Bose gas with generalized dispersion relation plus an energy gap

Bose–Einstein condensation in a Bose gas is studied analytically, in any positive dimensionality (d > 0) for identical bosons with any energy-momentum positive-exponent (s > 0) plus an energy gap Δ between the ground state energy ε ₀ and the first excited state, i.e. ε = ε ₀ for k = 0 and $\epsilon ={\epsilon }_0+\Delta +c_s{k}^s$, for k > 0, where $\hslash k$ is the particle momentum and c _s a constant with dimensions of energy multiplied by a length to the power s > 0. Explicit formula with arbitrary d/s and Δ are obtained and discussed for the critical temperature and the condensed fraction, as well as for the equation of state from where we deduce a generalized Δ independent thermal de Broglie wavelength. Also the internal energy is calculated from where we obtain the isochoric specific heat and its jump at T _c. When Δ > 0, for any d > 0 exists a Bose–Einstein critical temperature $T_c\ne 0$ where the internal energy shows a peak and the specific heat shows a jump. Both the critical temperature and the specific heat jump increase as functions of the gap but they decrease as functions of d/s. At sufficiently high temperatures Δ-independent classical results are recovered while for temperatures below the critical one the gap effects are predominant. For Δ = 0 we recover previous reported results. (paper)