is provided by a short-range pairwise interaction acting between two particles confined in an harmonic potential, a problem inspired by the physics of quantum dots. The basic quantum mechanics of exotic atoms will be briefly summarised, in particular with a discussion about the Deser–Trueman formula that gives the energy shift of exotic atoms in terms of the scattering length of the nuclear potential. A pedestrian derivation of this formula will be given in Appendix, which extents its validity beyond the case of exotic atoms. The link from the Coulomb to the harmonic cases will also be discussed in light of the famous Kustaanheimo–Stiefel (KS) transformation, which is reviewed in several papers (see, e.g., [@Mavromatis:1998] and refs. there) and finds here an interesting application.

The discussion is mainly devoted to one-dimensional problems or to S-states ($\ell=0$) in three dimensions. In Sec. \[se:order\], it is extended to the first P-state (2P), and it is shown that the rearrangement is much sharper for P and higher $\ell$ states than for S states.

Coulomb potential plus short-range attraction {#se:Coul-sr}
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The simplest model of exotic atoms corresponds to the Hamiltonian $$\label{basic-H}
 H=-\Delta -{1\over r} +\lambda v(r)~,$$ where $v(r)$ has a range that is very 