CNO: 1873185
                      Handy, C. R.; Wang, Xiao Qian. Extension of a spectral bounding method to
                      complex rotated Hamiltonians, with application to $p\sp 2-ix\sp 3$. J. Phys. A
                      34 (2001), no. 40, 8297--8307.

In the extended abstract of the preceding paper of the series
(CNO: 1855381) we already summarized the history and essence of
solving certain ''next-to-solvable" Schroedinger equtions by means
of the so called spectral bounding method proposed  by Carlos
Handy many years ago. A further natural step of development of
this packet of methods (all being related to a less standard
linear-programming approach to the current quantum bound state
problem) is presented now by Handy and Wang. Once more, they pay
their attention to the imaginary cubic interaction and once more,
they succeeded in breaking the apparent natural limitations of the
scope of the method, this time contemplating their Schroedinger
equation after its complex rotation which breaks the PT symmetry
(that's why the test of the method would be very stringent).

Within the new framework the paper clarifies several technical
aspects of the extended approach and resolves several underlying
mathematical questions concerning, e.g., the effect of the
emergence of the singularities within the Hamburger moment
language. The thorough discussion of the subtleties of the related
fourth-order re-formulation and thirteen-diagonal band-matrix
re-arrangement of the original problem is presented in a concise
manner. The method is verified and illustrated by the numerical
table which confirms the a priori expectations that the best rate
of convergence may be achieved in the standard setting, at the
vanishing complex re-scaling angle.

The choice of the particular cubic Hamiltonian example is based on
its enormous popularity. Without success I tried to list some key
references in the Los Alamos preprint arXiv: math-ph/0101027 but
one of them updates the Handy' sand Wang's introductory remarks
and deserves an explicit citation here. In J. Phys. A: Math. Gen.
34 (2001) 5679,  Dorey et al succeeded in proving that the energy
spectrum (under present consideration) is real, indeed.