  We show an explicit connection between the solution to the equations of
motion in the Gaussian functional approximation and the minimum of the
(Gaussian) effective potential/action of the linear $\Sigma$ model, as well as
with the N/D method in dispersion theory. The resulting equations contain
analytic functions with branch cuts in the complex mass squared plane.
Therefore the minimum of the effective action may lie in the complex mass
squared plane. Many solutions to these equations can be found on the second,
third, etc. Riemann sheets of the equation, though their physical
interpretation is not clear. Our results and the established properties of the
S-matrix in general, and of the N/D solutions in particular, guide us to the
correct choice of the Riemann sheet. We count the number of states and find
only one in each spin-parity and isospin channel with quantum numbers
corresponding to the fields in the Lagrangian, i.e. to Castillejo-Dalitz-Dyson
(CDD) poles. We examine the numerical solutions in both the strong and weak
coupling regimes and calculate the Kallen-Lehmann spectral densities and then
use them for physical interpretation.
