E.2 Quaternions in the GOE
461
The matrix elements in Eq. (E.27) are defined in terms of harmonic oscillator
eigenstates as (for N even)
S
N
1,1 (x i , x j ) =
N/2−1
k=0
φ 2k (x i )φ 2k (x j ) − φ
2k (x i )
x j
0
dt φ 2k (t)
,
(E.28)
S
N
1,2 (x i , x j ) =
N/2−1
k=0
φ
2k (x i )φ 2k (x j ) − φ 2k (x i )φ
2k (x j )
,
(E.29)
S
N
2,1 (x i , x j ) =
N (x i , x j ) − i , x j ),
(E.30)
where
N (x i , x j ) =
N/2−1
k=0
x i
0
dt φ 2k (t)φ 2k (x j ) − φ 2k (x i )
x j
0
dt φ 2k (t)
(E.31)
and
i , x j ) = i − x j ) −
1
2
(E.32)
( i − x j ) is the Heaviside function and has the property that = 0 for x < 0,
=
1
2 for x = 0, and = 1 for x > 0). The functions in Eqs. (E.29)–(E.31)
satisfy the conditions
S
N
1,2 (x j , x j ) = 0, S
N
2,1 (x j , x j ) = 0, and
N (x j , x j ) = 0.
(E.33)
Also,
S
N
1,2 (x i , x j ) = −S
N
1,2 (x j , x i ), ,
N (x i , x j ) = −
N (x j , x i ),
and S
N
2,1 (x i , x j ) = −S
N
2,1 (x j , x i ).
(E.34)
The quaternion ¯
Q N is self-dual (see Eq. (E.12)). Therefore the quaternion determinant and the determinant of ¯
Q N will have the form
Qdet[ ¯
Q N ] = A
1 0
0 1
and Det[ ¯
Q N ] = A
2 ,
(E.35)
where A is some function of energies, x 1 , . . ., x N .
We next establish a relation between the matrix ¯
K N (Eq. (E.24)) and the quaternion matrix ¯
Q N . Using the definition of S N
1,1 (x i , x j ), S N
1,2 (x i , x j ), and S N
2,1 (x i , x j )
in terms of harmonic oscillator wave functions, it is straightforward to show that