Stability of Three Unit Charges. Necessary Conditions

TL;DR

This paper establishes a physical condition on the masses of three Coulomb charges that guarantees the absence of bound states, demonstrating instability of certain negative muonic ions and extending previous results.

Contribution

It provides a new simple mass condition ensuring stability or instability of three-charge systems, extending Thirring's classical results to real mass values.

Findings

Certain negative muonic ions are proven unstable.

A simple mass condition guarantees the absence of bound states.

The problem is reduced to a one-particle binding in an effective potential.

Abstract

We consider the stability of three Coulomb charges $\{+1, -1, -1 \}$ with finite masses in the framework of nonrelativistic quantum mechanics. A simple physical condition on masses is derived to guarantee the absence of bound states below the dissociation thresholds. In particular this proves that certain negative muonic ions are unstable, thus extending the old result of Thirring \cite{thirring} to the actual values of all masses. The proof is done by reducing the initial problem to the question of binding of one particle in some effective potential.