https://doi.org/10.1140/epjp/s13360-023-04155-2
Regular Article
Infinite-dimensional representations of cubic and quintic algebras and special functions
School of Mathematics and Physics, The University of Queensland, 4072, Brisbane, QLD, Australia
b
junze.zhang@uqconnect.edu.au
Received:
2
March
2023
Accepted:
2
June
2023
Published online:
16
June
2023
Finite and Infinite-dimensional representations of symmetry algebras play a significant role in determining the spectral properties of physical Hamiltonians. In this paper, we introduce and apply a practical method to construct infinite dimensional representations of certain polynomial algebras which appear in the context of quantum superintegrable systems. Explicit construction of these representations is a non-trivial task due to the non-linearity of the polynomial algebras. Our method has similarities with the induced module construction approach in the context of Lie algebras and allows the construction of states of the superintegrable systems beyond the reach of the separation of variables. Our primary focus is the representations of the polynomial algebras underlying superintegrable systems in 2D Darboux spaces. As a result, we are able to construct a large number of states in terms of complicated expressions of Airy, Bessel and Whittaker functions which would be difficult to obtain in other ways.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
