A spin 1/2 particle and two mass parameters in an external Coulomb field

A generalized relativistic wave equation for afermion with two mass parameters is studied in thepresence of an external Coulomb field. After separating the variables, the problem is reduced to a system of eight differential equations of the first order.Taking into account the diagonalization of the spatial reflection operator, we derive two independentsystems of four equations, referring to states of opposite parity. At a large distance from the center,they take the form of two subsystems for two ordinary Dirac particles in external Coulomb field, withmasses M1and M2respectively. To simplify theproblem, we perform transition to the nonrelativistic description of the system. In this way, we derivetwo systems of linked second order equations, referring to states with different parities, from whichthe corresponding fourth order differential equations for separate functions are found. Solutionsof the Frobenius type are constructed, they involvepower series with 10-term recurrent relations. Twoof the four solutions are appropriate for describingbound states. As a quantization rule, we apply theknown transcendency condition, in this way we obtained two analytical formulas for energy spectra.They are similar to nonrelativistic spectra for ordinary spin 1/2 particle, but being governed by massesM1and M2.