6.5 Inclusion of Continuum Couplings in the Pairing Model
281
where E η are the pair energies in the eigenstate K. The normalization constants c η
can be calculated from the following equation:
1
(c η G)
2
=
1
(C η )
2
=
N pair
q
w q
(2 q − E η )
2
.
(6.49)
In order to simplify notations, it is convenient to define
B
†
η = B
†
η;norm /C η ,
(6.50)
so that
|Ψ norm =
N pair
η=1
C η B
†
η |ν = C|Ψ ,
(6.51)
where
C =
N pair
η=1
C η and |Ψ =
N pair
η=1
B
†
η |ν .
The operators ˆ ˜
n, B η , and B 0 :
B
†
0 =
N
q
˜
b
†
q
√
w q
(6.52)
satisfy fundamental commutator relations borne by
ˆ ˜
n q , B †
η
,
ˆ ˜
n q , B
†
0
,
B η , B
†
η
,
B 0 , B †
η
, and
B
†
0 , B †
η
(see Exercise I).
Exercise I
Calculate the commutators
ˆ ˜
n q , B †
η
,
ˆ ˜
n q , B
†
0
,
B η , B
†
η
,
B 0 , B †
η
, and
B
†
0 , B †
η
using the commutation relations for operators ˜ ˆ
n q , ˜
b
†
q , ˜
b q (see
Eq. (6.46)).
The Hamiltonian of the generalized rational Gaudin model (6.44) expressed in these
operators reads
ˆ
H =
N
q
q ˆ ˜
n q + GB
†
0 B 0 .
(6.53)