On the Stability Domain of Systems of Three Arbitrary Charges

We present results on the stability of quantum systems consisting of a negative charge $-q_1$ with mass $m_{1}$ and two positive charges $q_2$ and $q_3$, with masses $m_{2}$ and $m_{3}$, respectively. We show that, for given masses $m_{i}$, each instability domain is convex in the plane of the variables $(q_{1}/q_{2}, q_{1}/q_{3})$. A new proof is given of the instability of muonic ions $(\alpha, p, \mu^-)$. We then study stability in some critical regimes where $q_3\ll q_2$: stability is sometimes restricted to large values of some mass ratios; the behaviour of the stability frontier is established to leading order in $q_3/q_2$. Finally we present some conjectures about the shape of the stability domain, both for given masses and varying charges, and for given charges and varying masses.