[@kolomeisky:022721].

The limit of a point interaction {#se:pt-int}
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The simplest solvable model of exotic atoms is realised with a zero-range interaction. The formalism of the so-called “point-interaction” is well documented, see, e.g., [@alb], where the case of a point-interaction supplementing the Coulomb potential is also treated, without, however, a detailed discussion of the resulting spectrum.

It is known that an attractive delta function leads to a collapse in the Schrödinger equation. In more rigorous terms, the Hamiltonian should be redefined to be self-adjoint. For S-wave, a point interaction of strength $g=1/a$, located at $r=0$, changes the usual boundary conditions $u(0)=0$, $u'(0)=1$ (possibly modified by the normalisation) by $u'/u=1/a$ at $r=0$. Note that $a$ is the Coulomb-corrected scattering length.

In this model, the S-wave eigenenergies are given by $u'(0)/u(0)=1/a$ applied to the reduced radial wave function of the pure Coulomb problem, which results into [@alb] $$\label{eq:basiceq}
F(-2/k)=1/a~,\quad F(x)=\Psi(1+x)-{1\over2}\ln(x^{2})-{1\over 2x}~,$$ in terms of the digamma function $\Psi(x)=\Gamma'(x)/\Gamma(x)$. Using the reflection formula [@Abramovitz64], the function $F$ can be rewritten as $$F(-x)=\pi\cot(\pi x)+\Psi(x)-{1\over2}\ln(x^{2})+{1\over 2x}~,$$ explaining the behaviour observed on the left-hand side of Fig. \[FunctionF\].

Equation (\[eq:basiceq\]) shows that $a \to -\infty$ corresponds to the plain Coulomb interaction, where $E_{n}=E_{0,n}$. For small deviations, 