In the recent literature on applications of the standard formalism
of quantum mechanics of unitarily evolving systems one often finds
papers speaking about ``non-Hermitian" Hamiltonians $H$ with real
spectra which generate the evolution. In a way which I thoroughly
explained elsewhere (cf. M. Znojil, ``Three-Hilbert-space
formulation of Quantum Mechanics",  SIGMA 5 (2009), 001, doi:
10.3842/SIGMA.2009.001, arXiv: 0901.0700), this is a pure
misunderstanding. In ALL of these cases, the operator $H$ merely
appears non-self-adjoint in some ill-chosen, auxiliary, ``false"
representation of the Hilbert space of states (denoted as ${\cal
H}_D$ by Ghosh). In parallel, our $H$ remains safely self-adjoint in
a certain ``standard" textbook representation of the physical
Hilbert space of states (denoted as ${\cal H}_{\eta_+}$ by Ghosh).

Let us emphasize that the main and only source of misunderstandings
of this type is that for certain strange reasons people really love
working in the ``friendly", auxiliary representation space ${\cal
H}_D$ which is manifestly unphysical.

Via his paper, Ghosh tries to make these people happy by listing
several (roughly counting, six different) illustrative examples of
$H$ for which the correct representation space ${\cal H}_{\eta_+}$
still seems to remain known and sufficiently simple. In the language
of physics, these examples are based on the use of specific
Bogoliubov transformations and include single particle models
(harmonic oscillator and Stark and Zeeman effects) as well as
many-body systems including also various spin chains and/or, last
but not least, Lipkin–Meshkov–Glick model. Involuntarily, the author
performs, in this manner, a full-circle return to the many-body
example given also as illustrative in the 1992 paper [4] by
``fathers founders" of the whole business.

The paper is well written and instructive. Its author selects and
defines his space ${\cal H}_{\eta_+}$ in advance (in fact, via eqs.
(2) and (6)). In the language of mathematics this would immediately
lead to some doubts and questions about the (mathematically
necessary though often ignored conditions of) boundedness of the
metric $\eta_+$ of eq. (6) and of its inverse. A further recommended
reading and a more extensive discussion related to the latter and
similar subtleties may be recommended, say, in its first formulation
``Quasi-Hermiticity in infinite-dimensional Hilbert spaces" by R.
Kretschmer and L. Szymanowski in Phys. Lett. A 325 (2004) 112 - 117.

MR2599802 Ghosh, Pijush K. On the construction of a pseudo-Hermitian
quantum system with a pre-determined metric in the Hilbert space. J.
Phys. A 43 (2010), no. 12, 125203, 19 pp. 81Qxx