non-relativistic potentials, though exotic atoms have been more recently studied in the framework of effective field theory  [@Holstein:1999nq]. Ordinary perturbation theory is not applicable here. For instance, a hard core of radius $b$ much smaller than the Bohr radius $B$ produces a tiny upward shift of the level, while first-order perturbation theory gives an infinite contribution! The expansion parameter here is *not* the strength of the potential, but the ratio $b/B$ of its range to the Bohr radius, and more precisely, the ratio $a/B$ of its scattering length to the Bohr radius. The scheme of this “radius perturbation theory” is outlined in [@Mandelzweig:1977cu]. For the sake of this paper, the first order term of this new expansion is sufficient. It is due to Deser et al. [@Deser:1954vq], Trueman [@Trueman61], etc., and reads $$\label{eq:Trueman}
{E_n-E_{0,n}\over E_{0,n}}  \simeq-{4\over n}\left({a\over B}\right)~,$$ where $a$ is the scattering length in the potential $\lambda v(r)$. Here, $E_{0,n}$ ($=-1/(4 n^2)$ in our units) is the pure Coulomb energy, and $E_n$ the energy of $n$S level of the modified Coulomb interaction ($n=1, 2,\ldots$). Only in the case where $\lambda v(r)$ is very weak, the scattering length is given by the Born approximation, i.e., $a\propto \lambda$, and ordinary perturbation 