Upper Limits for the Number of Bound States Associated with the Yukawa Potential

The number of bound-state solutions of the Schrödinger equation for the screened Coulomb potential (Yukawa potential), -(C/r) exp( - αr), occurs frequently in theoretical discussions concerning, for example, gas discharges, nuclear physics, and semiconductor physics. The number of bound states is a function of (C/α). Three upper limits for the number of bound states associated with the Yukawa potential are evaluated and compared. These three limits are those given by Bargmann, Schwinger, and Lieb. In addition, the Sobolev inequality states that whenever (C/α) < 1.65 no bound state occurs. This agrees to within a few percent of the numerical calculations of Bonch-Bruevich and Glasko. The Bargmann and Lieb limits and the Sobolev inequality are substantially easier to evaluate than the Schwinger limit. Among the three limits, the Schwinger limit gives the most restrictive limit for the existence of only one bound state and, therefore, is the best one to use for the approach to no binding, i.e., 1.65 < (C/α) ≤ 1.98. The Lieb limit is the best among the three when (C/α) > 1.98. The Bargmann limit is the least restrictive.