I recommend this paper to all the potential readers who could miss
it being discouraged by its partially misleading title and/or
abstract. This well organized and comprehensible text pays main
attention to matrices rather than operators (that's why we see
"Hermitian" in place of the expected "essentially self-adjoint",
etc). The quantum dynamics is just mentioned, marginally, on the
first and last pages. Still, there is no doubt concerning the
relevance of this review for quantum mechanics in which, due to
the current conventional wisdom, the assumption of the unitary
evolution seemed always "necessary" as leading to the
time-independence of the probability density (cf. Stone's
theorem). In this sense, the paper contributes to the recent
project of innovation of quantum theory initiated by Bender and
Boettcher (cf. Phys. Rev. Lett. 24 (1998) 5243) admitting
pseudounitary evolution (the concept itself first appeared in the
title of my own LANL arXive preprint math-ph/0104012 in April
2001) but this is already the history covered/hidden by the mere
reference to the long series of the previous Mostafazadeh's own
papers.

An important merit of this new paper is that it is concise. On the
small space of 15 pages the author characterizes his pseudounitary
operators by their spectrum in main Theorem 2 (where an
unfortunate misplacement of the conditions is not difficult to be
put in place by the reader) and relates his pseudounitary matrices
to the (more standard concept of) elements of U(p,q) and discusses
the main differences. As a Corollary 1 of Theorem 3, the
exponential mapping between his pseudounitary and pseudoHermitian
matrices becomes bijection. For illustration, he classifies the
two-by-two cases. A concrete example is finally provided by the
classical equation of motion of a simple harmonic oscillator.

One appreciates that the text is quickly but carefully written and
that it compactly reviews and/or fills gaps in the literature. A
nonzero concentration is needed to follow the story. Fortunately,
all becomes clear after the reader imagines that in the author's
not entirely standard language, the usual pseudo-unitary matrix
should be called $\eta_{p,q}$-pseudo-unitary, with $\eta_{p,q}$
defined by eq. (30). In contrast, his n-dimensional pseudounitary
matrices do not form a group but just a subgroup of the group
defined by eq. (28). The author felt having good reasons for using
such an innovated terminology, optimally tailored for his
purposes, and some readers (including me) may concur.



MR2036172 Mostafazadeh, Ali . Pseudounitary operators and
pseudounitary quantum dynamics. J. Math. Phys. 45 (2004), no. 3,
932--946.