Determinants and Their Applications in Mathematical Physics

276 6. Applications of Determinants in Mathematical Physics

wxxxx=2Vn− 3 ,n,n+1+3Vn− 3 ,n− 1 ,n+2+3Vn− 2 ,n− 1 ,n+1+Vn− 3 ,n− 2 ,n+3,

wz=−Vn− 3 ,n− 1 ,n+Vn− 3 ,n− 2 ,n+1,

wzz=2Vn− 3 ,n,n+1−Vn− 3 ,n− 1 ,n+2−Vn− 2 ,n− 1 ,n+1,

wt=−4(Vn− 2 ,n− 1 ,n−Vn− 3 ,n− 1 ,n+1+Vn− 3 ,n− 2 ,n+2),

wxt=4(Vn− 3 ,n,n+1−Vn− 3 ,n− 2 ,n+3). (6.7.64)

Each of the sections in the second line of (6.7.60) simplifies as follows:

wt+4wxxx=12Vn− 3 ,n− 1 ,n+1,

(wt+4wxxx)x= 12(Vn− 2 ,n− 1 ,n+1+Vn− 3 ,n,n+1+Vn− 3 ,n− 1 ,n+2),

wxxxx−wzz=4(Vn− 2 ,n− 1 ,n+1+Vn− 3 ,n− 1 ,n+2),

(wt+4wxxx)x−3(wxxxx−wzz)=12Vn− 3 ,n,n+1

w

2

xx

−w

2

z

=4Vn− 3 ,n− 1 ,nVn− 3 ,n− 2 ,n+1. (6.7.65)

Hence,

1

12

F=Vn− 3 ,n− 2 ,n− 1 Vn− 3 ,n,n+1+Vn− 3 ,n− 2 ,nVn− 3 ,n− 1 ,n+1

+Vn− 3 ,n− 1 ,nVn− 3 ,n− 2 ,n+1. (6.7.66)

Let

Cn+1=

[

α 1 α 2 ...αn

]T

,

Cn+2=

[

β 1 β 2 ...βn

]T

,

where

αr=D

n

x

(ψr)

βr=D

n+1

x (ψr).

Then

Vn− 3 ,n− 2 ,n− 1 =An,

Vn− 3 ,n− 2 ,n=

∑

r

αrA

(n)

rn,

Vn− 3 ,n− 1 ,n+1=−

∑

s

βsA

(n)

r,n− 1

,

Vn− 3 ,n− 2 ,n+1=

∑

s

βsA

(n)

sn

,

Vn− 3 ,n− 1 ,n=−

∑

r

αrA

(n)

r,n− 1

,

Vn− 3 ,n,n+1=

∑

r

∑

s

αrβsA

(n)

rs;n− 1 ,n. (6.7.67)

Hence, applying the Jacobi identity,

1

12

F=An

∑

r

∑

s

αrβsA

(n)

rs;n− 1 ,n+

∑

r

αrA

(n)

rn

∑

s

βsA

(n)

s,n− 1