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