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, $V\left(\mathcal{r}\right)=-\left(\frac{e^2}{\kappa \mathcal{r}}\right)e^{-qr}cos\left(q\mathcal{r}\right)$, 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 ${\delta }_c$ above which no bound states with negative energy exist. The value of ${\delta }_c$ 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 $\delta$ greater than zero, and is found to be approximately linearly dependent on $\frac{1}{{\delta }_c}$, The square of the number of bound $s$ states is also shown to be linear with $\frac{1}{{\delta }_c}$, Positive-energy states are also discussed.