Quantum Mechanics of Multi-Prong Potentials

Some aspects of the bound state and scattering properties of a quantum mechanical particle in an arbitrary scalar N-prong potential are considered. Such a study is relevant in applications to mesoscopic devices. Multi-prong potentials are in some way intermediate between one- and two-dimensional systems. In contrast to the one-dimensional situation, where there is no degeneracy, the energy levels for the case of N identical prongs exhibit an alternating pattern of non-degeneracy and (N-1)-fold degeneracy. We generalize the techniques of supersymmetric quantum mechanics to multi-prong systems and generate solutions to new N-prong systems. Solutions for prongs of arbitrary lengths are developed. Scattering on piecewise constant potentials and tunnelling in N-well potentials are discussed in detail. Since our treatment is for general values of N, the results can be studied in the large N-limit. A somewhat surprising result is that a free particle incident on an N-prong vertex undergoes continuously increased backscattering as the number of prongs is increased.