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

The nonrelativistic motion of a charged particle around a dyon in (9 + 1) spacetime is known as the nine-dimensional McIntosh-Cisneros-Zwanziger-Kepler problem. This problem has been solved exactly by the variable-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. Published under an exclusive license by AIP Publishing

Automated summary

The nine-dimensional McIntosh-Cisneros-Zwanziger-Kepler problem, which describes the nonrelativistic motion of a charged particle around a dyon in (9 + 1) spacetime, has been solved exactly using the variable-separation method in spherical, parabolic, and prolate spheroidal coordinate systems. This study establishes a relationship between variable separation and the algebraic structure of SO(10) symmetry. Additionally, it demonstrates that each coordinate system corresponds to a set of eigenfunctions of a nonuplet of algebraically independent integrals of motion, and allows for the calculation of important integrals using algebraic methods.