Interaction between subbands in a quasi-one-dimensional superconductor

In the framework of the Bogoliubov–de Gennes equation, we study the spinless $p$-wave superconductor in an infinite strip in the presence of some impurity. We analytically determine the wave functions of stable bound states with energies close to edge points of the energy gap. We prove that for a small impurity potential, the contribution of the nearest subbands to the wave functions in the case of energy values close to edge points is very small, and these energy levels are significantly closer to the gap edge than in the one-dimensional case. We also study the bound states with nearly zero energy values; in contrast to the one-dimensional case, they do not have the “particle–hole” symmetry. In the cases under study, in addition to the bound states, there also exit resonance states related to them.