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The author investigates a simple method of constructing potentials which is related to the work of Bhattacharjie and Sudarshan (1962) and for which the Schrodinger equation can be solved in terms of known special functions. It turns out that this method can be related to supersymmetric quantum mechanics and this relationship can help to decide which special functions, satisfying linear homogeneous second-order differential equations, can be solutions of the Schrodinger equation with potentials of the form V(x)=W2(x)-W'(x). The author illustrates this procedure with the example of orthogonal polynomials and obtains explicit expressions of wavefunctions of a wide class of shape-invariant potentials.
G. Lévai (Tue,) studied this question.
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