Extension of a moment-problem minimax quantization procedure to anharmonic potentials

Recently, Handy, Appiah, and Bessis (HAB) [Phys. Rev. A 50, 988 (1994)] showed that the local maxima in the energy variable E of the function V(E)≡max_umin_σλ_E^σ[u→] (where λ_E^σ[u→] are the smallest eigenvalues of modified Hankel moment matrices) approximate the discrete energy states of Schrödinger Hamiltonians. Their theoretical result was demonstrated by way of two zero-missing-moment problems, the harmonic oscillator and the x²+λ[x²/(1+gx²)] potentials, for which the corresponding eigenvalue functions λ_E^σ are solely energy dependent. We examine the general n-missing-moment problem through an application of gradient optimization techniques that are suitable for piecewise differentiable functions, thereby enabling us to test HAB’s results for anharmonic potentials of missing-moment order 1, 2, and 3, corresponding to the quartic, sextic, and octic anharmonic potentials, respectively.