Andreev reflection in the $p$-wave superconductor–normal metal contact

In this paper, the Andreev reflection is mathematically rigorously studied for the matrix differential Bogolyubov–de Gennes Hamiltonian. This Hamiltonian describes electrons and holes in a one-dimensional hybrid structure normal metal–$p$-wave superconductor. In this case, the physically correct symmetrized form of the Hamiltonian is used, which is described in the article. The Hamiltonian contains two delta-shaped potentials, one of which models an impurity in a superconductor, and the second characterizes the “transparency” of the junction normal metal–superconductor. It is proved that in the case of the topological phase there is an ideal Andreev reflection, i.e., an electron incident from the side of a normal metal (this electron has energy in the lacuna (superconducting gap), which is in the spectrum of the Bogolyubov-de Gennes Hamiltonian) with probability one, is reflected as a hole, regardless of the parameters of potentials describing the impurity and the “transparency” of the junction. For the nontopological phase, the formulas for probabilities of hole (Andreev) reflection and electron (normal) reflection are found. As is common in the study of hybrid structures, the matching method is used.