The factorization method and particular solutions of the relativistic Schrödinger equation of nth order (n=4,6) Original Research Article

The Schrödinger equation in the relativistic configuration space for a relativistic function Ψ(r) has the form of an infinite-order linear differential equation with an inherent small parameter ϵ=1/c at the higher derivatives. In the formal limit c→∞ this equation degenerates to the standard nonrelativistic Schrödinger equation.To simplify the problem, we have considered the nth order differential equation (n=4,6) which corresponds to a truncation of the higher order derivative contributions. The linear nth order differential operator can be expressed in a factorized form: Ĥ=Ĥn/2…Ĥ2 Ĥ1, where Ĥi are differential operators of second order. Solving the differential equation of second order, Ĥ1Ψ(r)=0, we can obtain a particular solutions of the nth order equation.