Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 35C / d / Solution

In the continuum limit the number outside the ground state isAt the largest allowed chemical potential, $\mu =0$, the integrand behaves near zero as $E^{d/\alpha -2}$. The integral is finite exactly when $d/\alpha >1$. Excess particles must then occupy the ground state, so a Bose-Einstein condensate exists precisely for $\alpha <d$.