Abstract
The 1D nonrelativistic Schrödinger equation possessing an irregular singularpoint is investigated. We apply a general theorem about existence and structureof solutions of linear ordinary differential equations to the Schrödinger equationand obtain suitable ansatz functions and their asymptotic representations for alarge class of singular potentials. Using these ansatz functions, we work out allpotentials for which the irregular singularity can be removed and replaced by aregular one. We obtain exact solutions for these potentials and present sourcecode for the computer algebra system Mathematica to compute the solutions. Forall cases in which the singularity cannot be weakened, we calculate the mostgeneral potential for which the Schrödinger equation is solved by the ansatzfunctions obtained and develop a method for finding exact solutions.
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Schulze-Halberg, A. Exact Solutions of the Schrödinger Equation with Irregular Singularity. International Journal of Theoretical Physics 39, 2305–2325 (2000). https://doi.org/10.1023/A:1003720300015
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DOI: https://doi.org/10.1023/A:1003720300015