  The auxiliary field method is a new technique to obtain closed formulae for
the solutions of eigenequations in quantum mechanics. The idea is to replace a
Hamiltonian $H$ for which analytical solutions are not known by another one
$\tilde H$, including one or more auxiliary fields. For instance, a potential
$V(r)$ not solvable is replaced by another one $P(r)$ more familiar, or a
semirelativistic kinetic part is replaced by an equivalent nonrelativistic one.
The approximation comes from the replacement of the auxiliary fields by pure
real constants. The approximant solutions for $H$, eigenvalues and
eigenfunctions, are then obtained by the solutions of $\tilde H$ in which the
auxiliary parameters are eliminated by an extremization procedure for the
eigenenergies. If $H=T(\bm p)+V(r)$ and if $P(r)$ is a power law, the
approximate eigenvalues can be written $T(p_0)+V(r_0)$, where the mean
impulsion $p_0$ is a function of the mean distance $r_0$ and where $r_0$ is
determined by an equation which is linked to the generalized virial theorem.
The general properties of the method are studied and the connections with the
envelope theory presented. This method is first applied to nonrelativistic and
semirelativistic two-body systems, with a great variety of potentials. Closed
formulae are produced for energies, eigenstates, various observables and
critical constants, with sometimes a very good accuracy. The method is then
used to solve nonrelativistic and semirelativistic many-body systems with
one-body and two-body interactions. For such cases, analytical solutions can
only be obtained for systems of identical particles, but several systems of
interest for atomic and hadronic physics are studied. General results
concerning the many-body critical constants are presented, as well as duality
relations existing between approximate and exact eigenvalues.
