We present a theoretical analysis of some unexplored aspects of relaxed Bose-Einstein condensate dark matter (BECDM) haloes. This type of ultralight bosonic scalar field dark matter is a viable alternative to the standard cold dark matter (CDM) paradigm, as it makes the same large-scale predictions as CDM and potentially overcomes CDMs small-scale problems via a galaxy-scale de Broglie wavelength. We simulate BECDM halo formation through mergers, evolved under the Schrodinger-Poisson equations. The formed haloes consist of a soliton core supported against gravitational collapse by the quantum pressure tensor and an asymptotic $r^{-3}$ NFW-like profile. We find a fundamental relation of the core=to-halo mass with the dimensionless invariant $Xi equiv lvert E rvert/M^3/(Gm/hbar)^2$ or $M_{rm c}/M simeq 2.6 Xi^{1/3}$, linking the soliton to global halo properties. For $r geq 3.5 ,r_{rm c}$ core radii, we find equipartition between potential, classical kinetic, and quantum gradient energies. The haloes also exhibit a conspicuous turbulent behavior driven by the continuous reconnection of vortex lines due to wave interference. We analyse the turbulence 1D velocity power spectrum and find a $k^{-1.1}$ power-law. This suggests the vorticity in BECDM haloes is homogeneous, similar to thermally-driven counterflow BEC systems from condensed matter physics, in contrast to a $k^{-5/3}$ Kolmogorov power-law seen in mechanically-driven quantum systems. The mode where the power spectrum peaks is approximately the soliton width, implying the soliton-sized granules carry most of the turbulent energy in BECDM haloes.
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