Determinants and Their Applications in Mathematical Physics

6.2 Brief Historical Notes 245

The substitution

ζ=

1 −ξ

1+ξ

(6.2.21)

transforms equation (6.2.15) into the Ernst equation, namely

(ξξ

∗

−1)∇

2

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

ξn=

pxun−ωqyvn

wn

(ω

2

=−1), (6.2.23)

wherexandyare prolate spheroidal coordinates andun,vn, andwnare

determinants of arbitrary ordernin which the elements in the first columns

ofunandvnare polynomials with complicated coefficients. In 1983, Vein

showed that the Yamazaki–Hori solutions can be expressed in the form

ξn=

pUn+1−ωqVn+1

Wn+1

(6.2.24)

whereUn+1,Vn+1, andWn+1are bordered determinants of ordern+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

̈

Rn=

(

1+

̇

Rn− 1

c

)(

1+

̇

Rn

c

)

exp(Rn− 1 −Rn)

1+(1/c

2

) exp(Rn− 1 −Rn)

−

(

1 −

̇

Rn

c

)(

1+

̇

Rn+1

c

)

exp(Rn−Rn+1)

1+(1/c

2

) exp(Rn−Rn+1)

,(6.2.25)

where

̇

Rn=dRn/dt, etc., was introduced by Rujisenaars in 1990. In the

limit asc→∞, (6.2.25) degenerates into the equation

̈

Rn= exp(Rn− 1 −Rn)−exp(Rn−Rn+1). (6.2.26)

The substitution

Rn= log

{

Un− 1

Un

}

(6.2.27)

reduces (6.2.26) to (6.2.3).