Generating converging eigenenergy bounds for the discrete states of the -ix(3) non-Hermitian potential

Recent investigations by Bender and Boettcher and by Mezincescu have argued that the discrete spectrum of the non-Hermitian potential V(x) = -ix(3) sh ould be real. We give further evidence for this through a novel formulation which transforms the general one-dimensional Schrodinger equation (with co mplex potential) into a fourth-order linear differential equation for /Psi (x)/(2). This permits the application of the eigenvalue moment method, deve loped by Handy, Bessis and coworkers, yielding rapidly converging lower and upper bounds to the low-lying discrete state energies. We adapt this forma lism to the pure imaginary cubic potential, generating tight bounds for the first five discrete state energy levels.