(Quasi)-Convexity Formulation of a Multi-Minima, Variational, Quantization Method

Barta's ground state energy bounding theorem states that the E$_{L}$ = Inf ( H$\Phi $(x)/ $\Phi $ (x)) and E$_{U}$ = Sup ( H$\Phi $(x)/ $\Phi $ (x)) for an arbitrary, positive, bounded, configuration, $\Phi $(x), define lower and upper bounds, respectively, to the bosonic ground state energy: E$_{L} <$ E$_{ground} <$ E$_{U}$. Searching for the x-values corresponding to the infimum (Inf) and supremum (Sup) is a multi-extrema plagued process, particularly in multidimensions. We can reformulate Barta's configuration space analysis in terms of the \textit{Moment Problem}, via a Generalized Eigenvalue Problem representation. This removes all multi-extrema considerations, recasting the original variational problem as one involving (quasi)-convexity optimization. We outline the procedure, and apply it to some, illustrative problems.