**NONLINEAR PHENOMENA IN COMPLEX SYSTEMS**

An Interdisciplinary Journal
* 2018, Vol.21, No.4, pp.309 - 325*

**
Geometrization for a Quantum-Mechanical Problem of a
Vector Particle in an External Coulomb Field
**

*
N. G. Krylova, E. M. Ovsiyuk, V. Balan, and V. M. Red'kov
*
The quantum mechanical problem for a relativistic spin 1 particle is studied. With the use
of the space reflection operator, the radial system of ten equations is splitted into independent
subsystems, consisting of 4 and 6 equations, respectively. The last one reduces to a system of
4 linked first order differential equations for the complex radial functions *f *^{ i } (r), i = 1, ..., 4.
We investigate this system by using the tools of the Jacobi stability theory, namely, the
Kosambi-Cartan-Chen (KCC) theory. It has been shown that the second KCC-invariant is
a function of the radial coordinate r, and it does not depend on *x*^{ i } and *y*^{ i } = *dx*^{ i }/*dr*,
*f *^{ i }(r), i = 1, ..., 8.
Due to this, the remaining 3 KCC-invariants identically vanish. In accordance with the
general theory, a pencil of geodesic curves from the point r0 converges (or diverges) if the
real parts of all eigenvalues of the 2-nd KCC-invariant *P*^{ i}_{j } are negative (or positive). We
determine the expressions for the matrix *P*^{ i}_{j }(r)
for *r*→0 and for *r*→∞, and examine
the asymptotic behavior of the eigenvalue problem *PΨ = λ Ψ*. The established behavior of
eigenvalues correlates with the existence of two solutions which may be associated with the
bound states of a particle in a Coulomb field. We further describe a method which permits
to examine projections of the whole set of solutions onto different 2-planes of the 4-space.
In each case, such a projection consists of two branches, which are characterized by different
2-nd order differential equations.

*Key words: *
quantum mechanics, spin 1 particle, Coulomb field, differential geometry, Kosambi-
Cartan-Chern invariants

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Last updated: *December 28, 2018*