The choice of the title of the book is certainly unfortunate.
Indeed, when I showed this title to a number of my colleagues, many
of them immediately associated the book with the two years old
Strocchi's excellent monograph on non-perturbative quantum field
theory. No surprise that they felt frustrated by the
misunderstanding. Moreover, those of them who still, with obvious
curiosity opened the book, reported just an additional
disappointment. What they did expect was something like functions
$e^{-1/x^2}$ but what they truly got was just the split of
Hamiltonian $H$ into the very traditional sum $H=H_0+V$. From their
only slightly remote perspective, nothing beyond perturbation
theory.

The use of the misleading title acquires additional flavor of
disbelief when one notices that the authors dedicated their very
carefully prepared, well organized review to the memory of Lev
Ivanovich Komarov who was one of the pioneers of the development of
what the book is truly about, viz., of the so called (and cleverly
called) ``operator method'' (OM) which is a fairly universal tool
for the solution of an impressively broad class of various
bound-state Schroedinger equations $H\,|\psi_n \rangle= E_n|\psi_n
\rangle$.

Probably, the overambitious authors intended to promise more than
they could deliver. It is a pity. Indeed, the book is comprehensive.
It deserves full attention of the appropriate readership and, first
of all, of all of the researchers who work in the field of
perturbation theory and of its practical applications. Naturally, in
order to attract their attention a better choice of the title would
also deserve to be paralleled by a more open-minded preface and
Chapter 1. Frankly: in place of reading the highly philosophical
introductory text about the ``Capabilities of Approximate Methods in
Quantum Theory'' in Chapter 1 I would personally prefer to start my
reading from a concise outline of the basic OM ideas.

Fortunately, in this context I feel strongly privileged by my age
(in 1982 I read the pioneering Feranchuk-Komarov paper in Physics
Letters immediately after it appeared), by my interests (I read this
text - and many of its follow-ups - with genuine appreciation) and
by my scientific contacts (let me add here my special thanks, for
numerous debates and for his prevailingly enthusiastic praise of the
merits of the subject, to Marcelo Fern\'{a}ndez). So let me now try
to save time of the readers {\it in spe} (i.e., of many theoretical
physicists and quantum chemists) and to fill, partially, the gap.
With apologies that my recommendation to read the book will be too
concise (I shall skip a few rather essential subtleties like, e.g.,
a transition from wave functions to statistical operators, etc) and
not always sufficiently rigorous (after all, we have the book!).

In OM, in essence (i.e., in the sufficiently simple applications at
least), the Hamiltonian $H$ is assumed given in a generic
coordinate-, momentum- and coupling-dependent form $H(x,p,\lambda)$.
Immediately, this (i.e., mostly, partial differential) operator is
being transformed into an algebraic form defined in terms of the
usual creation and annihilation operators $a^\dagger$ and $a$,
respectively.

The latter step allows one to introduce a new, auxiliary,
``mass-like'' free parameter $\omega$, to be assigned a
variational-like optimization role later on. In the subsequent, key
step of the construction one separates $H=H_0+V$ where $H_0$ is
``diagonal'' (i.e., it depends solely on the number operator
$n=a^\dagger a$) while the rest of $H$ is a ``non-perturbative
perturbation'' $V = H-H_0$.

The rest of the story is to be found in the book. There are lots of
details, lots of clever tricks and lots of illustrative examples
including many numerical tables and colored figures. Interested
(plus, hopefully, hereby converted) readers may either swallow all
the text or pick up some raisins. These are presented, in a
comparative independence, in Chapters number two (about the method),
three (mainly about various 1D anharmonicities), four (where the
temperature enters the game), five (here, more degrees of freedom
are taken into account), six (remarkable: magnetic field is now
admitted) plus 7 and 8 (about atoms) and the last one (mainly about
polaron, perceived as a representative of systems possessing
infinitely many degrees of freedom).

Having read Chapter 9 the reader might miss  a Summary or Outlook.
But this is just proper time to return and indulge in the
Introduction. One may now properly appreciate the statements about
the prominent features of OM in which the mathematical concept of
smallness of perturbations is certainly {\em different} from the
traditional physicist's perception of what should be small in the
Hamiltonian. With a final advice added at the end: the readers
should take the dramatic description of the drawbacks of the
existing alternative methods {\it cum grano salis}.


MR3308409 Feranchuk, Ilya; Ivanov, Alexey; Le, Van-Hoang;
Ulyanenkov, Alexander Non-perturbative description of quantum
systems. Lecture Notes in Physics, 894. Springer, Cham, 2015.
xvi+362 pp. ISBN: 978-3-319-13005-7; 978-3-319-13006-4 81-02 (35J10
35Q40 81Q05 81Q35 81S20)