278
8 Manifestations of Chaos in Quantum Scattering Processes
a real symmetric matrix, the ensemble is called the Lorentzian orthogonal ensemble
(OE).
To find the normalization constant C N , if we make a change of variables and
introduce a new N×N real symmetric matrix, ¯
X N , such that ¯
X N = ( ¯
H N −
e ¯
1 N )/λ. Both ¯
H N and ¯
X N are diagonalized by the same N ×N orthogonal matrix,
¯
O N . If the eigenvalues of ¯
H N are e j for j = 1, . . . , N, then the eigenvalues of
¯
X N are x j = (e j − −e)/λ for j = 1, . . . , N. We can now write the normalization
condition in the form
P
N ( ¯
H N )dd H N = ˜
C N
dd X N
Det[ ¯
1 N + ¯
X 2
N ] (N +1)/2
= 1,
(8.128)
where dd X N = 2 N(N−1)/4 dX 1,1 . . . dX N,N dX 1,2 . . . dX N −1,N . The value of the
integral can be obtained from Eq. (8.119) and yields the normalization constant for
the Lorentzian orthogonal ensemble,
˜
C N = 2
−N(N−1)/4 π
−N(N+1)/4
N
j =1
(j )
(8.129)
(Hua 1963).
8.6.3 The Relation Between COE and OE
The circular orthogonal ensemble gives the distribution of matrix elements of an
N ×N symmetric unitary matrix, ¯
U N , that has eigenvalues e iθ 1 , . . . , e iθ N and is
diagonalized by an N×N orthogonal matrix, ¯
O N . We can relate the unitary matrix
¯
U N to a real symmetric Hermitian matrix, ¯
X N , through the expression
¯
U N =
¯
1 N + i ¯
X N
¯
1 N − i ¯
X N
,
(8.130)
so that ¯
X N is also diagonalized by the orthogonal matrix ¯
O N . If ¯
X N has eigenvalues
x j with j = 1, . . . , N, then the eigenvalues of ¯
U N and ¯
X N are related via the
expressions
e
iθ j =
1 + ix j
1 − ix j
, θ j = 2tan
−1 (x j ), so dθ j =
2dx j
1 + x 2
j
.
(8.131)
Let us remember that the probability distribution in COE is uniform so that
P C
N ( ¯
U N ) =
1
U N
(see Eq. (8.120)). Therefore,
8 Manifestations of Chaos in Quantum Scattering Processes
a real symmetric matrix, the ensemble is called the Lorentzian orthogonal ensemble
(OE).
To find the normalization constant C N , if we make a change of variables and
introduce a new N×N real symmetric matrix, ¯
X N , such that ¯
X N = ( ¯
H N −
e ¯
1 N )/λ. Both ¯
H N and ¯
X N are diagonalized by the same N ×N orthogonal matrix,
¯
O N . If the eigenvalues of ¯
H N are e j for j = 1, . . . , N, then the eigenvalues of
¯
X N are x j = (e j − −e)/λ for j = 1, . . . , N. We can now write the normalization
condition in the form
P
N ( ¯
H N )dd H N = ˜
C N
dd X N
Det[ ¯
1 N + ¯
X 2
N ] (N +1)/2
= 1,
(8.128)
where dd X N = 2 N(N−1)/4 dX 1,1 . . . dX N,N dX 1,2 . . . dX N −1,N . The value of the
integral can be obtained from Eq. (8.119) and yields the normalization constant for
the Lorentzian orthogonal ensemble,
˜
C N = 2
−N(N−1)/4 π
−N(N+1)/4
N
j =1
(j )
(8.129)
(Hua 1963).
8.6.3 The Relation Between COE and OE
The circular orthogonal ensemble gives the distribution of matrix elements of an
N ×N symmetric unitary matrix, ¯
U N , that has eigenvalues e iθ 1 , . . . , e iθ N and is
diagonalized by an N×N orthogonal matrix, ¯
O N . We can relate the unitary matrix
¯
U N to a real symmetric Hermitian matrix, ¯
X N , through the expression
¯
U N =
¯
1 N + i ¯
X N
¯
1 N − i ¯
X N
,
(8.130)
so that ¯
X N is also diagonalized by the orthogonal matrix ¯
O N . If ¯
X N has eigenvalues
x j with j = 1, . . . , N, then the eigenvalues of ¯
U N and ¯
X N are related via the
expressions
e
iθ j =
1 + ix j
1 − ix j
, θ j = 2tan
−1 (x j ), so dθ j =
2dx j
1 + x 2
j
.
(8.131)
Let us remember that the probability distribution in COE is uniform so that
P C
N ( ¯
U N ) =
1
U N
(see Eq. (8.120)). Therefore,