St¨uckelberg particle in external magnetic field. The method of projective operators

We study the St¨uckelberg equation for a relativistic particle 
with two spin states S = 1 and S = 0 in the presence of 
an external uniform magnetic field. The particle is described 
by an 11-component wave function consisting of a scalar, a 
vector, and an antisymmetric tensor. On the solutions of the 
equation, the operators of energy, the third projection of the 
total angular momentum, and the third projection of the linear 
momentum along the direction of the magnetic field are 
diagonalized. After separation of variables, a system for 11 
radial functions is obtained. Its solution is based on the use of 
the Fedorov-Gronsky method, in which all 11 radial functions 
are expressed in terms of three main functions. Exact solutions 
with cylindrical symmetry are constructed. Three series 
of energy levels are found.