Tight bounds for the critical screening parameters of the generalized exponential cosine screened Coulomb potential

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Abstract
Analytic expressions have been obtained for the lower and upper bounds to the critical screening parameter λc associated with the ground state of the generalized exponential cosine screened Coulomb potential, V(r)=−e−λr cos(ελr)/r employing the methods of Hulthén and Laurikainen and of Hemmer. It is found that the predicted numerical results for the range 0≤ε≤1 for the lower bound for λc are within 0.08% of the exact results, and for the upper bound, within 0.7% of the exact results. It emerges further that the critical screening parameters for the excited s‐states can be determined as well in an approximate way. For the static screened Coulomb potential, we obtain λnsc =1.283 664n−2 −0.092 793n−3, where n is the principal quantum number. The predicted λc for various quantum states (n=1 to 9) are in excellent agreement with the values obtained numerically by Rogers et al. [Phys. Rev. A 1, 1577 (1970)].