Contact symmetry of time-dependent Schr\"odinger equation for a two-particle system: symmetry classification of two-body central potentials

TL;DR

This paper classifies symmetries of the two-particle Schrödinger equation with central potentials, revealing that contact transformations mirror point transformations and detailing their algebraic structures.

Contribution

It provides a comprehensive symmetry classification of two-body central potentials using contact transformations, showing their equivalence to point transformations and analyzing their algebraic structures.

Findings

Contact transformations are equivalent to point transformations in this context.

Four symmetry classes are identified for the two-particle Schrödinger equation.

Detailed Lie algebra structures and invariants are derived for each class.

Abstract

Symmetry classification of two-body central potentials in a two-particle Schr\"{o}dinger equation in terms of contact transformations of the equation has been investigated. Explicit calculation has shown that they are of the same four different classes as for the point transformations. Thus in this problem contact transformations are not essentially different from point transformations. We have also obtained the detailed algebraic structures of the corresponding Lie algebras and the functional bases of invariants for the transformation groups in all the four classes.