 260 - Determinants and Their Applications in Mathematical Physics

6.2 Brief Historical Notes 245

The substitution
1 − ξ
ζ = (6.2.21)
1+ ξ
transforms equation (6.2.15) into the Ernst equation, namely
2
∗
∗
(ξξ − 1)∇ ξ =2ξ (∇ξ · ξ) (6.2.22)
which appeared in 1968.
In 1977, M. Yamazaki conjectured and, in 1978, Hori proved that a
solution of the Ernst equation is given by
2
ξ n = pxu n − ωqyv n (ω = −1), (6.2.23)
w n
where x and y are prolate spheroidal coordinates and u n , v n , and w n are
determinants of arbitrary order n in which the elements in the first columns
of u n and v n are polynomials with complicated coefficients. In 1983, Vein
showed that the Yamazaki–Hori solutions can be expressed in the form
pU n+1 − ωqV n+1
ξ n = (6.2.24)
W n+1
where U n+1 , V n+1 , and W n+1 are bordered determinants of order n+1 with
comparatively simple elements. These determinants are defined in detail in
Section 4.10.3.
Hori’s proof of (6.2.23) is long and involved, but no neat proof has yet
been found. The solution of (6.2.24) is stated in Section 6.10.6, but since
it was obtained directly from (6.2.23) no neat proof is available.

6.2.9 The Relativistic Toda Equation

The relativistic Toda equation, namely

˙ ˙ exp(R n−1 − R n )
R n−1
¨
R n = 1+ 1+ R n
2
c c 1+(1/c ) exp(R n−1 − R n )

˙ R n+1 exp(R n − R n+1 )
˙
− 1 − R n 1+ 2 ,(6.2.25)
c c 1+(1/c ) exp(R n − R n+1 )
˙
where R n = dR n /dt, etc., was introduced by Rujisenaars in 1990. In the
limit as c →∞, (6.2.25) degenerates into the equation
¨
R n = exp(R n−1 − R n ) − exp(R n − R n+1 ). (6.2.26)
The substitution

U n−1
R n = log U n (6.2.27)
reduces (6.2.26) to (6.2.3).