Properties of the weakly-bound (1,1)-states and rotationally excited (2,0)-states in the muonic molecular $d d \mu, d t \mu$ and $t t \mu$ ions

Total energies and other bound state properties of the weakly-bound (1,1)-states and rotationally excited (2,0)-states in the three-body muonic molecular $d d \mu, d t \mu$ and $t t \mu$ ions are determined to high numerical accuracy and investigated. Our current numerical accuracy achieved for the total and binding energies of the weakly-bound (1,1)-states in the both $d d \mu$ and $d t \mu$ ions significantly exceeds similar accuracy obtained in earlier computations of these weakly-bound states in these two ions. The bound state properties of the weakly-bound (1,1)-states and (2,0)-states in the $d d \mu, d t \mu$ and $t t \mu$ muonic ions have never been determined to high accuracy in earlier studies. We also briefly discuss the current status of muon-catalyzed nuclear fusion and develop the new universal variational expansion which can be used for extremely accurate bound state calculations of arbitrary three-body systems, including the truly adiabatic ${}^{\infty}$H$^{+}_{2}$ ion and close systems.