Lie algebras of differential operators and Lie-algebraic potentials

An explicit characterisation of all second order differential operators on the line which can be written as bilinear combinations of the generators of a finite-dimensional Lie algebra of first order differential operators is found, solving a problem arising in the Lie-algebraic approach to scattering theory and molecular dynamics. One-dimensional potentials corresponding to these Lie algebras are explicitly classified, which include the harmonic oscillator, Morse, one-soliton (Pöschl-Teller), Mathieu, Lamé, confluent hypergeometric, and Bessel potentials.