8.3 Scattering Theory
255
The complex values of energy E at which the S-matrix has poles are given by the
condition
Det[E ¯
1 N − ¯
H in + i ¯
w· ¯
w
T
] = 0.
(8.56)
Note that ¯
w· ¯
w T depends on the energy E. One can obtain an approximate idea of
where poles may be found in the neighborhood of a particular energy E = E 0 (away
from energy thresholds where new channels open). This is done by fixing the value
of the energy E in ¯
w· ¯
w T to be E = E 0 and replacing the matrix ¯
w· ¯
w T by the matrix
¯
w o · ¯
w T
o ≡( ¯
w· ¯
w T ) E=E 0 . The matrix ¯
H eff ≡ ¯
H in − i ¯
w o · ¯
w T
o can be diagonalized via a
unitary transformation, ¯
U . Then we find
¯
U
†
·(E ¯
1 N − ¯
H in + i ¯
w o · ¯
w
T
o )· ¯
U = E ¯
1 N − ¯
E eff ,
(8.57)
where ¯
E eff has matrix elements ( ¯
E eff ) n,n = (E n − ii n )δ n,n . Equation (8.56) can
then be written
det[E ¯
1 N − ¯
H in + i ¯
w o · ¯
w
T
o ] =
∞
n=1
[E − E n + ii n ] = 0.
(8.58)
Thus, the poles of the S-matrix, in this approximation, are just the eigenvalues of
the “effective” Hamiltonian (non-Hermitian Hamiltonian), ¯
H eff = ¯
H in − i ¯
w o · ¯
w T
o .
Based on this formalism, there have been several studies of the behavior of S-matrix
poles as parameters are varied (Rotter et al. 2000; Rotter 2001), and (Stöckmann
et al. 2002). One must be careful, however. If evanescent modes are important, then
their contributions must be included and this simple approximation may not apply.
The matrix ¯
w· ¯
w T is an N×N real symmetric matrix, and the matrix ¯
w T · ¯
w is
an M×M real symmetric matrix. The matrix ¯
w T · ¯
w is diagonalized by an M×M
orthogonal matrix, ¯
O M , so that
¯
O
T
M · ¯
w
T
· ¯
w· ¯
O M = ¯
M ,
(8.59)
where ( ¯
M ) i,j = v 2
i δ i,j for i, j = 1, . . . , M. The matrix ¯
w· ¯
w T is diagonalized by
an N ×N orthogonal matrix, ¯
O N , so that
¯
O
T
N · ¯
w· ¯
w
T
· ¯
O N = ¯
N ,
(8.60)
where ( ¯
N ) i,j = v 2
i δ i,j for i, j = 1, . . . , M, and ( ¯
N ) i,j = 0 otherwise. Thus,
the matrix ¯
w T · ¯
w has M eigenvalues (v 2
1 , v 2
2 , . . . , v 2
M ) and the matrix ¯
w· ¯
w T has N
eigenvalues (v 2
1 , v 2
2 , . . . , v 2
M , 0, 0, . . . , 0).
Example 8.2 (S-matrix for 1-d Scattering System)
For the one-dimensional scattering system considered in Example 8.1, the scattering
“matrix” is the function S(E). It is related to the reaction function, R(E), obtained in
Example 8.1, through the relation
255
The complex values of energy E at which the S-matrix has poles are given by the
condition
Det[E ¯
1 N − ¯
H in + i ¯
w· ¯
w
T
] = 0.
(8.56)
Note that ¯
w· ¯
w T depends on the energy E. One can obtain an approximate idea of
where poles may be found in the neighborhood of a particular energy E = E 0 (away
from energy thresholds where new channels open). This is done by fixing the value
of the energy E in ¯
w· ¯
w T to be E = E 0 and replacing the matrix ¯
w· ¯
w T by the matrix
¯
w o · ¯
w T
o ≡( ¯
w· ¯
w T ) E=E 0 . The matrix ¯
H eff ≡ ¯
H in − i ¯
w o · ¯
w T
o can be diagonalized via a
unitary transformation, ¯
U . Then we find
¯
U
†
·(E ¯
1 N − ¯
H in + i ¯
w o · ¯
w
T
o )· ¯
U = E ¯
1 N − ¯
E eff ,
(8.57)
where ¯
E eff has matrix elements ( ¯
E eff ) n,n = (E n − ii n )δ n,n . Equation (8.56) can
then be written
det[E ¯
1 N − ¯
H in + i ¯
w o · ¯
w
T
o ] =
∞
n=1
[E − E n + ii n ] = 0.
(8.58)
Thus, the poles of the S-matrix, in this approximation, are just the eigenvalues of
the “effective” Hamiltonian (non-Hermitian Hamiltonian), ¯
H eff = ¯
H in − i ¯
w o · ¯
w T
o .
Based on this formalism, there have been several studies of the behavior of S-matrix
poles as parameters are varied (Rotter et al. 2000; Rotter 2001), and (Stöckmann
et al. 2002). One must be careful, however. If evanescent modes are important, then
their contributions must be included and this simple approximation may not apply.
The matrix ¯
w· ¯
w T is an N×N real symmetric matrix, and the matrix ¯
w T · ¯
w is
an M×M real symmetric matrix. The matrix ¯
w T · ¯
w is diagonalized by an M×M
orthogonal matrix, ¯
O M , so that
¯
O
T
M · ¯
w
T
· ¯
w· ¯
O M = ¯
M ,
(8.59)
where ( ¯
M ) i,j = v 2
i δ i,j for i, j = 1, . . . , M. The matrix ¯
w· ¯
w T is diagonalized by
an N ×N orthogonal matrix, ¯
O N , so that
¯
O
T
N · ¯
w· ¯
w
T
· ¯
O N = ¯
N ,
(8.60)
where ( ¯
N ) i,j = v 2
i δ i,j for i, j = 1, . . . , M, and ( ¯
N ) i,j = 0 otherwise. Thus,
the matrix ¯
w T · ¯
w has M eigenvalues (v 2
1 , v 2
2 , . . . , v 2
M ) and the matrix ¯
w· ¯
w T has N
eigenvalues (v 2
1 , v 2
2 , . . . , v 2
M , 0, 0, . . . , 0).
Example 8.2 (S-matrix for 1-d Scattering System)
For the one-dimensional scattering system considered in Example 8.1, the scattering
“matrix” is the function S(E). It is related to the reaction function, R(E), obtained in
Example 8.1, through the relation