Response of the near-threshold resonances of the molecular ion to an effective atomic $2s$--$2p$ splitting

For the nonrelativistic three-body Coulomb Hamiltonian, near-threshold resonances of the deuterium–tritium muonic molecular ion ($dt\mu$) are supported by the long-range dipole coupling between degenerate atomic $2s$ and $2p$ states. We study the effect of atomic $2s$–$2p$ splitting on their positions and decay widths. Three-body complex-absorbing-potential calculations resolve additional members of the $L^{\pi }=0^{+}$ and $1^{-}$ series. Their geometric spacing agrees with that expected from the $1\mathrm{/}{R}^2$ dipole attraction, where $R$ is the atom–nucleus separation. To assess the effect of splitting, we scan a prescribed $2s$–$2p$ interval using an effective potential in a spin-independent, nonrelativistic Hamiltonian. This models the atomic splitting without an explicit Lamb-shift calculation. Within this model, the studied resonances lie closer to the $d\mu (2s)+t$ threshold and have smaller Coulomb decay widths than in the degenerate case. At an effective interval of $0.2\text{eV}$, the seventh $1^{-}$ member lies within $2\text{meV}$ of this threshold with a full width of about $4\text{meV}$, compared with $48$ and $15.6\text{meV}$ at degeneracy. These values characterize the response within the tested numerical representations. Its width decreases, but its separation from the threshold decreases more strongly. The long-range analysis attributes these shifts and width reductions to modified dipole mixing. It further suggests that atomic $2s$–$2p$ splitting may truncate the infinite accumulation of resonances predicted for degenerate atomic $2s$ and $2p$ levels.