Bound and Resonant Spectra of Few-Lepton Coulomb Systems

We present a unified calculation of bound and resonant states in purely leptonic Coulomb systems: $e^{-}{e}^{-}{e}^{+}$ (${\mathrm{P}\mathrm{s}}^{-}$), ${\mu }^{+}{e}^{-}{e}^{-}$ (${\mathrm{M}\mathrm{u}}^{-}$), ${\mu }^{+}{\mu }^{+}{e}^{-}$ (${\mathrm{M}\mathrm{u}}_2^{+}$), $e^{+}{e}^{+}{e}^{-}{e}^{-}$ (${\mathrm{P}\mathrm{s}}_2$), and ${\mu }^{+}{\mu }^{+}{e}^{-}{e}^{-}$ (${\mathrm{M}\mathrm{u}}_2$). Using an extended stochastic variational method combined with the complex scaling method, we resolve the natural-parity $S$- and $P$-wave spectra. Near their respective $n=2$ thresholds, all three trilepton systems exhibit Gailitis–Damburg sequences generated by $2S$ and $2P$ Stark mixing and the resulting inverse-square attraction. Although microscopically distinct from the Efimov effect, this mechanism produces the same inverse-square asymptotics and geometric scaling. The ${\mathrm{P}\mathrm{s}}^{-}$ and ${\mathrm{M}\mathrm{u}}^{-}$ systems exhibit resonance sequences of comparable density, whereas the ${\mathrm{M}\mathrm{u}}_2^{+}$ produces a much denser spectrum, with 20 resolved members in the ${}^3{P}^o$ channel. In ${\mathrm{M}\mathrm{u}}_2^{+}$, the deeper states follow molecular Born–Oppenheimer configurations, while the near-threshold spectrum is governed by the atomic $\mathrm{M}\mathrm{u}(2)+{\mu }^{+}$ structure. In ${\mathrm{P}\mathrm{s}}_2$, coupling between threshold-degenerate configurations is essential for a near-threshold bound state. In ${\mathrm{M}\mathrm{u}}_2$, the Born–Oppenheimer organization of the resonance spectrum is channel dependent.