Excitation spectrum of Bose-Einstein condensate The Gross-Pitaevskii equation
$$
i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^2\nabla^2}{2m}\psi+V\psi+U_0|\psi|^2\psi
$$
($V$ is the external potential, $U_0$ is the interaction constant) in conventionally used to describe dynamics of a Bose-Einstein condensate.
Also it is often used to find excitation spectrum. For example, in the uniform case $V=0$, assuming small perturbations [following C.J. Pethick, H. Smith, Bose-Einstein Condensation in Dilute Gases]
$$
\psi=\left\{\sqrt{n}+[ue^{i(\mathbf{qr}-\omega t)}-v^*e^{-i(\mathbf{qr}-\omega t)}]\right\}e^{-i\mu t/\hbar}
$$
and taking the equilibrium concentration $n$ and chemical potential $\mu=U_0n$, we get
$$
\hbar\omega=\sqrt{\left(\frac{\hbar^2q^2}{2m}+U_0n\right)^2-(U_0n)^2.}
$$
This is usual expression for the spectrum of Bogolyubov excitations which can also be found from microscopic mean-field theory. Excitation energies in more complicated nonuniform cases are also often found using the linearized Gross-Pitaevskii equation.
Now the question:

*

*Why does the linearized Gross-Pitaevskii equation, which is aimed to describe coherent motion of the condensate, at the same time describe the spectrum of incoherent thermal excitations, which do not belong to the condensate?


*Does any qualitative difference between coherent oscillations of the condensate and thermal excitations exist, while they are described by the same Gross-Pitaevskii equation?
 A: Thanks for the reference. A tentative answer:
I believe the resolution can be found at the beginning of Ch. 8 of the same book (Pethick and Smith). The authors expand the state around the ground state with fluctuations:
$$\hat{\psi}=\psi_0+\delta \hat{\psi}$$
If the fluctuations are not included, then one gets a classical field theory for the ground state, which follows the Gross-Pitaevskii equations and indeed cannot account for non-condensed atoms. However, the fluctuation term, since it is describing deviations from the ground state, describes non-condensed bosons. Indeed, later on (in eq. 8.35 and following) the authors show how much the condensate fraction is reduced for a given amount of excitations.
In general, both time-dependent fluctations of the condensate and time-dependent scattering out of the condensate are possible. The difference is essentially adiabatic versus diabatic evolution. If you imagine a time-varying external potential that changes sufficiently slowly that the condensate can evolve adiabatically, then the resulting evolution of the condensate may be described using the G-P equation. However, an external potential that causes diabatic transitions that scatter particles out of the condensate must be analyzed with the quantum treatment described above.
