  Recent investigations by Bender and Boettcher (Phys. Rev. Lett 80, 5243
(1998)) and Mezincescu (J. Phys. A. 33, 4911 (2000)) have argued that the
discrete spectrum of the non-hermitian potential $V(x) = -ix^3$ 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 $|\Psi(x)|^2$. This permits the
application of the Eigenvalue Moment Method, developed by Handy, Bessis, and
coworkers (Phys. Rev. Lett. 55, 931 (1985);60, 253 (1988a,b)), 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.
