Variables separation and superintegrability of the nine-dimensional MICZ-Kepler problem

Abstract

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.

Normed Division Algebras Application to the Monopole Physics

“…Here, the quantum number Q is an integer because it has to close the abstract monopole space S 7 . Following our works [69][70][71][72], we expand the Hamiltonian of the 9D MICZ-KP into a more specific formĤ…”

“…The analytical solutions with the wavefunctions were obtained by the variable separation method in spherical [69], parabolic, and prolate spheroidal coordinates [71]. This problem is also maximally superintegrable as the 3D and 5D MICZ-KP [70]. Within the analytical approach in Ref.…”

“…where the notation [Â,B] ± =ÂB ±B is used for commutator/anticommutator between two oper-ators andB. SinceD-matrix obeys the SO(10) algebra [70,72,74], the 9D MICZ-KP is SO(10) symmetric. This symmetry is higher than the trivial geometric symmetry SO(9); thus, the quantization of the 9D MICZ-KP is degenerated so high that its energy solely depends on the principal quantum number n only.…”

“…To examine the superintegrability of the 9D MICZ-KP, in the study [70], we aimed to construct at most 2 × 9 − 1 = 17 algebraically independent constants of motion. If we reach the number 17, the 9D MICZ-KP is called maximally superintegrable.…”

“…As in Ref. [70], we chose the ninth constant of motion as the total generalized angular mometumD 2 9 =Λ 2 . We took other seven block-matrices from the bottom right ofD and built their Casimir operators: Collecting seventeen constants of motionD 2 1 ,D 2 2 , .…”