MR2117176 Caliceti, Emanuela ; Graffi, Sandro . On a class of non
self-adjoint quantum nonlinear oscillators with real spectrum. J.
Nonlinear Math. Phys. 12 (2005), suppl. 1, 138--145.




The necessity of the proofs of the reality of spectra for certain
non-Hermitian Hamiltonians emerged in connection with the
emergence of certain very strange (viz., asymptotically imaginary
cubic) potentials $V(x) = {\rm i}\,x^3$ in relativistic field
theory. The recent resolution of this particular problem (cf.
refs. [19] and [17]) re-attracted attention to many similar
Schr\"{o}dinger equations. For their interesting subset with
polynomial $V(x)$ [such that,  at the large $x$, one has ${\rm
Im}\ V(x) \approx g\,x^{2r-1}$ while the real part of $V(x)$
happens to be sufficiently quickly growing, ${\rm Re}\ V(x)
\approx x^{4r+2}$ (or more)], the authors noticed that the
imaginary part of $V(x)$ is relatively bounded. This alowed them
to prove, using perturbative construction with non-vanishing
circle of convergence, the reality of the spectrum for not too
large strengths $g$ of non-Hermiticity.