Relativistic particle with spin $1$ in presence of coulomb field, quantum states with minimal angular momentum $j = 0$ in Lobachevsky an Riemann models

Previously, the nonrelativistic problems of a vector particle in the Coulomb field on the background of Lobachevsky and Riemann space models were studied, when a s modified Duffin–Kemmer equation was applied with additional spin-orbital interaction, which permitted to get three independent second order differential equations in nonrelativistic Pauli-like approximation and obtain three series of energies levels. However, the states with minimal value for quantum number of total angular momentum $j = 0$ were not considered, in the present paper these states are investigated. Conventional and modified Duffin–Kemmer equations provide us with solutions in hypergeometric and transcendental general Heun functions, even without transition to the nonrelativistic limit. These equations lead to slightly different energy spectra. Because the difference between two energy spectra at $j = 0$ turn out to be insignificant, we may expect similar property for states with larger values of $j$, even though solutions for conventional equation in Pauli approximation are not known.