G.3 Circular Orthogonal Ensemble (COE)
499
obtain JS N (θ ) from the Fourier series expansion of the square wave function N (θ ),
which is given by
N (θ ) =
1
π
∞
n=0
sin[(2n + 1) θ/2]
[(2n + 1)/2]
.
(G.36)
In Eq. (G.36), let n =
N
2 + k. Then it can be written (for N even)
N (θ ) =
1
π
−1
k=−
N
2
sin[(N + 2k + 1) θ/2]
[(N + 2k + 1)/2]
+
1
π
∞
k=0
sin[(N + 2k + 1) θ/2]
[(N + 2k + 1)/2]
.
(G.37)
In the first term on the right, let k = −p − 1. Then Eq. (G.37) takes the form
N (θ ) =
1
π
N
2 −1
p=0
sin[(N − 2p − 1) θ/2]
[(N − 2p − 1)/2]
+
1
π
∞
k=0
sin[(N + 2k + 1) θ/2]
[(N + 2k + 1)/2]
.
(G.38)
The first term on the right is IS N (θ ). Therefore, the second term on the right is
−JS N (θ ), and we obtain
JS N (θ ) = −
1
π
∞
k=0
sin[(N + 2k + 1) θ/2]
[(N + 2k + 1)/2]
.
(G.39)
The matrix elements of ¯
σ N 1 (i, j ) in Eqs. (G.32)–(G.35) satisfy the conditions
DS N (θ j , θ j ) = 0, JS N (θ j , θ j ) = 0, and IS N (θ j , θ j ) = 0.
(G.40)
Also,
DS N (θ i , θ j ) = −DS N (θ j , θ i ), IS N (θ i , θ j ) = −IS N (θ j , θ i ),
and JS N (θ i , θ j ) = −JS N (θ j , θ i ).
(G.41)
From Appendix E, we see that the quaternion ¯
Q
(1)
N is self-dual. This means that the
quaternion determinant and the determinant of ¯
Q
(1)
N will have the form
Qdet[ ¯
Q
(1)
N ] = A
1 0
0 1
and Det[ ¯
Q
(1)
N ] = A
2 ,
(G.42)
respectively, where A is some function of the angles θ 1 , . . ., θ N .