Complex band-structure analysis and topological physics of Majorana nanowires

We review applications of complex band structure theory to describe Majorana states in nanowires and nanowire junctions. The dimensionality of the considered wires is gradually increased, from strictly 1D to quasi-1D with one and two transverse dimensions. The complex wave number analysis is applied to two main types of Majorana modes: end states in hybrid semiconductor wires and chiral edge states in hybrid wires of quantum-anomalous Hall insulators. In topological NS and NSN junctions with Majorana modes conductance oscillations and spatial distributions of density and current are described. Finally, the optical absorption in finite systems with both types of Majorana modes is considered.