Algebraic Theory of the Three-Body Problem

The three-body problem appears in many branches of physics (Fig.1). Many techniques have been developed to solve the non-relativistic quantum mechanical problem for 3 particles interacting through a two-body or three-body force. These techniques are designed to solve the differential or integro-differential Schrodinger equation. In this article, an alternative formulation of the three-body is given, in terms of Lie algebras. This formulation provides a framework for detailed calculations and, most importantly, allows one to classify all solutions that can be obtained in closed form (exactly solvable problems).