Within the context of Supersymmetric Quantum Mechanics and its related
hierarchies of integrable quantum Hamiltonians and potentials, a general
programme is outlined and applied to its first two simplest illustrations.
Going beyond the usual restriction of shape invariance for intertwined
potentials, it is suggested to require a similar relation for Hamiltonians in
the hierarchy separated by an arbitrary number of levels, N. By requiring
further that these two Hamiltonians be in fact identical up to an overall shift
in energy, a periodic structure is installed in the hierarchy of quantum
systems which should allow for its solution. Specific classes of orthogonal
polynomials characteristic of such periodic hierarchies are thereby generated,
while the methods of Supersymmetric Quantum Mechanics then lead to generalised
Rodrigues formulae and recursion relations for such polynomials. The approach
also offers the practical prospect of quantum modelling through the engineering
of quantum potentials from experimental energy spectra. In this paper these
ideas are presented and solved explicitly for the cases N=1 and N=2. The latter
case is related to the generalised Laguerre polynomials, for which indeed new
results are thereby obtained. At the same time new classes of integrable
quantum potentials which generalise that of the harmonic oscillator and which
are characterised by two arbitrary energy gaps are identified, for which a
complete solution is achieved algebraically.Comment: 1+19 page