short as compared to the Bohr radius of the pure Coulomb problem. Throughout this paper, the energy units are set such that $\hbar^2/(2\mu)=1$, where $\mu$ is the reduced mass. In (\[basic-H\]) the scaling properties of the Coulomb interaction are also used to fix the elementary charge $e=1$, without loss of generality. The study will be restricted here to S-wave states. The case of P-states or higher waves is briefly discussed in Sec. \[se:order\].

As an example, a simple square well $v(r)=-\theta(b-r)$ is chosen in Fig. \[CoulSW\], with a radius $b=0.01$ which is small compared to the Bohr radius, which is $B=2$ in our units. If alone, this potential $\lambda v(r)$ requires a strength $\lambda_n b^2=(2n-1)^2\pi^2/4$ to support $n$ bound states in S-wave, with numerical values $\{\lambda_n b^2\}=\{2.46, 22.2, \ldots\}$. These are precisely the values at which the atomic spectrum is rearranged in Fig. \[CoulSW\], with the $n$S state falling into the domain of nuclear energies and all other $i$S atomic states with $i>n$ experiencing a sudden change and drops to (but slightly above) the unperturbed $(i-1)$S energy.

The theory of level shifts of exotic atoms is rather well established, see e.g., [@Ericson:1988gk Ch. 6]. The discussion is restricted here to 