Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

4. Disorder—Results and Limiting Cases
4.1.1. Direct derivation from hydrodynamic equations of
motion
We start from the equations of motion (2.18) for density and phase in the
limit of negligible quantum pressure. For the ground state this means that
the Thomas-Fermi formula (2.20) applies, n0(r) = nTF = (µ − V (r))/g. 1
The speed of sound characterizes the dynamics of small deviations δn(r, t) =
n(r, t) − n0(r) and δϕ(r, t) = ϕ(r, t) − ϕ0 from the ground state in the long
wave-length regime kξ ≪ 1. In terms of density and superfluid velocity, the
linearized equations of motion (2.18) read
∂tδn + ∇ · [n0(r)v] = 0, (4.1)
∂tv = − g
∇δn, (4.2)
m
and are recognized as the linearized versions of continuity equation and
Euler’s equation for an ideal compressible fluid, respectively. These can be
combined to a single classical wave equation
� � 2 2 2 1
c ∇ − ∂t δn = m∇ · [V (r)∇δn] . (4.3)
Translation invariance of the free equation suggests using a Fourier representation
in space and time,
� 2 2 2
ω − c k � � d ′ d k
δnk =
(2π) d � Vkk ′δnk ′. (4.4)
The disorder potential causes scattering k → k ′ of plane waves with an
amplitude
�V kk ′ = − 1
m (k · k′ ) V k−k ′. (4.5)
The factor k·k ′ originates from the mixed gradient in (4.3) and implies pure
p-wave scattering of sound waves as discussed in section 2.4, in contrast to
s-wave scattering of independent particles [23].
In contrast to the perturbation theory in section 3.3, the starting point
(4.3) is a second-order equation of motion and does not have the blockmatrix
structure. Apart from that, the diagrammatic perturbation theory
works exactly the same way. Analogously to (3.28), the following equation
defines the Green function � G
� �G −1
0 − � V � � G = 1. (4.6)
1 Note that within the Thomas-Fermi approximation, the average particle density does not change. Thus,
we need not distinguish Λµ and ΛN (section 3.4.5) in the present section 4.1.
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