Very interesting and compact material (in fact, a PhD thesis) is
presented, alas, under a not too informative title. Indeed, all of
us might immediately feel discouraged by the fact that there is
nothing particularly interesting, say, on the exactly solvable
quantum Kepler problem where the interaction potential $V(\vec{r})$
is trivially ``unbounded". Even worse: the author certainly knew
many much better titles himself. Unfortunately, he used one of the
best-fitting eligible ones (viz., ``Relatively bounded perturbations
in Krein spaces") just to give the name to the first chapter of his
book. Feeling, presumably, the point in the minimality of the
subsequent replacement of ``bounded" by ``form-bounded" to give the
name (and contents) to the second chapter (there are no more
chapters in effect, since the third one - in fact, a half-sized last
one - just offers illustrative examples).

The book is aimed at, and may be warmly recommended to, all of the
mathematicians and/or mathematical physicists who felt, recently,
addressed by the upgrade of the formalism of quantum theory
carrying, in the current literature, the almost unanimously accepted
name which more or less coincides with the first half of the title
of the book under review (cf. also the dedicated webpages
http://ptsymmetry.net/ or
http://gemma.ujf.cas.cz/%7Eznojil/conf/index.html in this context).
Although the young author himself does not seem to have contributed
to the (or at least to the here-cited) journal-published literature
yet, he was already able to offer a very concise and perfectly
mathematically formulated account of the selected
bounded-perturbation subfield of the subject. Having built a
vocabulary, translation and a bridge between the cultures and
languages of the physicists and mathematicians.

It is certainly true that for mathematicians, the subject of the
relatively bounded perturbations of self-adjoint operators in Krein
spaces is not a new topic. This is emphasized by this book, in which
the mere review of the classical related literature as well as of
its recent important perturbation-theory upgrades (cf., in
particular, the papers by Caliceti with coauthors between 2005 and
2008) did the very good job. With the upgrade, compactification and
completion of the existing spectral-stability results and extension
theory in Krein spaces of course - the PhD supervision by Christiane
Tretter is well felt behind the scenes.

This being said, it should be added that the physicists working in
the field might be disappointed by the scope of the book which
provides just a thorough account of the conditions under which the
spectrum of the perturbed operators remains real. No account is
added of the very recent shift of attention of the community (cf.,
e.g., the webpage http://ptqm.physi.uni-heidelberg.de/ of the recent
dedicated conference for more details) to the dynamical regime in
which the spectrum of the PT-symmetric quantum Hamiltonians
degenerates and becomes complex.

MR2830591 Nesemann, Jan ${PT}$-symmetric Schroedinger operators with
unbounded potentials. Dissertation, Universitaet Bern, Bern, 2010.
Vieweg + Teubner, Wiesbaden, 2011. viii+83 pp. ISBN:
978-3-8348-1762-4 81Qxx (34L40 47B25 47N50)