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.

Normed Division Algebras Application to the Monopole Physics

“…Fortunately, this is true for the 9D MICZ-KP, and hence, the 9D MICZ-KP is a good example that the definition of a maximally superintegrable system works for a vector degenerated system [70]. In the series of our studies, we have variable separate the 9D MICZ-KP in spherical [69], parabolic [70], and prolate spheroidal [71] coordinates, respectively.…”

“…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 SO(10, 2) dynamical symmetry was used to obtain the algebraic solutions [73]. 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. [71], the interbasis transformation between the parabolic and spherical bases has been constructed. However, a similar transformation between the prolate spheroidal and spherical bases was not built in an analytical approach since it required a complex integration between confluent Heun [75], associated Laguerre, and generalized Jacobi polynomials [36].…”

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

“…In the last two decades, many works had both analytically [31,32,33,34] and algebraically [35,36,37,38] examine the fivedimensional MICZ-Kepler problem. Simultaneously, after first being introduced a decade ago [18,19], nine-dimensional MICZ-Kepler problem (9D MICZ-KP) has also been examined up to now in various aspects such as its (dynamical) symmetry [39,40], its algebraic solutions [39], its analytical solutions with the wavefunctions in spherical [41], parabolic and prolate spheroidal coordinates [42], and also its superintegrability [43]. Also, in Ref.…”