Exact solutions, critical parameters and accidental degeneracy for the hydrogen atom in a spherical box

We derive some properties of the hydrogen atom inside a box with an impenetrable wall that have not been discussed before. Suitable scaling of the Hamiltonian operator proves to be useful for the derivation of some general properties of the eigenvalues. The radial part of the Schr\"{o}dinger equation is conditionally solvable and the exact polynomial solutions provide useful information. There are accidental degeneracies that take place at particular values of the box radius, some of which can be determined from the conditionally-solvable condition. Some of the roots stemming from the conditionally-solvable condition appear to converge towards the critical values of the model parameter. This analysis is facilitated by the Rayleigh-Ritz method that provides accurate eigenvalues.