Critical radius and dipole polarizability for a confined system

The analytical transfer matrix method (ATMM) is applied to calculating the critical radius r_\rm c and the dipole polarizability \alpha_\rm d in two confined systems: the hydrogen atom and the Hulthén potential. We find that there exists a linear relation between r_\rm c^1/2 and the quantum number n_r for a fixed angular quantum number l, moreover, the three bounds of \alpha_\rm d (\alpha_\rm d^K, \alpha_\rm d^B, \alpha_\rm d^U) satisfy an inequality: \alpha_\rm d^K\leq\alpha_\rm d^B\leq\alpha_\rm d^U. A comparison between the ATMM, the exact numerical analysis, and the variational wavefunctions shows that our method works very well in the systems.