Nonlinear Phenomena in Complex Systems, 1999, Vol.2, No.2, pp.102--116

NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal

1999, Volume 2, Number 2, pp.102--116

Discretized Schrödinger Eigenfunctions and q-Hypergeometric Series on Deforming Geometric Progressions.
Andreas Ruffing

Discretizations of the Schrödinger equation are introduced on geometric progressions R_q := {+qⁿ, -qⁿ, | n in Z}, q > 1. The symmetries of the geometric progressions are elaborated. We investigate the influence of this discretization to q-deformations of Hermite polynomials. The limits q -> 1 and q -> infinity as deformations of R_q are considered. The first limit, q -> 1, is related to an approximation theoretical problem for step functions in L²(R). The second limit, q -> infinity, is related to the theory of topological deformations on compact Riemann surfaces. Both limits are related to each other. Proceeding into this direction, one obtains the fascinating fact that quantum group structures can be related to topological degeneration effects. The results finally contribute to a better mathematical understanding of quantum models with dilatative supersymmetry.

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