The quasi-nonassociative exceptional $F(4)$ deformed quantum oscillator

We present the deformed (for the presence of Calogero potential terms) one-dimensional quantum oscillator with the exceptional Lie superalgebra $F(4)$ as spectrum-generating superconformal algebra. The Hilbert space is given by a $16$-ple of square-integrable functions. The energy levels are $\frac{2}{3}+n$, with $n=0,1,2,\dots$. The ground state is $7$ times degenerate. The excited states are $8$ times degenerate. The $(7,8,8,8,\dots )$ semi-infinite tower of states is recovered from the $(7;8;1)$ supermultiplet of the $\mathcal{N}=8$ worldline supersymmetry. The model is unique, up to similarity transformations, and admits an octonionic-covariant formulation which manifests itself as “quasi-nonassociativity”. This means, in particular, that the Calogero coupling constants are expressed in terms of the octonionic structure constants. The associated $F(4)$ superconformal quantum mechanics is also presented.