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One-loop omega-potential of quantum fields with ellipsoid constant-energy surface dispersion law

Rapidly convergent expansions of a one-loop contribution to the partition function of quantum fields with ellipsoid constant-energy surface dispersion law are derived. The omega-potential is naturally decomposed into three parts: the quasiclassical contribution, the contribution from the branch cut of the dispersion law, and the oscillating part. The low- and high-temperature expansions of the quasiclassical part are obtained. An explicit expression and a relation of the contribution from the cut with the Casimir term and vacuum energy are established. The oscillating part is represented in the form of the Chowla-Selberg expansion for the Epstein zeta function. Various resummations of this expansion are considered. The developed general procedure is applied to two models: massless particles in a box both at zero and non-zero chemical potential; electrons in a thin metal film. The rapidly convergent expansions of the partition function and average particle number are obtained for these models. In particular, the oscillations of the chemical potential of conduction electrons in graphene and a thin metal film due to a variation of sizes of the crystal are described.