Bound-state calculations of Coulomb three-body systems

Various geometrical and energetical properties in the symmetric muonic molecular ions $\mathrm{pp}\mu ,dd\mu ,tt\mu$, molecular ions $ppe,dde,tte$, and exotic system $\mu \mu e$ $\left({\mu }^{+}{\mu }^{+}{e}^{-}\right)$ are determined with high accuracy by using the two-stage strategy proposed by Frolov [Phys. Rev. A 57, 2436 (1998)]. The significant difference between bound-state spectra in muonic molecular ions $\mathrm{pp}\mu ,dd\mu ,tt\mu$ and molecular ions $ppe,dde,tte$ ions is explained by using the general theory of bound-state spectra in Coulomb three-body systems, which is closely related with the general theory of compact operators. In particular, the principal classification of the bound-state spectra in such systems can be made in the same manner as for compact operators. For instance, the discrete spectrum of a Coulomb three-body system may have the Hilbert-Schmidt, nuclear or finite-dimensional structure. Moreover, this structure can be changed by varying some of the physical parameters (e.g., masses or charges) of the system. The developed theory is applied to the case of symmetric Coulomb three-body systems with unit charges.