Some problems of the theory of quantum statistical systems with an energy spectrum of the fractional-power type

We consider the problem of the effective interaction potential in a quantum many-particle system leading to the fractional-power dispersion law. We show that passing to fractional-order derivatives is equivalent to introducing a pair interparticle potential. We consider the case of a degenerate electron gas. Using the van der Waals equation, we study the equation of state for systems with a fractional-power spectrum. We obtain a relation between the van der Waals constant and the phenomenological parameter $\alpha$, the fractional-derivative order. We obtain a relation between energy, pressure, and volume for such systems: the coefficient of the thermal energy is a simple function of $\alpha$. We consider Bose–Einstein condensation in a system with a fractional-power spectrum. The critical condensation temperature for $1<\alpha<2$ is greater in the case under consideration than in the case of an ideal system, where $\alpha=2$.