The title of the book ``Non-Hermitian Quantum Mechanics" (NHQM) is
like the new dress for an old lady. Under this name the book under
review truly does identify and fill the lasting gap in the market.
Still, by the personal opinion of the reviewer (which need not
necessarily be shared by the author - and probably even is not), the
book does not certainly describe any new Quantum Theory. It rather
summarizes some of the most recent and advanced applications of the
standard formalism, with the title making the subject just looking
more attractive. A more conservative author would call this text
``Contemporary theory of resonances" or something like that.

I am not saying that the latter title would sound better or that it
would attract a broader audience but certainly, the word
``resonance" could have then been omitted from the titles of
chapters (out of the total of ten, it occurs in four). Plus: the
word ``resonance" would more precisely target the potential readers
with predominant interest in atomic, molecular, chemical (and,
perhaps, also nuclear and condensed matter and optical) physics.

This being said, this book really is top class. Being written by an
extremely experienced active scientist it summarizes, first of all,
the results of his own research performed during the last thirty or
so years. The reading of this text should be recommended warmly. The
message should be praised for being timely and well organized.

The text is well written and compact. Its main strength is not in
the listing of references. Just a short list of the most relevant
ones is added, to each chapter, as a ``further reading", rather
exceptionally not (co-)authored by N.M. Moreover, the text is
self-characterized as a ``textbook" and this name fits well (note
just that almost one quarter of pages is devoted to exercises and
their solutions).

Perhaps, as ``the first book ever written that presents NHQM" (cited
from the first line of preface and/or from the author's homepage)
the text should have been less centered around applications in
quantum chemistry and around the subset of these applications which
are better known as the ``complex scaling method". On the other
hand, such an author's decision and strategy protects this review
from getting separated from the reality of physics of the
surrounding-us world. The author never breaks this contact and, in
analogy with classical drama, his book keeps the unity of place and
action and preserves the integrity of the style without ceasing to
be useful.

According to the present reviewer's taste (and in my full personal
admiration towards the author's achievements!), a sprinkle of a more
abstract framing of the material would still help. In places, it
could even simplify and shorten the explanations (typically, in
chapter 6, where the introduction of the elementary notion of
biorthogonal systems is too much practice-motivated and, hence,
rather clumsy and too long). The abstract mathematical details of
``NHQM theory" also come rather late -- too late, perhaps (mainly in
chapter 7).

In consequence, the less informed readers (or naive undergraduates,
at which the book is also aimed) might feel quite puzzled by many
physics-explaining commentaries about NHQM, appearing in the first
and/or last chapter but lacking any sufficiently explicit or
mathematical specification of what the notion really means. For
example: Is it really so that (according to chapter 10) ``QM
branches into two formalisms"? And: should we believe that
(according to chapter 1) ``there is no (known) transformation which
enables one to map results which were obtained using one formalism
to the other one"?

At the same time, having passed through the bulk text, it is a
pleasure of reader to get rewarded, in by far the longest chapter 8,
by the author's particularly personal account of the extension of
the formalism to the scattering regime (here, also the list of
``further reading" gets particularly long and multifaceted), with
several extremely interesting applications -- let me mention just
the so called high-order harmonic generation and above-threshold
ionization of atoms in laser fields for illustration.

If we omit comments on the rather off-stream and comparatively
incomplete account of the well known Feshbach's projection-operator
method in chapter 10 (which would deserve to be longer) and if we
politely pretend that we did not notice the absence of any global
summary (a ``missing chapter 11"?) we will definitely feel pleased
also by the effectively last chapter 9. This chapter might be
perceived as a certain climax of the story, by having dealt with one
of the true highlights of the formalism, viz., with the concept and
properties of the so called ``exceptional points".

By its origin, the term ``exceptional point" (EP) belongs to the
theory of analytic functions. In the context of mathematical physics
its introduction should be attributed to T. Kato who recognized its
role in Rayleigh-Schr\"{o}dinger perturbation theory. Unfortunately
(i.e., to my small personal disappointment), the title of chapter 9
does not carry any immediate reference to EPs. Still, the contents
reflect several truly newest, EP-related developments in the NHQM
field.

In PS let me add that (1) in the list of references appended to
chapter 9, the papers by N.M. just take the exceptionally small
share of cca 25 percent; apparently, more people are currently
entering the N.M.'s traditional NHQM field at present; (2) in
November 2010, Dieter Heiss organized, in Stellenbosch, the first
special international meeting devoted strictly to the physics of
exceptional points (during which, incidentally, I personally met the
author); the details may still be googled out (or clicked:
http://www.nithep.ac.za/2g6.htm); (3) the NHQM subject appeared
(newly and, presumably, under the influence of the N.M.'s years-long
publication output) in the MR Mathematics Subject Classification
scheme, released in the year 2010, as the separate item Nr. 81Q12.


MR2768169 Moiseyev, Nimrod Non-Hermitian quantum mechanics.
Cambridge University Press, Cambridge, 2011. xiv+394 pp. ISBN:
978-0-521-88972-8 81Q12