On the solution of the asymptotic differential equations in quantum mechanics

The solution of the asymptotic quantum mechanical differential equations in the neighbourhood of singular points is suggested to overcome some complexities occurring with the technique of the indicial equations. After separation of the variables, the differential equations are solved in their asymptotic regions, the general regular solutions in the whole interval being then found by replacing integration constants by functions of the variables and finding the differential equations determining these functions. The resulting differential equations are now free from singularities and can be solved by the usual series expansion, giving simple polynomial solutions for the harmonic oscillator and the hydrogen atom and more complicated solutions for H 2 + .