On the three-body Schr\"{o}dinger equation with decaying potentials

TL;DR

This paper analyzes the three-body Schrödinger equation with decaying potentials, revealing solutions within specific algebraic structures and providing analytic solutions for certain Coulombic systems, including bound states at threshold.

Contribution

It characterizes the three-body Schrödinger operator as an extension of exponential unitary groups and finds solutions in commutator subalgebras, offering new analytic solutions for Coulomb systems.

Findings

Solutions exist in commutator subalgebras for decaying potentials.

Analytic solutions for lower bound states in Coulomb systems are derived.

Bound state eigenvalues at threshold are obtained for three-unit-charge systems.

Abstract

The three-body Schr\"{o}dinger operator in the space of square integrable functions is found to be a certain extension of operators which generate the exponential unitary group containing a subgroup with nilpotent Lie algebra of length $\kappa+1$, $\kappa=0,1,...$ As a result, the solutions to the three-body Schr\"{o}dinger equation with decaying potentials are shown to exist in the commutator subalgebras. For the Coulomb three-body system, it turns out that the task is to solve - in these subalgebras - the radial Schr\"{o}dinger equation in three dimensions with the inverse power potential of the form $r^{-\kappa-1}$. As an application to Coulombic system, analytic solutions for some lower bound states are presented. Under conditions pertinent to the three-unit-charge system, obtained solutions, with $\kappa=0$, are reduced to the well-known eigenvalues of bound states at threshold.