8.6 COE and GOE
273
approach in more detail in the next subsection. The second approach begins with
the reaction matrix formulation of the scattering matrix given in Eq. (8.48). The
Hamiltonian, ¯
H in , is then assumed to belong to one of the Gaussian ensembles,
and the consequence of this for scattering processes is investigated. This second
approach is the one we follow in this subsection.
When the incident energy of an incident particle, E, is far enough from the
threshold energy at which a new channel opens (so evanescent modes can be
neglected), the scattering matrix can be written in the form
¯
S M = U
†
k ·
¯
1 M − i ¯
K M
¯
1 M + i ¯
K M
·U
†
k ,
(8.100)
where ¯
1 M is an M×M unit matrix and ¯
K M is the reaction matrix,
¯
K M = ¯
w
T
·
1
E ¯
1 N − ¯
H N
· ¯
w.
(8.101)
In Eq. (8.101), ¯
1 N is an N×N unit matrix, and the N ×N Hamiltonian matrix, ¯
H N ,
describes the dynamics inside the cavity if N cavity eigenstates are kept. Note that
Eq. (8.101) reduces to Eq. (8.50) if we write ¯
H N = ¯
O· ¯
H in · ¯
O T and ¯
w = ¯
O· ¯
w, where
¯
O is the orthogonal matrix that diagonalizes the Hamiltonian ¯
H N . The number of
channels open in the leads (propagating modes) depends on the energy, E, of the
incident particle. Let us consider an energy E for which M channels are open in the
leads. Then the matrix ¯
w in Eq. (8.101) is an N×M matrix, and the matrices ¯
K M and
¯
S M are M×M matrices. If the Hamiltonian ¯
H N , which governs the deterministic
dynamics in the cavity, is replaced by a random Hamiltonian matrix, then the Smatrix, ¯
S M , will also be a random matrix.
In the subsections below, we will determine the scattering properties of a system
whose dynamics is governed by a random Hamiltonian matrix, ¯
H N . We will only
discuss scattering systems whose Hamiltonians are invariant under time reversal
and are rotationally invariant, so the Hamiltonian matrix elements are distributed
according to the Gaussian orthogonal ensemble (see Chap. 6). Our discussion can
be generalized to systems whose Hamiltonians are distributed according to the
Gaussian unitary ensemble or the Gaussian symplectic ensemble.
It is natural to ask the following question. For a scattering system whose
dynamics is governed by a random Hamiltonian matrix, with matrix elements
distributed according to the Gaussian orthogonal ensemble, will the S-matrix be
distributed according to the circular orthogonal ensemble? In other words, what is
the connection between these two ensembles? In subsections below, we will show
that a GOE random Hamiltonian, ¯
H N , does indeed give rise to a COE random Smatrix, but only under very special conditions.
The proof showing how a GOE Hamiltonian gives a COE S-matrix involves
several steps. In Sect. 8.6.1 below, we first review some key properties of the
circular orthogonal ensemble (COE) (a more extensive discussion of COE is given
273
approach in more detail in the next subsection. The second approach begins with
the reaction matrix formulation of the scattering matrix given in Eq. (8.48). The
Hamiltonian, ¯
H in , is then assumed to belong to one of the Gaussian ensembles,
and the consequence of this for scattering processes is investigated. This second
approach is the one we follow in this subsection.
When the incident energy of an incident particle, E, is far enough from the
threshold energy at which a new channel opens (so evanescent modes can be
neglected), the scattering matrix can be written in the form
¯
S M = U
†
k ·
¯
1 M − i ¯
K M
¯
1 M + i ¯
K M
·U
†
k ,
(8.100)
where ¯
1 M is an M×M unit matrix and ¯
K M is the reaction matrix,
¯
K M = ¯
w
T
·
1
E ¯
1 N − ¯
H N
· ¯
w.
(8.101)
In Eq. (8.101), ¯
1 N is an N×N unit matrix, and the N ×N Hamiltonian matrix, ¯
H N ,
describes the dynamics inside the cavity if N cavity eigenstates are kept. Note that
Eq. (8.101) reduces to Eq. (8.50) if we write ¯
H N = ¯
O· ¯
H in · ¯
O T and ¯
w = ¯
O· ¯
w, where
¯
O is the orthogonal matrix that diagonalizes the Hamiltonian ¯
H N . The number of
channels open in the leads (propagating modes) depends on the energy, E, of the
incident particle. Let us consider an energy E for which M channels are open in the
leads. Then the matrix ¯
w in Eq. (8.101) is an N×M matrix, and the matrices ¯
K M and
¯
S M are M×M matrices. If the Hamiltonian ¯
H N , which governs the deterministic
dynamics in the cavity, is replaced by a random Hamiltonian matrix, then the Smatrix, ¯
S M , will also be a random matrix.
In the subsections below, we will determine the scattering properties of a system
whose dynamics is governed by a random Hamiltonian matrix, ¯
H N . We will only
discuss scattering systems whose Hamiltonians are invariant under time reversal
and are rotationally invariant, so the Hamiltonian matrix elements are distributed
according to the Gaussian orthogonal ensemble (see Chap. 6). Our discussion can
be generalized to systems whose Hamiltonians are distributed according to the
Gaussian unitary ensemble or the Gaussian symplectic ensemble.
It is natural to ask the following question. For a scattering system whose
dynamics is governed by a random Hamiltonian matrix, with matrix elements
distributed according to the Gaussian orthogonal ensemble, will the S-matrix be
distributed according to the circular orthogonal ensemble? In other words, what is
the connection between these two ensembles? In subsections below, we will show
that a GOE random Hamiltonian, ¯
H N , does indeed give rise to a COE random Smatrix, but only under very special conditions.
The proof showing how a GOE Hamiltonian gives a COE S-matrix involves
several steps. In Sect. 8.6.1 below, we first review some key properties of the
circular orthogonal ensemble (COE) (a more extensive discussion of COE is given