Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

(2)
k ′ q k
= ξ2
�
k ′′2 + k ′2 + k 2 + 3
2
+ � k ′2 + k 2 − k ′ · k � 2ξ −2 −(q−k ′ ) 2 −(q−k) 2
2ξ −2 +(k ′ −k) 2
� (q − k ′ ) 2 + (q − k) 2 �
� � �
2.3. Bogoliubov Excitations
2 � 1 + ξ2 (q−k ′ ) 2
2
�� 1 + ξ 2 (q−k) 2
2
� �
.
(2.48b)
Again, the scattering elements are transformed according to equation (2.41),
which brings us to the structure of the second-order scattering matrix
V (2)
k ′ k
1
=
L d
2
�
q
�
(2)
w k ′ q k y(2)
k ′ q k
y (2)
k ′ q k w(2)
k ′ q k
� Vk ′ −qVq−k
µ
=:
1
µL d
2
�
q
V k ′ −qv (2)
k ′ q k Vq−k = ⊛⊛ .
(2.49)
In particular, the diagonal element of W (2) is needed later. In this case, the
envelope function simplifies to
w (2)
k k+p k =
kξ (2k
�
2 + k2ξ2 2 + 3p2 + k · p)ξ2 (2 + p2ξ2 ) 2 . (2.50)
Higher orders. If needed, higher-order scattering vertices V (n) can be derived
systematically with the following prescription.
1. Compute the ground state (2.21) to the desired order, using the expansion
shown in box 2.1.
2. Compute the inverse field ¯ Φq = (Φ 2 0/Φ)q.
3. S (n) and R (n) are obtained by collecting all terms of order n in equations
(2.39) and (2.40).
4. Finally, apply the transformation (2.41) in order to obtain W (n) and
Y (n) .
V = V = ⊛ + ⊛⊛ + ⊛⊛⊛ + . . .
= V (1) + V (2) + V (3) + . . . (2.51)
The inhomogeneous Bogoliubov Hamiltonian is an essential cornerstone
for this entire work. It will be essential for both the single scattering in
section 2.4 and the disordered problem in chapter 3.
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