Generating converging bounds to the (complex) discrete states of the P-2+iX(3)+i alpha X Hamiltonian

The eigenvalue moment method (EMM) is applied to the H-alpha = P-2 + iX(3) + i alphaX Hamiltonian, enabling the algebraic/numerical generation of conv erging bounds to the complex energies of the L-2 states, as argued (through asymptotic methods) by Delabaere and Trinh. The robustness of the formalis m, and its computational implementation, suggest that the present non-negat ivity formulation implicitly contains the key algebraic relations by which to prove Bessis' conjecture that the eigenenergies of the H-o Hamiltonian a re real. The required algebraic analysis of the EMM procedure pertaining to this problem will be presented in a forthcoming paper.