Real linear quaternionic differential operators

The renewed interest in searching for quaternionic deviations of standard (complex) quantum mechanics resulted, in the last years, in a better understanding of the quaternionic mathematical tools needed to solve quantum mechanical problems. In particular, a relevant progress has been achieved in solving eigenvalue problems and differential equations for quaternionic operators. The practical methods recently proposed to solve quaternionic and complex linear second-order differential equations with constant coefficients represent a fundamental starting point to discuss quaternionic potentials in quantum mechanics and study possible violations from complex theories. Nevertheless, only for a restricted class of real linear quaternionic differential operators (namely, symmetric operators) the solution of differential problems was given. In this paper, we study real linear quaternionic differential equations. The proposed resolutionʹs method is based on the Jordan canonical form of (real linear) quaternionic matrices.