The nine-dimensional MICZ-Kepler problem (9D MICZ KP) considers a charged particle moving in the Coulomb field with the presence of a SO(8) monopole in a nine-dimensional space. This problem received much effort recently, for example, exact solutions of the Schrödinger equation of the 9D MICZ KP have been given in spherical coordinates. In this paper, we construct parabolic and prolate spheroidal basis sets of wave functions for the system and give the explicit interbasis transformations and relations between spherical, parabolic, and prolate spheroidal bases. To build the parabolic and prolate spheroidal bases, we show that the Schrödinger equation of the considered system is also variable separable in both parabolic and prolate spheroidal coordinates, and then, solve this equation exactly. The variable separability in different coordinate systems is actually a consequence of the superintegrability which has been proved recently for the 9D MICZ KP.
The nine-dimensional MICZ-Kepler problem is of recent interest. This is a system describing a charged particle moving in the Coulomb field plus the field of a SO(8) monopole in a nine-dimensional space. Interestingly, this problem is equivalent to a 16-dimensional harmonic oscillator via the Hurwitz transformation. In the present paper, we report on the multiseparability, a common property of superintegrable systems, and the superintegrability of the problem. First, we show the solvability of the Schrödinger equation of the problem by the variables separation method in different coordinates. Second, based on the SO(10) symmetry algebra of the system, we construct explicitly a set of seventeen invariant operators, which are all in the second order of the momentum components, satisfying the condition of superintegrability. The found number 17 coincides with the prediction of (2n − 1) law of maximal superintegrability order in the case n = 9. Until now, this law is accepted to apply only to scalar Hamiltonian eigenvalue equations in n-dimensional space; therefore, our results can be treated as evidence that this definition of superintegrability may also apply to some vector equations such as the Schrödinger equation for the nine-dimensional MICZ-Kepler problem.
We suggest one variant of generalization of the Hurwitz transformation by adding seven extra variables that allow an inverse transformation to be obtained. Using this generalized transformation we establish the connection between the Schrödinger equation of a 16-dimensional isotropic harmonic oscillator and that of a nine-dimensional hydrogen-like atom in the field of a monopole described by a septet of potential vectors in a non-Abelian model of 28 operators. The explicit form of the potential vectors and all the commutation relations of the algebra are given.
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