The message of this brief letter is straightforward saying,
basically, that via a chosen non-unitary similarity map $\rho$ you
get a non-Hermitian isospectral representation (say, $\Sigma$) of a
given Hermitian operator $\sigma$. A few elementary illustrative
examples are then added, with the final sentence saying that ``It is
to be seen whether or not any \ldots [models] considered in this
article have relevance in the realistic physical world.''
What makes this message next to illegible is its extremely
unfortunate terminology. I have to offer a translation of the title
at least. Firstly, the authors studies ``Dirac operators'' (=
standard meaning: relativistic Hamiltonians $H$) which are assumed
either ``Dirac-Hermitian'' ($H=H^\dagger$ in Hilbert space ${\cal
H}_D$) or not. The concept of the ``Dirac non-Hermitian'' (meaning,
simultaneously, also ``non-Dirac Hermitian'' written in the more
mind-boggling form ``non-Dirac-Hermitian'' with two hyphens)
concerns the Dirac operators $H_{1D}$, $H_{2D}$ and $H_{3D}$ studied
in respective sections 3, 4 and 5. Unfortunately, the mind is
further boggled since the subscript now abbreviates ``dimension''. I
almost do not dare to recall that these operators themselves are NOT
``Dirac-Hermitian''. Moreover: their ``Dirac-Hermitian'' partners
(denoted by the lower-case symbols $h_{1D}$ etc) are squared
yielding the upper-cased $H_S=h_{1D}^2$ (etc). The new subscript
$_S$ is NOT read as (expected and usual) ``Schr\"{o}dinger
operator'' but rather as - unnecessarily further puzzling -
``supersymmetric operator''. Finally, completing the translation of
the title, ``non-dissipative'' should read ``having real spectrum''
while the derridian ``deconstruction'' merely means that the
Hermitian operator and the isospectral mapping are chosen in
advance.
MR2824212 Ghosh, Pijush K. Deconstructing non-dissipative
non-Dirac-Hermitian relativistic quantum systems. Phys. Lett. A 375
(2011), no. 37, 3250--3254. 81Qxx