We show that the quasiclassical Green's Function for Fermi liquids can be constructed from the solutions of the Bogoliubov-de Gennes equation within the Andreev approximation and derive self-consistent relations to be satisfied by the quasiclassical Green's function at the surfaces. The so-called normalization condition for the quasiclassical Green's function is obtained from this self-consistent relation. We consider a specularly reflecting wall, a randomly rippled wall, and a proximity boundary as model surfaces. Our boundary condition for the randomly rippled wall is different from that derived by Buchholtz and Rainer and Buchholtz.
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Nagai, K., Hara, J. Boundary conditions for quasiclassical Green's Function for superfluid Fermi systems. J Low Temp Phys 71, 351–367 (1988). https://doi.org/10.1007/BF00116868
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DOI: https://doi.org/10.1007/BF00116868