The n-dimensional (isotropic and non-isotropic) harmonic oscillator is studied as a Wigner quantum system. In particular, we focus on the energy spectrum of such systems. We show how to solve the compatibility conditions in terms of 𝔬𝔰𝔭(1|2n) generators, and also recall the solution in terms of 𝔤𝔩(1|n) generators. A method is described for determining a spectrum generating function for an element of the Cartan subalgebra when working with a representation of any Lie (super)algebra. Here, the character of the representation at hand plays a crucial role. This method is then applied to the n-dimensional isotropic harmonic oscillator, yielding explicit formulas for the energy eigenvalues and their multiplicities.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc93-0-15,
author = {Stijn Lievens and Joris Van der Jeugt},
title = {Spectrum generating functions for oscillators in Wigner's quantization},
journal = {Banach Center Publications},
volume = {95},
year = {2011},
pages = {189-197},
zbl = {1248.81048},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc93-0-15}
}
Stijn Lievens; Joris Van der Jeugt. Spectrum generating functions for oscillators in Wigner's quantization. Banach Center Publications, Tome 95 (2011) pp. 189-197. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc93-0-15/