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We develop an efficient numerical method to directly calculate the critical screening parameters for one-electron systems with Hulthén and Debye-Hückel screened Coulomb potentials. Compared to indirect methods, which locate the critical screening parameters via searching the potential parameters with near-zero energy, the method developed in this work directly calculates the critical screening parameters as eigenvalues of a generalized eigenvalue problem. This feature allows us to simultaneously determine the critical parameters for bound states from low-lying excitation to high-lying Rydberg limit with high accuracy. The method is applied to screened Coulomb potentials to investigate the asymptotic behavior of critical screening parameters as the principal quantum number n approaches infinity. It has been shown that the critical screening parameters in Hulthén and Debye-Hückel potentials follow the 2n^{-2} and 4n^{-2}/π asymptotic laws, respectively, and that the orbital angular momentum affects the higher-order coefficients linearly.

We revisit the generalized pseudospectral (GPS) method to investigate the physical properties of shell‐confined atoms. Based on a general mapping function, the GPS method is developed for obtaining highly accurate bound state energies, wave functions, and radial quantities for the shell‐confined hydrogen atom. Besides the incidental and simultaneous degeneracies in the energy spectrum, we find that energy degeneracy appears very commonly in the shell‐confined system. The contour maps of the eigenenergies for some low‐lying bound states are obtained and the corresponding wave functions are analyzed. Radial expectation values with both positive and negative powers are calculated with high accuracy. By analyzing the Schwarz‐like inequalities, we show that the ground state density function for the outer‐confined hydrogen atom is second‐order monotonic, while for the shell‐confined hydrogen atom the radial densities for all bound states are zeroth‐order monotonic. In the extreme situation when the inner radius coalesces with the outer radius, the electron becomes a classical particle.