On exact solutions of the Schrodinger equation

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Abstract
In the one- and three-dimensional Schrodinger equation, a broad class of the regular potentials (rational functions of the rational powers of r) may admit exact bound-state solutions in the generalised harmonic-oscillator elementary form psi (r)=rsigma *polynomial*exp(-polynomial). The necessary and sufficient conditions of this phenomenon are derived in the form of coupled algebraic equations. The methods of their solution and a few examples are discussed. In particular, the well known Coulombic and oscillator solvability and the similar recent results of Singh et al. (1978) and Whitehead et al. (1982) are reproduced as the two simplest special cases.

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