Application of geometrical methods to study the systems of differential equations for quantum-mechanical problems

Abstract

A geometrical method based on the structural stability theory is used to study systems of differential equations which arise in quantum-mechanical problems. We consider a 1/2-spin particle in external Coulomb field or in the presence of magnetic charge on the background of the de-Sitter space, and a free 3/2-spin particle in spherical coordinates of the flat space. It turns out that the first and the second Kosambi-Cartan-Chern invariants are nontrivial for the corresponding systems, while the 3-d, 4-th and 5-th invariants identically vanish. From physical point of view, the second invariant determines how rapidly the different branches of the solution diverge from or converge to the intersection points, while the most interesting are the singular points. The convergence (divergence) near the singular points r = 0, 1 are shown to correlate with the behavior of solutions for quantum mechanical states (discrete and continuous spectra). The vanishing of the 3-d, 4-th and 5-th invariants geometrically implies the existence of a nonlinear connection on the tangent bundle, having zero torsion and curvature.