Critical value calculations for the screening parameter of Hulthen potential

The critical screening parameter values correspond to the zero energy states of a screened Coulomb potential and, generally, create great numerical instabilities depending on the type of the system and the methods used in the calculations. This is due to the eigenvalue problem weighted by the screening potential which becomes negligible for large arguments. Hence, the conversion of the weighted eigenvalue problem to a regular one with unit weight gains great importance. This can be accomplished via an appropriate coordinate transformation on the radial variable of the relevant Schrödinger's equation. The transformation is constructed in such a way that the resulting ordinary differential equation differs by an effective potential function from the one which is satisfied by extended Jacobi polynomials. In this form the eigenvalue parameter is proportional to the reciprocal of the screening parameter's critical value. This can be numerically solved via variational approximation methods. In this work, this approach is used to evaluate the critical values for the Hulthen potential systems.