Supersymmetric quantum mechanics: Engineered hierarchies of integrable potentials and related orthogonal polynomials

作者：Iyela Daddy Balondo^*; Govaerts Jan; Hounkonnou M Norbert

来源：Journal of Mathematical Physics, 2013, 54(9): 093502.

DOI：10.1063/1.4819497

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 which should allow for its resolution. 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. In the context of dressing chains and deformed polynomial Heisenberg algebras, some partial results for N >= 3 also exist in the literature, which should be relevant to a complete study of the N >= 3 general periodic hierarchies.

• 出版日期2013-9