  Turbulent fluid dynamics typically involves excitations on many different
length scales. Classical incompressible fluids can be cleanly represented in
Fourier space enabling spectral analysis of energy cascades and other
turbulence phenomena. In quantum fluids, additional phase information and
singular behaviour near vortex cores thwarts the direct extension of standard
spectral techniques. We develop a formal and numerical spectral analysis for
$U(1)$ symmetry-breaking quantum fluids suitable for analyzing turbulent flows,
with specific application to the Gross-Pitaevskii fluid. Our analysis builds
naturally on the canonical approach to spectral analysis of velocity fields in
compressible quantum fluids, and establishes a clear correspondence between
energy spectral densities, power spectral densities, and autocorrelation
functions, applicable to energy residing in velocity, quantum pressure,
interaction, and potential energy of the fluid. Our formulation includes all
quantum phase information and also enables arbitrary resolution spectral
analysis, a valuable feature for numerical analysis. A central vortex in a
trapped planar Bose-Einstein condensate provides an analytically tractable
example with spectral features of interest in both the infrared and ultraviolet
regimes. Sampled distributions modelling the dipole gas, plasma, and clustered
regimes exhibit velocity correlation length increasing with vortex energy,
consistent with known qualitative behaviour across the vortex clustering
transition. The spectral analysis of compressible quantum fluids presented here
offers a rigorous tool for analysing quantum features of superfluid turbulence
in atomic or polariton condensates.
