On exact solutions of the Schroedinger equation

In the one- and three-dimensional Schroedinger 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) = rsup(σ) x polynomial x 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 (Phys. Rev.; D 18: 1901 (1978) and Lett. Math. Phys.; 3:73 (1979) and Whitehead et al (J. Phys. A. 15: 1217 (1982)) are reproduced as the two simplest special cases. (author)