Harmonic-trap energy-zero convention

The exact three-dimensional oscillator ground energy is $3\hslash \omega /2$ above the potential minimum. Subtracting it sets the condensate energy to zero. The commonly used $g(ϵ)={ϵ}^2/[2(\hslash \omega {)}^3]$ is the leading continuum term; it does not include the finite-level corrections from shifting the energy origin.