A long-missing contribution to the branch of constructive quantum
mechanics a.k.a. cryptohermitian, quasi-Hermitian, PT-symmetric or
pseudo-Hermitian quantum mechanics. In this approach the building of
new tractable phenomenological models is based on their
simplification via a nonstandard (i.e., nontrivial, metric-mediated
and, in general, strongly ambiguous re-) definition of the dual
elements in the Hilbert space of states [i.e., in Dirac's
terminology, of the so called bra vectors - a more detailed version
of this abstract explanation can be found in [11] or in my text in
SIGMA 5 (2009), 001]. Concerning the particular, illustrative
anharmonic-oscillator example given in the title, some of its
puzzling features are already known for more than thirty years. The
history [partly told to me by Gabriel Alvarez, cf. also his
comprehensive text in J. Phys. A: Math. Gen. 27 (1995) 4589]
started, perhaps, by the half-forgotten perturbation-theory paper on
the asymptotically purely imaginary anharmonicity where,
asymptotically, $\epsilon=1$, cf. Caliceti et al, Comm. Math. Phys.
75 (1980) 51. Still, the paper under consideration seems to be the
first successful attempt introducing the above-mentioned definition
via perturbation series using the smallness of exponent $\epsilon$.
Psychologically, this is a slightly surprising fact since it was
precisely Bender et al who developed the appropriate method (called
$\delta-$expansions, cf. ref. [13]) five years earlier than their
attention has been re-directed to the apparently non-Hermitian
Hamiltonians in question (by Daniel Bessis, cf. footnote [12]).


MR2533897 Bender, Carl M.; Besseghir, Karim; Jones, Hugh F.; Yin,
Xinghui Small-$\epsilon$ behavior of the non-Hermitian $\scr P\scr
T$-symmetric Hamiltonian $H=p\sp 2+x\sp 2(ix)\sp \epsilon$. J. Phys.
A 42 (2009), no. 35, 355301, 10 pp. 81Qxx