$\mathbb{C}{\mathbb{P}}^N$-Rosochatius system, superintegrability, and supersymmetry

We propose a new superintegrable mechanical system on the complex projective space ${\mathbb{CP}}^N$ involving a potential term together with coupling to a constant magnetic fields. This system can be viewed as a ${\mathbb{CP}}^N$-analog of both the flat singular oscillator and its spherical analog known as the “Rosochatius system.” We find its constants of motion and calculate their (highly nonlinear) algebra. We also present its classical and quantum solutions. The system belongs to the class of “Kähler oscillators” admitting $SU(2|1)$ supersymmetric extension. We show that, in the absence of magnetic field and with the special choice of the characteristic parameters, one can construct $\mathcal{N}=4,d=1$ Poincaré supersymmetric extension of the system considered.

Figure 1

Frequencies for $N=1$, 2, 3.