Superintegrability and Coulomb-Oscillator Duality

The wave functions are given for the eight-dimensional isotropic oscillator, the five-dimensional Coulomb and $$SU(2)$$ Yang–Coulomb monopole problems, the three-dimensional MIC–Kepler problems, the generalized MIC–Kepler problem and generalized Kepler–Coulomb and oscillator systems, as well as the four-dimensional isotropic and double singular oscillator in coordinates systems in which separation of variables is allowed in the corresponding Schrödinger equations. Using the condition of orthogonality of radial wave functions with respect to the non-energy quantum number, which we have proven, the coefficients of the interbasis expansions of the systems given above are found. Duality transformations are established that transfer the problem of an eight-dimensional isotropic oscillator into five-dimensional Coulomb and $$SU(2)$$ Yang–Coulomb monopole problems (Hurwitz transformation), as well as the MIC–Kepler problem and the generalized MIC–Kepler system into a four-dimensional isotropic and double singular oscillator (Kustaanheimo–Stiefel transformation), respectively. The coefficients are found that relate the bases of these systems to each other. For the five-dimensional $$SU(2)$$ model of the Yang–Coulomb monopole, the hidden $$SO(6)$$ symmetry group is found and the energy spectrum is calculated in a purely algebraic way. The Schrödinger equation for a “free” particle is solved in a ring-shaped model and the formula is derived that generalizes the expansion of a plane wave into spherical waves. Quantum-mechanical scattering problems are considered in five-dimensional Coulomb systems and $$SU(2)$$ Yang–Coulomb monopole systems, as well as in the fields of the MIC–Kepler problem.