Some aspects of the bound state and scattering properties of a quantum
mechanical particle in an arbitrary scalar N-prong potential are cons
idered. 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 id
entical 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 develo
ped. Scattering on piecewise constant potentials and tunnelling in N-w
ell potentials are discussed in detail. Since our treatment is for gen
eral values of N, the results can be studied in the large N-limit. A s
omewhat surprising result is that a free particle incident on an N-pro
ng vertex undergoes continuously increased backscattering as the numbe
r of prongs is increased.