  The recent work of Delabaere and Trinh (2000 J. Phys. A 33 8771) discovered
the existence of ${\cal P}{\cal T}$-symmetry breaking, complex energy, $L^2$
solutions for the one dimensional Hamiltonian, $P^2+iX^3+i\alpha X$, in the
asymptotic limit, $\alpha \to -\infty$. Their asymptotic analysis produced
questionable results for moderate values of $\alpha$. We can easily confirm the
existence of ${\cal P}{\cal T}$-symmetry breaking solutions, by explicitly
computing the low lying states, for $|\alpha| < O (10)$. Our analysis makes use
of the Multiscale Reference Function (MRF) approach, developed by Tymczak et al
(1998 Phys. Rev. Lett. 80 3678; 1998 Phys. Rev. A 58, 2708). The MRF results
can be validated by comparing them with the converging eigenenergy bounds
generated through the Eigenvalue Moment Method, as recently argued by Handy
(2001a,b). Given the reliability of the MRF analysis, its fast numerical
implementation, high accuracy, and theoretical simplicity, the present
formalism defines an effective and efficient procedure for analyzing many
related problems that have appeared in the recent literature.
