In the case of a weakly interacting Bose-Einstein condensate the Gross-Pitaevskii equation is the motion equation that governs the evolution of the N-particle wavefunction. Starting from the Time-Dependent Gross-Pitaevskii equation: $$i\hbar\dot{\Psi}(x,t) = \left(-\frac{\hbar^2}{2m}\nabla^2_x + V(x,t) + g|\Psi(x,t)|^2 \right) \Psi(x,t)$$

Assuming the system to be time-independent the solution can be expressed as: $\Psi(x,t) = \psi(x) \phi(t)$, where, both fields satisfy: $$i\hbar\dot{\phi(t)} = -\mu \phi(t)$$ $$\mu\psi(x) = \left(-\frac{\hbar^2}{2m}\nabla^2_x + V(x,t) + g|\psi(x)|^2 \right)\psi(x)$$

My question is very simple but I don't see the point: how do we know that the constant of the separation of variables method corresponds to the chemical potential? Is related with the fact that $N=\int dx |\psi(x)|^2$ ?