Collect. Czech. Chem. Commun. 2005, 70, 941-950

Notes on the Riccati Equation

Eugene S. Kryachko

Bogoliubov Institute for Theoretical Physics, Kiev, 03143 Ukraine and Department of Chemistry, Bat. B6c, University of Liège Sart-Tilman, B-4000 Liège 1, Belgium


The relationship between the Riccati and Schrödinger equations is discussed. It is shown that the transformation converting the Riccati equation into its normal form is expressed in terms of the roots of its algebraic part treated as a second-order polynomial. Together with the well-known Riccati transformation, a new transformation which also links the Riccati equation to the second-order linear differential equation is introduced. The latter is actually the Riccati transformation applied to an "inverse" Riccati equation. Two specific forms of the Riccati equation admitting the explicit particular rational solutions are obtained.

Keywords: Schrödinger equation; Riccati transformation; Differential equations; Quantum chemistry.

References: 45 live references.