Determinants and Their Applications in Mathematical Physics

252 6. Applications of Determinants in Mathematical Physics

=2

∑

r

brcrφr

∑

j

e

cju

A

rj

+2

∑

r

brcr

∑

j

φre

cju

A

rj

cj− 1

=2

∑

r

brcrφ

2

r−F

′

= (logA)

′′

−F

′

, (6.4.18)

R=2

∑

j

e

cju

cj− 1

∑

r

brcrφ

′

rA

rj

=2

∑

j

e

cju

cj− 1

∑

r

brcrφr[cjA

rj

−φrφj]

=Q−P. (6.4.19)

Hence, eliminatingP,Q, andRfrom (6.4.16)–(6.4.19),

d

2

F

du

2

− 2

dF

du

+2F(logA)

′′

=0. (6.4.20)

Put

F=e

u

y. (6.4.21)

Then, (6.4.20) is transformed into

d

2

y

du^2

−y+2y

d

2

du^2

(logA)=0. (6.4.22)

Finally, putu=ωεx,(ω

2

=−1). Then, (6.4.22) is transformed into

d

2

y

dx

2

+ε

2

y+2y

d

2

dx

2

(logA)=0,

which is identical with (6.4.1), the Kay–Moses equation. This completes

the proof of the theorem.

6.5 The Toda Equations......................

6.5.1 The First-Order Toda Equation...........

Define two Hankel determinants (Section 4.8)AnandBnas follows:

An=|φm|n, 0 ≤m≤ 2 n− 2 ,

Bn=|φm|n, 1 ≤m≤ 2 n− 1 ,

A 0 =B 0 =1. (6.5.1)

The algebraic identities

AnB

(n+1)

n+1,n−BnA

(n+1)

n+1,n+An+1Bn−^1 =0, (6.5.2)

Bn− 1 A

(n+1)

n+1,n

−AnB

(n)

n,n− 1

+An− 1 Bn= 0 (6.5.3)