Local approximation to the critical parameters of quantum wells

We calculate the critical parameters for some simple quantum wells by means of the Riccati-Padé method. The original approach converges reasonably well for nonzero angular-momentum quantum number l but rather too slowly for the s states. We therefore propose a simple modification that yields remarkably accurate results for the latter case. The rate of convergence of both methods increases with l and decreases with the radial quantum number n. We compare RPM results with WKB ones for sufficiently large values of l. As illustrative examples we choose the one-dimensional and central-field Gaussian wells as well as the Yukawa potential. The application of perturbation theory by means of the RPM to a class of rational potentials yields interesting and baffling unphysical results.