Algebraic structure underlying spherical, parabolic and prolate spheroidal bases of the nine-dimensional MICZ-Kepler problem

TL;DR

This paper explores the algebraic structure of different bases in the nine-dimensional MICZ-Kepler problem, linking variable separation methods to SO(10) symmetry and enabling computation of complex integrals.

Contribution

It establishes a direct relationship between variable separation solutions and the algebraic SO(10) symmetry in the nine-dimensional MICZ-Kepler problem, revealing new connections between bases.

Findings

Each basis is an eigenfunction set of integrals of motion.

Connections between bases enable calculation of complex special function integrals.

The algebraic structure facilitates understanding of the problem's symmetry.

Abstract

The nonrelativistic motion of a charged particle around a dyon in (9+1) spacetime is known as the nine-dimensional MICZ-Kepler problem. This problem has been solved exactly by the variables-separation method in three different coordinate systems, spherical, parabolic, and prolate spheroidal. In the present study, we establish a relationship between the variable separation and the algebraic structure of SO(10) symmetry. Each of the spherical, parabolic, or prolate spheroidal bases is proved to be a set of eigenfunctions of a corresponding nonuplet of algebraically-independent integrals of motion. This finding also helps us establish connections between the bases by the algebraic method. This connection, in turn, allows calculating complicated integrals of confluent Heun, generalized Laguerre, and generalized Jacobi polynomials, which are important in physics and analytics.