Moment-problem formulation of a minimax quantization procedure

The eigenvalue moment method (EMM) is a general theory for generating converging lower and upper bounds to the discrete, low-lying spectrum of Schr\"odinger Hamiltonians. Recently, Handy, Giraud, and Bessis [Phys. Rev. A 44, 1505 (1991)] developed a dynamical systems EMM formulation through the discovery of a fundamental convex function, FE[u]=Min_σ=0,1〈V^(σ,E) [u]{}M^(σ,E)[u]{}V^(σ,E)[u]〉. By incorporating this within the c-shift EMM theory of Handy and Lee [J. Phys. A 24, 1565 (1991)], there results an alternative quantization procedure involving the function V(E)=Maxᵤ{Min}_{{{σ}}=0,1}$〈${V}^{{{σ}},E,S)}[u]{S}_{{{σ}}}^{{{-}}1}M^(σ,E)[u]S_σ^-1{} V^(σ,E,S)〉, whose local maxima converge to the discrete energy states of the system. We discuss the relevant theory and present several examples.