Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

3. Disorder
V 2
0 σ d Cd(qσ). We compute the first block of the Born approximation Σ B :=
Σ (2)
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Σ B (k, �ω) =
= V 2
�
V 2 d 1
0 σ
L d
2
0 σ d
L d
2
q
�
q
�
Cd(qσ) v (2)
k (k+q) k + v(1)
k k+qG0(k + q)v (1)
�
k+q k
�
⎡
�
Cd(qσ) ⎣w (2)
k (k+q) k +
� (1) �2 w k(k+q)
�ω − ɛk+q + i 0 −
� y (1)
k(k+q)
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�2 ⎤
⎦ .
�ω + ɛk+q
(3.39)
The infinitesimally shifted pole in the propagator G0(ɛk) ′
k is split into real
and imaginary part, using the Sokhatsky-Weierstrass theorem [103] [�ω −
ɛk ′ + i0]−1 = P[�ω − ɛk ′]−1 − iπδ(�ω − ɛk ′). Thus, elastic scattering enters
in the imaginary part and inelastic processes are captured by the Cauchy
principal value P of the integral (3.39). For the actual evaluation of the
integral, the first-order and second-order envelope functions w ′
kk and ykk ′,
given in (2.45c) (2.45d) and (2.50), are needed.
Transferring the results to other types of disorder
In the Born approximation, the self-energy (3.39) depends only on the twopoint
correlator of the disorder potential
VkV −k ′ = (2π) d δ(k − k ′ ) V 2
0 σ d C(kσ). (3.40)
The field correlators (3.7) occur only in higher-order diagrams. The selfenergy
in the Born approximation is valid also for other types of disorder, like
for example a Gaussian model C G d (kσ) = (2π)d/2 exp(−k 2 σ 2 /2), as employed
e.g. in [122]. In subsection 4.1.4, we will come back to this and compare the
effect of speckle disorder to the effect of Gaussian disorder on the disorderaveraged
density of states.
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