Parabolic, prolate spheroidal bases and relation between bases of the nine-dimensional MICZ-Kepler problem

Abstract

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.