Perturbation of relativistic hydrogen Is there anyone that can provide a source for a relativistic hydrogen atom that is placed in a finite potential well? I am trying to find the new ground state energy for a relativistic electron that has the normal coulomb interaction inside a confinement radius $R$ and the coulomb interaction plus a scalar potential beyond the confinement radius.
 A: Quite possibly this has not been done, yet.
Here is quite recent paper (December 2017):

Rojas, R. A., N. Aquino, and A. Flores‐Riveros. Fine structure in the hydrogen atom boxed in a spherical impenetrable cavity. International Journal of Quantum Chemistry (2017). DOI:10.1002/qua.25584.
The spectrum of the hydrogen atom confined in a spherical impenetrable box of radius $R_c$ has been investigated by many authors up to date, but not at the level of relativistic corrections. It is well known that, as $R_c$ diminishes, all energy levels and the pressure increase very rapidly, whereas the polarizability goes to zero. In this report, we have computed the relativistic corrections that underlie the fine structure of the confined hydrogen atom, as a function of $R_c$. Such corrections correspond to relativistic kinetic energy, spin-orbit coupling and the Darwin term, which are calculated in the frame of time-independent perturbation theory, for which, use was made of the exact confined hydrogen atom wave functions. We show that for a confinement radius of 0.5 au the relativistic corrections increase up to three orders of magnitude with respect to those corresponding to the free atom. As $R_c$ decreases, the kinetic energy correction and the spin-orbit coupling for $j=1/2$ become negative whereas their absolute value and the Darwin term, which is positive, increase very rapidly.

Authors use already known exact nonrelativistic wavefunctions for infinite potential well written in terms of confluent hypergeometric function, with its parameters determined by numerical solution of boundary condition.  These wavefunctions are plugged into the first order perturbation theory to determine relativistic corrections.
And authors promise in conclusion:

Analysis of the effects on the fine structure when the atom is placed inside soft spherical walls is in progress.

So, either you have to wait, or do it yourself. The problem does not seem to be too difficult. Wavefunctions for nonrelativistic confined hydrogen atom in a finite well are also known:

Ley‐Koo, E., & Rubinstein, S. (1979). The hydrogen atom within spherical boxes with penetrable walls. The Journal of Chemical Physics, 71(1), 351-357. DOI:10.1063/1.438077.

As one can expect, they are written in terms of the same confluent hypergeometric functions both inside and outside the well and its parameters are determined by matching boundary condition at $R_c$. So, once you have the wavefunctions it would be straightforward to compute relativistic corrections in the first order perturbation expansion.
One thing to note: you might have to include Darwin term ($\sim \delta'(r-R_c)$) for the border of the well. It was not an issue for impenetrable wall but might be relevant for well of finite depth.
A: Here this issue is discussed in detail. But the article is in Russian.
https://inis.iaea.org/search/search.aspx?orig_q=RN:47070864
