Transcendental momentum quantization in semiconducting Rashba nanowires and zero energy states in their normal and superconducting phase

We study finite system properties of the canonical low energy model for a semiconducting nanowire with Rashba spin-orbit coupling. The case of an isolated wire as well as of one proximitized by an s-wave superconductor are considered. Already for the normal wire, the presence of spin-orbit coupling leads to eigenstates of the finite system composed of more than two momentum eigenstates. The quantization condition for the wavevectors is not that of a quantum box, but given instead by a transcendental equation linking the involved wavevectors. For the wire with superconducting pairing, the presence of electron and hole channels complicates the composition of the eigenstates. In this case we derive an approximate quantization condition close to the phase boundary, and a condition for the appearance of exact zero energy states. It can be satisfied both in the topological and in the trivial phase. Both the trivial and topological zero energy states contribute to the linear transport through Andreev reflection and direct transmission processes, with their relative importance depending on the degree of the states' localization at the boundary.