From the point of view of relativistic quantum mechanics, the key
target of this study is the so called Klein paradox: Everybody knows
-- or may read in refs. [50] - [53] -- that for the apparently
confining potential $V(x)= \omega^2x^2$ (and the like -- in units
$m/2=1$) the Dirac or Klein-Gordon spectrum of ``energies" appears,
paradoxically, continuous and unbounded. Still, in a way explained
by the first coauthor in ref. [27] one can construct the so called
metastable states and arrive at many useful experimental predictions
(typically, of the pair-production rates).

From the point of view of pure mathematics, the main methodical tool
offered by the paper under review (which may be also read as a
certain continuation of their previous paper [28]) lies in the
change of perspective mediated, in essence, by certain suitable
isospectral as well as non-isospectral changes of the Hamiltonian.
This idea dates back to the 18 years old ref. [37] coauthored by the
second present coauthor and, by the way, not too often known or
cited in a recent series of rediscoveries of the concept of
PT-symmetry. In this context let me also add, marginally (cf. ref.
[1]), that it was a true pleasure of mine to see the new example of
efficiency of the concept of the -- here, ``Titchmarsh-operator" or
``associated" -- energy-dependent and non-Hermitian though mutually
conjugate PT-symmetric Hamiltonians $H_\pm$.

The resulting message is extremely interesting: the use of auxiliary
(in some sense, just partially or asymptotically isospectral)
$H_\pm$ (as well as the use of sophisticated methods of its
numerical analysis) is shown to provide a truly reliable approximate
localization of the relativistic resonances as well as the strict
correspondence of the metastable states to the bound states in
non-relativistic limit.

An important encouragement is provided for all of the recent efforts
aimed at extraction of information from the analyticity (or, more
explicitly, from a complex-dilatability) of certain ``useful"
operators with continuous spectra. In this sense the paper is also
remarkable and non-main-stream by its {\em admitting} the natural
{\em dissipative} nature of quantum evolution. Last but not least,
the mathematically rigorous and exhaustive though still legible
style of the text might offer a very promising bridge between the
physics- and mathematics-oriented communities.


MR2771874 Giachetti, Riccardo; Grecchi, Vincenzo $PT$-symmetric
operators and metastable states of the 1D relativistic oscillators.
J. Phys. A 44 (2011), no. 9, 095308, 13 pp. 81Qxx