Distribution functions of the degenerate Fermi gas taking account of pair correlations

In order to extend the Thomas-Fermi statistical method to superconducting-type pairing correlation systems within the framework of the Hartree-Fock-Bogolyubov theory, the single-particle distribution function and pair correlation distribution function are calculated for the degenerate Fermi gas up to terms of the h3 order by the quasiclassical method. In connection with the solution of this problem, a method of obtaining quasiclassical estimates of operator functions is developed, which is more general than the Kirzhnits method and is also of an independent interest