Let us rewrite Hamiltonian (1.1) in terms
of the creation and annihilation operators
and
in momentum representation
(2.9)
where we have included the external potential . We use the
Bogoliubov prescription
and we consider
as small perturbations.
To second order in
for
the external potential term can be written as
(2.10)
The term
must be calculated in the second Born
approximation in order to obtain an expression which is correct up
to second order in the particle-impurity scattering amplitude.
(2.11)
The part of the Hamiltonian which is independent of the external
potential can be diagonalized by the Bogoliubov transformation
(1.30). The Hamiltonian takes the form
(2.12)
where and are defined by (1.36) and (1.35) respectively.
The linear term in the quasiparticle operators can be eliminated by
means of the following transformation (analogous transformation, but for
a different model of the disorder was introduced in [16])
(2.13)
with defined by
(2.14)
The transformation (2.13) does not change the commutation
rules and the new quasiparticle operators
,
satisfy the usual bosonic commutation relations.
Finally, the Hamiltonian takes the form
(2.15)
The creation and annihilation particle operators
,
are obtained from the corresponding
quisiparticle operators
,
in the
following way