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 $\Delta$ between the ground state energy $\varepsilon_0$ and the first excited state, i.e., $\varepsilon=\varepsilon_0$ for $k=0$ and $\varepsilon=\varepsilon_0 +\Delta+ c_sk^s$, for $k>0$, where $\hbar \mathbf{k}$ 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 $\Delta$ 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 $\Delta$ 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 $\Delta > 0$, a Bose-Einstein critical temperature $T_c \neq 0$ exists for any $d > 0$ at which 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 of $d/s$. At sufficiently high temperatures $\Delta$- independent classical results are recovered. However, for temperatures below the critical one the gap effects are predominant. For $\Delta = 0$ we recover previous reported results.