Bound eigenstates for the superposition of the Coulomb and the Yukawa potentials

The eigenvalue problem for two particles interacting through the potential being the superposition of the attractive Coulomb potential (-A/r) and the Yukawa potential B exp(-Cr)/r of arbitrary strength B and screening parameter C is solved by variational means. The energy levels $E_{\mathrm{nl}}$ for the states 1s through 7i are calculated as functions of B and C. It is shown that for a given principal quantum number n the energy eigenvalues increase (decrease) with increasing azimuthal quantum number l if the Yukawa potential is attractive (repulsive), i.e., for l>l’: $E_{\mathrm{nl}}$≥$E_{\mathrm{nl}\mathcal{’}}$ if B<0, and $E_{\mathrm{nl}}$≤$E_{\mathrm{nl}\mathcal{’}}$ if B>0. It leads to the crossing of the energy levels with n≥2. For B>0 the levels with larger n and l become lower than those with smaller n and l, e.g., $E_{3d}$<$E_{2s}$, $E_{4f}$<$E_{2s}$, and $E_{4f}$<$E_{3p}$. For B<0 and certain intervals of C the levels with larger n but smaller l lie below those with smaller n and larger l, e.g., $E_{4s}$<$E_{3d}$, $E_{5s}$<$E_{4f}$, and $E_{5p}$<$E_{4f}$. The values of B and C for which the lowest-energy levels cross over are estimated. Moreover, the splitting of the 2s and 2p levels (the Lamb shift) is discussed.