280
8 Manifestations of Chaos in Quantum Scattering Processes
The density of energy levels can be obtained from the one-level cluster function,
T
N (e 1 ). Because eigenphases are uniformly distributed in COE, the one-level COE
cluster function is T C
N (θ 1 ) =
N
2π , and the average spacing of levels in the COE is
D = 2π/N. Now consider the cluster function T
N (e 1 ) in the neighborhood e 1 ≈≈e,
lim
e 1 →→e
T
N (e 1 ) ≈ lim
e 1 →→e
T
C
N (2x 1 )
2λ
λ 2 + (e 1 − −e) 2
=
N
πλ
.
(8.136)
Thus, for the OE, the average spacing of levels, at the center of the distribution, is
= πλ/N = Dλ/2.
We can now take the final step in our proof. First consider the COE cluster
function, T C
N (θ 1 , . . . , θ n ). Let ξ j = θ j /D = Nθ j /2π , and take the limit N → ∞
so ξ j remains finite. Then
lim
N →∞
π
−π
dθ 1 . . .
π
−π
dθ n T
C
N (θ 1 , . . . , θ n )
= lim
N →∞
π/D
−π/D
dξ 1 . . .
π/D
−π/D
dξ n D
n T
C
N (Dξ 1 , . . . , Dξ n )
≡
∞
−∞
dξ 1 . . .
∞
−∞
dξ n Y
C (ξ 1 , . . . , ξ n ).
(8.137)
Next consider the OE cluster function, T
N (e 1 , . . . , e n ). We can let 2(e j −−e)/λ =
Dξ j so (e j − −e) = ξ j . Then
lim
N →∞
∞
−∞
de 1 . . .
∞
−∞
de n T
N (e 1 , . . . , e n )
= lim
N →∞
∞
−∞
de 1 . . .
∞
−∞
de n T
C
N (2tan
−1 (x(e 1 )), . . . , 2tan
−1 (x(e n )))
×
n
j =1
2λ
λ 2 + (e j − −e) 2
=
∞
−∞
dξ 1 . . .
∞
−∞
dξ n Y
C (ξ 1 , . . . , ξ n ).
(8.138)
Thus, the n-body cluster functions for COE and OE become identical in the
limit N→∞ and e j →→e so that ξ j = (e j − −e)// remains finite. Therefore
the statistical properties of the two ensembles become identical in this limit. It
is important to note that in this limit we are focusing on the equivalence of the
ensembles only in the neighborhood of the peak of the probability distribution.
The probability distribution characterizing the OE does not have a finite variance,
whereas those for COE and GOE do have a finite variance.