Exact solutions of the Schrodinger equation with irregular singularity

The 1D nonrelativistic Schrodinger equation possessing an irregular singular point is investigated. We apply a general theorem about existence and structure of solutions of linear ordinary differential equations to the Schrodinger equation and obtain suitable ansatz functions and their asymptotic representations for a large class of singular potentials. Using these ansatz functions, we work out all potentials for which the irregular singularity can be removed and replaced by a regular one. We obtain exact solutions for these potentials and present source code for the computer algebra system Mathematica to compute the solutions. For all cases in which the singularity cannot be weakened, we calculate the most general potential for which the Schrodinger equation is solved by the ansatz functions obtained and develop a method for finding exact solutions.