also of interest to study how the system behave, as a function of the coupling factors, if two or more attractive holes are envisaged inside a single wide well.

Trueman–Deser formula
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We give here a pedestrian derivation of the Trueman–Deser formula. Consider a repulsive interaction added to a long-range attractive potential $V_0(r)$ in unit such that $\hbar^2/(2m)=1$. This short-range repulsion, at energy $E\simeq0$ is equivalent to a hard core potential of radius $a$, where $a$ is the scattering length of $V$. Hence the pure Coulomb and the modified Coulomb problems results for orbital momentum $\ell=0$ into $$\begin{aligned}
&-u_0''(r)+ V_0(r) u_0(r) =E_0 u_0(r)~,\quad & u_0(0)=0~,\quad u_0(\infty)=0~,\nonumber\\
&-u''(r)+V_0(r) u(r)=E u(r)~,\quad &u(a)=0~,
\quad u (\infty)=0~.\end{aligned}$$ After multiplication by $u$ and $u_0$, respectively, the difference leads to $$(E-E_0)\int_a^\infty u_0(r)u(r)\,\mathrm{d}r=u'(a)u_0(a)~.$$ In the LHS, the integral is close to the normalisation integral of $u_0$ or $u$, i.e., close to unity. If $V_0(r)$ is smooth, then $u(r)$ does not differ much from the shifted version $u_0(r+a)$ of the unperturbed solution. Hence $u'(a)\simeq u_0(0)$. Also $u_0$ is nearly linear near $r=0$, and $u_0(a)\simeq u'(0) a$, and eventually $$E-E_0\simeq u'(0)^2 a~.$$ which reduces to (\[eq:Trueman\]) if $V_0(r)=-1/r$. For a moderately attractive potential, $a$ is negative, but the formula 