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We develop Green's function formalism to describe continuous multi-layeredquasi-one-dimensional setups described by piece-wise constant single-particleHamiltonians. The Hamiltonians of the individual layers are assumed to bequadratic polynomials in the momentum operator with matrix-valued(multichannel) coefficients. This, in particular, allows one to study transportin heterostructures consisting of multichannel conducting, superconducting, orinsulating components with band structures of arbitrary complexity. We find ageneral expression for the single-particle Green's function of the combinedsetup in terms of the bulk (translationally invariant) Green's functions of itsconstituents. Furthermore, we provide the expression for the global density ofstates of the combined system and establish the bound state equation in termsof bulk Green's functions. We apply our formalism to investigate the spectrumand current-phase relations in ordinary and topological Josephson junctions,additionally showing how to account for the effects of static disorder andlocal Coulomb interaction.

Green's functions of quasi-one-dimensional layered systems and their application to Josephson junctions