Heat kernel approach for confined quantum gas

In this paper, based on the heat kernel technique, we calculate equations of state and thermodynamic quantities for ideal quantum gases in confined space with external potential. Concretely, we provide expressions for equations of state and thermodynamic quantities by means of heat kernel coefficients for ideal quantum gases. Especially, using an analytic continuation treatment, we discuss the application of the heat kernel technique to Fermi gases in which the expansion diverges when the fugacity $z>1$. In order to calculate the modification of heat kernel coefficients caused by external potentials, we suggest an approach for calculating the expansion of the global heat kernel of the operator $-\mathrm{\Delta }+U\left(x\right)$ based on an approximate method of the calculation of spectrum in quantum mechanics. At last, we discuss the properties of quantum gases under the condition of weak and complete degeneration, respectively.