Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2. The Inhomogeneous Bogoliubov Hamiltonian
now determined within the Born approximation, i.e. on the right hand side
of (2.54), γ k ′′ is replaced with γ (0)
k ′′ .
In order to single out the elastically scattered wave, which is the most
relevant part, we apply the Sokhatsky-Weierstrass theorem [103] [x+i0] −1 =
P x −1 − iπδ(x) to the Green function, with P denoting the Cauchy principal
value. We consider the imaginary part of the Green function
ImG0(k ′ �
�∂ɛk
, ωk) = −πδ(�ωk − ɛk ′) = −π �
�
′
∂k ′
�
�
�
�
−1
δ(|k ′ | − k). (2.55)
From the first equality, it is seen that there is no contribution (to order
V ) from the anomalous scattering term, because � γ (0)
−k ′′
�∗ has a negative
frequency. Only elastic scattering to k ′ with |k ′ | = k is permitted (second
equality). In order to account for the finite system size, the Dirac δ-function
has to be averaged over the k-space resolution ∆k = 2π/L, which yields a
factor L/(2π). Thus, the elastic scattering amplitude at angle θ is given by
Im γ(s) (θ)
γ0
= − 1 2−d
L 2 W ′
k k
2
� �
�∂ɛk
�
� �
� ∂k �
−1
. (2.56)
The derivative of the dispersion relation can be expressed in terms of the
density of states (3.63) and evaluates as follows
� �
�∂ɛk
�−1
� �
� ∂k � = 2π ρ0(ɛk)
�
1 2 + k2ξ2 =
�k 2µξ 1 + k2ξ2 . (2.57)
The other quantity needed for the elastic scattering formula (2.56) is the
scattering element W k ′ k. For consistency with the Born approximation, the
first order in the impurity strength is sufficient.
First-order scattering element
The first-order scattering element (2.45c) is the product of the momentum
transfer of the external potential V ′
k −k and an envelope function w (1)
k ′ . The k
potential factor is the same as for free particles scattered at the potential
and provides the momentum transfer for scattering from k to k ′ . The envelope
function comprises the Bogoliubov dynamics and interaction effects.
It depends only on the dimensionless momenta kξ and k ′ ξ. Remarkably,
the envelope function features a node which separates forward scattering
from backscattering with opposite sign, figure 2.6(a). Let us come back to
elastic scattering. For ɛk = ɛk ′ and ɛ0 k = ɛ0 k ′, the scattering amplitude (2.45c)
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