On q-Deformations and Dunkl-Deformations of Harmonic Oscillators

• Andreas Ruffing
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 543)

Abstract

It is a well established fact that there are deeper connections between the theory of orthogonal polynomials on the one hand and properties of Schrödinger operators on the other hand. Those operators are assumed to act in conventionally used Hilbert spaces like for example $$\mathcal{L}^2 \left( {R^n } \right)$$. A prominent example for these connections are the classical continuous Hermite polynomials which correspond to Schrödinger operators with a quadratic po- tential. In the one dimensional case, the support of these polynomials is the real line. When dealing with a discretization of this support, one meets the next ingredient which enriches the investigation of orthogonal polynomials: It is the aspect of deformation. The idea of deforming polynomials plays a crucial role in the context of special functions. In the case of a q-deformation one sees that the deformation itself can be associated with discretizing the support for orthogonal polynomials. This allows to switch from polynomials defined in the continuum to polynomials defined on a geometric progression. Behind these observations seems to lie a more general concept: Deformati- ons can be related to discretizations or quantizations and vice versa. This concept has also turned out to be an important guideline in structures of non-commutative geometry, thus also in areas quite far away from the theory of orthogonal polynomials.

References

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