Bound Eigenstates of the Exponential Cosine Screened Coulomb Potential

A number of properties of the bound eigenstates of an electron in an exponential cosine screened Coulomb potential, $V(\mathcal{r})=\ensuremath{-}(\frac{{e}^{2}}{\ensuremath{\kappa}\mathcal{r}}){e}^{\ensuremath{-}qr}cos(q\mathcal{r})$, are studied. Perturbation and variation methods are used to calculate the eigenvalues. Detailed results are presented for the first four $s$ states. For each state there is a critical value of the screening parameter ${\ensuremath{\delta}}_{c}$ above which no bound states with negative energy exist. The value of ${\ensuremath{\delta}}_{c}$ for the ground state is obtained from a two-parameter variational calculation. The total number of different energy levels is finite for any value of the screening parameter $\ensuremath{\delta}$ greater than zero, and is found to be approximately linearly dependent on $\frac{1}{{\ensuremath{\delta}}_{c}}$, The square of the number of bound $s$ states is also shown to be linear with $\frac{1}{{\ensuremath{\delta}}_{c}}$, Positive-energy states are also discussed.