Generating converging eigenenergy bounds for the discrete states of the -ix³ 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) = -ix³ should be real. We give further evidence for this through a novel formulation which transforms the general one-dimensional Schrodinger equation (with complex potential) into a fourth-order linear differential equation for |Ψ(x)|². This permits the application of the eigenvalue moment method, developed by Handy, Bessis and coworkers, yielding rapidly converging lower and upper bounds to the low-lying discrete state energies. We adapt this formalism to the pure imaginary cubic potential, generating tight bounds for the first five discrete state energy levels.