Determinants and Their Applications in Mathematical Physics

286 6. Applications of Determinants in Mathematical Physics

Hence, the first term ofF is given by

AxA

∗

x=(−1)

n+1

Qn+1,n+2Qn+2,n+1. (6.9.28)

Differentiating (6.9.26) and referring to (6.9.6),

Axx=ω

∑

r

cr

∂Arr

∂x

=ω

∑

r

cr

∑

s

∂Ass

∂θs

∂θs

∂x

=−

∑

r

∑

s

crcsArs,rs

=

∑

r

∑

s

†

crcsArs, (6.9.29)

At=

∑

r

∂A

∂θr

∂θr

∂t

=−ω

∑

r

c

2

r

Arr. (6.9.30)

Hence, applying (6.9.13) and (6.9.19),

Axx+ωAt=

∑

r

∑

s

†

crcsArs+

∑

r

c

2

rArr

=

∑

r

∑

s

crcsArs

=Pn+2,n+2

=Qn+2,n+2+Q. (6.9.31)

Hence, the second term ofFis given by

A

∗

(Axx+ωAt)=(−1)

n

(B−Qn+1,n+1)(Qn+2,n+2+Q). (6.9.32)

Taking the complex conjugate of (6.9.31) and applying (6.9.20) and

(6.9.15),

(Axx+ωAt)

∗

=P

∗

n+2,n+2

=(−1)

n+1

(Qn+2,n+2−Q). (6.9.33)

Hence, the third term ofFis given by

A(Axx+ωAt)

∗

=(−1)

n+1

(B+Qn+1,n+1)(Qn+2,n+2−Q). (6.9.34)

Referring to (6.9.14),

1

2

(−1)

n

[

A

∗

(Axx+ωAt)+A(Axx+ωAt)

∗

]

=BQ−Qn+1,n+1Qn+2,n+2

=QQn+1,n+2;n+1,n+2−Qn+1,n+1Qn+2,n+2.