  The eigenenergies $\epsilon^{(N)}(m;\{n_i,l_i\})$ of a system of $N$
identical particles with a mass $m$ are functions of the various radial quantum
numbers $n_i$ and orbital quantum numbers $l_i$. Approximations $E^{(N)}(m;Q)$
of these eigenenergies, depending on a principal quantum number
$Q(\{n_i,l_i\})$, can be obtained in the framework of the auxiliary field
method. We demonstrate the existence of numerous exact duality relations
linking quantities $E^{(N)}(m;Q)$ and $E^{(p)}(m';Q')$ for various forms of the
potentials (independent of $m$ and $N$) and for both nonrelativistic and
semirelativistic kinematics. As the approximations computed with the auxiliary
field method can be very close to the exact results, we show with several
examples that these duality relations still hold, with sometimes a good
accuracy, for the exact eigenenergies $\epsilon^{(N)}(m;\{n_i,l_i\})$.
