The ultimate purpose of studies like this lies in the constructive
build-up of certain new quantum field theories. A few technical
aspects of this key idea are tested here via one of the best
understood quantum-mechanical models with a non-Hermitian
PT-symmetric quantum Hamiltonian $H= (PT) H (PT)^{-1} \neq
H^\dagger$ controlled by a repulsive quartic interaction term at
large values of coordinates which are (tacitly but very importantly)
assumed complex. In a continuation of one of previous papers on this
subject (viz., of ref. [5] coauthored by one of the present authors)
an innovative insight is gained in several aspects of the puzzle.
The $N-$dimensional version of the model is considered in which its
$O(N)$ symmetry enables the authors to separate the two-dimensional
radial part of Schr\"{o}dinger equation. In a preparatory step the
absence of the so called scattering states is demonstrated in a way
paralleling one of proofs proposed, for toy-model potentials
$V(x,y)=x^2y^2$, by Simon [11]. Secondly, under the assumption that
$N$ is large, a few key features of the ground-state energy are
clarified. Variational, Born-Oppenheimer or zero-mass approximations
are shown to serve the purpose revealing the presence of a phase
transition at certain parameters. The methodical impact of this
large$-N$ result is only weakened by the puzzling fact that the
domain beyond the phase-transition boundary appears only accessible
indirectly, i.e., by means of transition from $H$ to its isospectral
Hermitian partner. Indeed, in the light and spirit of the
above-mentioned ultimate purpose of the analysis this observation is
slightly disappointing because the isospectral-partner operators
usually happen to become prohibitively complicated and non-unique in
similar physical theories. With the remarkable exception of the
particular model in question, brought in attention, implicitly, by
Ref. [4] and, explicitly, by Buslaev and Grecchi in J. Phys. A 26
(1993) 5541. Fortunately, one can stay optimistic since, in the
context of PT-symmetric theories, the large$-N$ method seems to
remain sufficiently robust and efficient not only in all the other
similar, exceptionally friendly or almost solvable models [{\it pars
pro toto} let me cite my own large$-N$ approach to
anharmonic-Calogero manybody model in J. Phys. A 36 (2003) 9929 plus
its PT-symmetry-oriented, Andreas Fring coauthored pendant in J.
Phys. A 41 (2008) 194010] but also in many PT symmetric systems with
a truly prohibitively complicated structure of their isospectral
Hermitian description [in this respect I may finally recommend an
illustrative example and more detailed commentary which I quite
recently presented in Phys. Lett. A 374 (2010) 807].

MR2525281 Nishimura, Hiromichi; Ogilvie, Michael The large-$N$ limit
of $PT$-symmetric $O(N)$ models. J. Phys. A 42 (2009), no. 2, 02202,
9 pp. 81Qxx