Spectrum generating functions for non-canonical quantum oscillators

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. After briefly recalling the notion of a Wigner quantum system, 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. We then go on to describe a general method for determining a spectrum generating function for an element of the Cartan subalgebra when working with a representation of any Lie (super)algebra. Herein, 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. This is done using various interesting computational results from the field of symmetric and supersymmetric Schur functions.