For a typical mathematician the title of this paper must sound
incomprehensible because he/she knows that in a consistent quantum
theory the Stone's theorem must be obeyed. In other words the
Hamiltonian $H$ of a (stable) quantum system must {\em always} be
(essentially) self-adjoint. An easy key to the resolution of this
apparent paradox lies in the (implicit but, recently, increasingly
popular) {\em parallel} representation of a given quantum system
in {\em several} (in general, unitarily non-equivalent) Hilbert
spaces ${\cal H}^{(1)},{\cal H}^{(2)},{\cal H}^{(3)},\dots$.
Typically, the operator $H$ acquires a particularly simple (i.e.,
say, easily diagonalizable) form in ${\cal H}^{(1)}$ where it
remains, by assumption, manifestly non-self-adjoint. The trick
(summarized and (re)explained, more thoroughly, elsewhere - cf.,
e.g., M. Znojil, SIGMA 5 (2009) 001 [19 pages]) is based on a
non-unitary generalization of the Fourier-type mappings. This
mapping (denoted as $exp(-{\cal Q}/2)$ in MS under consideration,
or by the symbol $\Omega$ in {\it loc. cit.}) ``pulls back" our
$H$ into a family of amended, ``physical" (although not always
mutually unitarily equivalent) Hilbert spaces ${\cal
H}^{(2)},{\cal H}^{(3)},\dots$. In these new spaces the
Hamiltonian becomes safely self-adjoint and Hermitian. For this
reason, {\em all} our Hamiltonians {\em are} Hermitian in a
suitable space (so, one shouldn't call them non-Hermitian but
rather ``cryptohermitian" at most).
In a typical ``computationally friendly" model of such a class we
start working in the ``false" space ${\cal H}^{(1)}$ chosen as the
Hilbert space of quadratically integrable functions of $N$ real
variables, ${\cal H}^{(1)}\equiv L^2(I\!\!R^N)$. In parallel we
restrict our attention to Hamiltonians $H = T+V$ where the first,
linear-differential-operator component $T = \sum
\,a_n\,\partial^2_n$ represents kinetic energy. In this context
the necessity of moving to ``physical" spaces ${\cal
H}^{(2)},{\cal H}^{(3)},\dots$ is, typically, evoked by the
transition to a broader class of interactions (given, say, by a
{\em complex} $V$ in ${\cal H}^{(1)}$).
In this context the present MS pays attention to several solvable
examples where the energies remain real for a weak coupling
between some well separated subspaces of ${\cal H}^{(1)}$ where
$V$ is chosen as real and complexified, respectively.
MR2455804 Bender, Carl M.; Jones, Hugh F. Interactions of
Hermitian and non-Hermitian Hamiltonians. J. Phys. A: Math. Theor.
41 (2008), no. 24, 244006, 8 pp. 81Q10