  We perform a rigorous analysis of the anelastic approximation for the
Gross-Pitaevskii equation with $x$-dependent chemical potential. For general
initial data and periodic boundary condition, we show that as $\eps\to 0$,
equivalently the Planck constant tends to zero, the density $|\psi^{\eps}|^{2}$
converges toward the chemical potential $\rho_{0}(x)$ and the velocity field
converges to the anelastic system. When the chemical potential is a constant,
the anelastic system will reduce to the incompressible Euler equations. The
resonant effects the singular limit process and it can be overcome because of
oscillation-cancelation.
