.

Hybrid form of quantum theory with non-Hermitian Hamiltonians

Miloslav Znojil

Department of Physics, Faculty of Science, University of Hradec Králové,

Rokitanského 62, 50003 Hradec Králové, Czech Republic

and

The Czech Academy of Sciences, Nuclear Physics Institute,

Hlavní 130, 250 68 Řež, Czech Republic

e-mail<EMAIL_ADDRESS>
## Abstract

In Schrödinger picture the unitarity of evolution is usually guaranteed by the
Hermiticity of the Hamiltonian operator $\mathfrak{h}=\mathfrak{h}^{\dagger}$
in a conventional Hilbert space ${\cal H}_{textbook}$. After a Dyson-inspired
operator-transformation (OT) non-unitary preconditioning
$\Omega:\mathfrak{h}\to H$ the simplified Hamiltonian $H$ is, in its
manifestly unphysical Hilbert space ${\cal H}_{auxiliary}$, non-Hermitian.
Besides its natural OT-based physical interpretation it can also be
“Hermitized” (i.e., made compatible with the unitarity) via a metric-amendment
(MA) change of the Hilbert space, ${\cal H}_{auxiliary}\to{\cal
H}_{physical}$. In our present letter we propose another, third, hybrid form
(HF) of the Hermitization of $H$ in which the change involves, simultaneously,
both the Hamiltonian and the metric. Formally this means that the original
Dyson map is assumed factorizable, $\Omega=\Omega_{M}\Omega_{H}$. A key
practical advantage of the new HF approach lies in the model-dependent
adaptability of such a factorization. The flexibility and possible optimality
of the balance between the MA-related (i.e., metric-amending) factor
$\Omega_{M}$ and the OT-related (i.e., Hamiltonian-changing) factor
$\Omega_{H}$ are explicitly illustrated via an elementary two-state quantum
model.

## Keywords

.

non-Hermitian quantum mechanics of unitary systems;

hiddenly Hermitian quantum Hamiltonians;

factorized Dyson map;

Hermitization using a combined amendment of the inner product and Hamiltonian;

##  1 Introduction

In the Dyson’s paper [1] the author had to deal with a specific realistic
quantum Hamiltonian

$\mathfrak{h}=\mathfrak{h}^{\dagger}\,$ (1)

which was defined as acting and self-adjoint in a conventional Hilbert space
${\cal H}_{textbook}$, During calculations such a representation of the system
appeared “user-unfriendly” and, for this reason, Schrödinger equation had to
be simplified. For the purpose, the author made use of an isospectral
preconditioning of the Hamiltonian,

$\mathfrak{h}\ \to\ H=\Omega^{-1}\,\mathfrak{h}\,\Omega\,.$ (2)

A remarkable as well as not quite expected success has been achieved via an
insightful choice of an unusual, manifestly non-unitary  preconditioning
operator

$\Omega\neq\left(\Omega^{-1}\right)^{\dagger}\,.$ (3)

The original Schrödinger equation containing the prohibitively complicated
initial Hamiltonian $\mathfrak{h}=\mathfrak{h}_{(Dyson)}$ appeared converted
into a more easily tractable evolution and eigenvalue problem.

In loc. cit., in spite of the related loss of the Hermiticity of $H\neq
H^{\dagger}$, the invertible operator-transformation (OT) change (2) of the
Hamiltonian based on a judicious choice or construction of a suitable Dyson
map (3) paid off. The first step towards the innovative use of non-Hermitian
Hamiltonians $H\neq H^{\dagger}$ with real spectra in the quantum mechanics of
unitary systems has been made (see Refs. [2, 3] and also a few more detailed
explanatory comments in section 2 below).

Unfortunately, a wider acceptance of the OT idea appeared slowed down by the
numerous emerging mathematical obstacles. In a way emphasized, almost
simultaneously, by mathematicians [4], the manifest non-Hermiticity of $H$ may
cause a serious conceptual concern because it often implies, among other, the
emergence of instabilities of the spectrum caused even by very small
perturbations of the Hamiltonian [5].

One of the remedies has been proposed, in 1992, by Scholtz et al [6]. In
essence, these authors emphasized another, alternative (viz., “metric-
amendment”, MA) aspect of the Dyson’s concept of the Hermitization (see also a
concise outline of the MA theory in section 3 below). From the new perspective
the above-mentioned mathematical obstacles have been shown “removable” via
additional assumptions like, typically, via the purely technically motivated
requirement that the admissible non-Hermitian Hamiltonian operators $H$ have
to be required bounded.

Naturally, the latter requirement (which proved not too essential, say, for
the model-building in nuclear physics [7]) appeared hardly acceptable in a
broader quantum-theoretical context. Fortunately, a truly sophisticated way
out of the dead end has been found by Bender with Boettcher [8]. In a way not
unknown in the older mathematical literature [9] these authors shifted
emphasis, in effect, from the mathematical, OT-mediated correspondence (2)
between $H$ and its isospectral partner $\mathfrak{h}$ to the underlying, MA-
mediated physics. In particular, there authors turned attention to the concept
of parity-times-time-reversal symmetry of $H$ (${\cal PT}-$symmetry of $H$).
As a consequence their study inspired the current enormous popularity of the
widespread use of the manifestly non-Hermitian Hamiltonians $H$ in the various
areas of physics (see, e.g., the early paper [10], the standard reviews [11,
12] or the more recent monographs [2, 13]).

In our present letter we intend to complement these developments by an
innovative proposal of a certain third, “hybrid-form” (HF) combination of the
two standard and apparently distinct OT and MA theory-building strategies. In
a way explained in section 4 and illustrated by a schematic application in
section 5 we will show that for a unitarily evolving quantum system
characterized by a preselected phenomenological Hamiltonian $H$ (with real
spectrum) an optimal representation of its unitary evolution might be based on
a mere partial (i.e., not too complicated) modification of the Hamiltonian
operator accompanied by a mere partial (i.e., not too complicated) amendment
of the related correct physical Hilbert-space metric.

##  2 The Dyson-inspired Hermitization $H\ \to\ \mathfrak{h}$

In the early nineties Scholtz et al [6] recalled the success of the
isospectral simplification by mapping (2) in nuclear physics [7]. In its light
they proposed an extension of the theory and a replacement of the de-
Hermitizing mapping $\Omega:\mathfrak{h}\to H$ by its Hermitizing inversion,

$H\ \to\ \mathfrak{h}=\Omega\,H\,\Omega^{-1}=\mathfrak{h}^{\dagger}\,.$ (4)

This means that one could consider certain less usual quantum models which are
controlled by a manifestly non-Hermitian candidate $H$ for the Hamiltonian.
The transition (4) to its isospectral self-adjoint partner $\mathfrak{h}$ can
be then interpreted as an OT-mediated Hermitization and a guide to the
standard probabilistic interpretation of the system in question. The unitarity
of the evolution of the underlying quantum system as required by the Stone
theorem [14] in Schrödinger picture would then be restored.

In this spirit an innovative model-building process can start from any
sufficiently user-friendly non-Hermitian candidate $H$ for the Hamiltonian.
Still, a necessary guarantee of the unitarity of the evolution must be
provided, but this can be achieved by the mere reference to the isospectrality
(4) between $H$ and its self-adjoint partner $\mathfrak{h}$ [6]. What is
obtained is an apparently non-Hermitian or, in the current mathematical
terminology, quasi-Hermitian [4] formulation of quantum mechanics (QHQM).
According to its reviews in the newer literature the QHQM theory can be also
assigned alternative names of ${\cal PT}-$symmetric quantum mechanics [11],
three-Hilbert-space quantum mechanics [15] or pseudo-Hermitian quantum
mechanics [12].

In the latter, quantum-mechanics-representation context let us emphasize that
in the three reviews [6, 11] and [12] (as well as in our present forthcoming
text, for the sake of brevity) the Dyson map itself is always assumed
stationary,

$\Omega\neq\Omega(t)\,.$ (5)

Here, such a stationarity assumption is important. I.a., this will enable us
to follow, in section 4 below, the methodical guidance as provided by a truly
exceptional stationary-model study [9] by Buslaev and Grecchi. Indeed, Buslaev
with Grecchi were probably the first authors who opened the question of having
the Dyson map factorized. In this sense, their paper can be read as an
immediate predecessor and support of our present fundamental factorizaton
ansatz (cf. Eq. (11) in section 4 below). In addition, their paper can also be
read as complementing our present illustrative HF model of section 5 by a non-
numerical (i.e., rather rare) and constructive exemplification of the
preference of the extreme, purely OT, Dyson-map-based Hermitization (4) of a
sufficiently realistic and, at the same time, mathematically friendly and
analytic ordinary-differential Hamiltonian $H$.

##  3 The Hilbert-space metric-operator-based Hermitization ${\cal
H}_{auxiliary}\ \to\ {\cal H}_{physical}$

The distinguishing feature of the most recent implementations of the QHQM
formalism is that the necessary Hermitization of the observables, i.e., the
guarantee of the unitarity of the evolution is not provided by a
reconstruction of the textbook Hamiltonian $\mathfrak{h}$ defined as acting in
a textbook Hilbert space ${\cal H}_{textbook}$ but rather by the replacement
of the Hermiticity requirement (1) imposed upon $\mathfrak{h}$ in ${\cal
H}_{textbook}$ by its equivalent form imposed directly upon $H$. It is only
necessary to imagine that the latter operator is assumed introduced and
defined as manifestly non-Hermitian in a manifestly unphysical Hilbert space
(say, ${\cal H}_{auxiliary}$). In the light of its definition in terms of
self-adjoint $\mathfrak{h}$ [cf. Eq. (2)] it is easy to see that the
conventional Hermiticity postulate (1) imposed upon $\mathfrak{h}$ is formally
equivalent to relation

$H^{\dagger}\,\Theta=\Theta\,H\,,\ \ \ \ \ \Theta=\Omega^{\dagger}\,\Omega\,,$
(6)

i.e., to the quasi-Hermiticity requirement imposed upon $H$ in ${\cal
H}_{auxiliary}$. Now, what remains to do is to recognize that the quasi-
Hermiticity (6) of $H$ in ${\cal H}_{auxiliary}$ is equivalent to the
Hermiticity of the same operator in another, non-equivalent Hilbert space
(say, ${\cal H}_{physical}$) which only differs from ${\cal H}_{auxiliary}$ by
an appropriate ad hoc  redefinition of the inner product,

$(\psi_{a},\psi_{b})_{auxiliary}=\langle\psi_{a}|\psi_{b}\rangle\ \to\
(\psi_{a},\psi_{b})_{physical}=\langle\psi_{a}|\,\Theta\,|\psi_{b}\rangle\,.$
(7)

In other words, all of the mathematical calculations may stay represented in
${\cal H}_{auxiliary}$. For the purposes of physics (i.e., typically, for the
evaluation of predictions) the auxiliary bra-vectors $\langle\psi|\in{\cal
H}_{auxiliary}^{\prime}$ (where the prime ′ marks the dual vector space [16])
have only to be replaced by the amended, different bra-vector elements

$\langle\psi_{physical}|=\langle\psi|\Theta\equiv\langle\psi_{\Theta}|\
\in\,{\cal H}_{physical}^{\prime}\,$ (8)

spanning the correct and physical dual vector space ${\cal
H}_{physical}^{\prime}$. The triplet of the relevant Hilbert spaces may be
then arranged in a diagram

$\begin{array}[]{c}\begin{array}[]{|c|}\hline\cr{\cal H}_{auxiliary}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm QHQM\ metric\
}\Theta}}{{\longrightarrow}}\ \begin{array}[]{|c|}\hline\cr{\cal
H}_{physical}\\\ \hline\cr\end{array}\\\ \ \ \ \ \ \ \ \ \
\stackrel{{\scriptstyle{\rm Dyson^{\prime}s\ map\ }\Omega}}{{}}\ \searrow\ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \nearrow\\!\\!\swarrow\ \stackrel{{\scriptstyle{\rm
equivalent\ physics\,.}\ }}{{}}\\\ \ \ \ \begin{array}[]{|c|}\hline\cr\ \ \ \
{\cal H}_{textbook}\\\ \hline\cr\end{array}\\\ \end{array}$ (9)

This diagram shows that the operator-transformation Hermitization (OT
Hermitization) of Eq. (4) which changes the Hamiltonian finds its alternative
in the Hamiltonian-preserving and metric-amending Hermitization (MA
Hermitization) of Eq. (7). In both of these scenarios, all of the ket- and
bra-vector elements of the relevant Hilbert spaces can be represented in one,
“mathematical” Hilbert space ${\cal H}_{auxiliary}$ in a way summarized by the
three-Hilbert-space diagrams (9) and

$\begin{array}[]{c}\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\\\ \hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm MA\
path}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,\Omega^{\dagger}\Omega\\\ \hline\cr\end{array}\\\ \ \ \
\ \ \ \stackrel{{\scriptstyle{\rm OT\ path}}}{{}}\ \searrow\ \ \ \ \ \ \ \ \ \
\nearrow\\!\swarrow\ \stackrel{{\scriptstyle{\rm equivalence\,}\ }}{{}}\\\
\begin{array}[]{|c|}\hline\cr\Omega\,|\psi_{m}\rangle,\
\langle\psi_{n}|\,\Omega^{\dagger}\\\ \hline\cr\end{array}\\\ \end{array}$
(10)

(cf. also review [15]).

##  4 A hybrid, partial-Dyson-map plus partial-metric-operator Hermitization

In our present paper we intend to complement the latter two forms of
alternative Hermitizations by a new, third, hybrid one. A basic motivation of
our proposal resulted from the following two observations.

  * •

In the case of the Dyson-inspired Hamiltonian-transformation Hermitization
$H\to\mathfrak{h}=\mathfrak{h}^{\dagger}$ there exist several important weak
points of the strategy. The main one is that whenever one is able to construct
$\mathfrak{h}$ in closed form, the major part of the motivation of working
with its non-Hermitian partner is lost. In such a case it makes good sense to
forget about $H$ and to relocate all of the necessary calculations directly to
${\cal H}_{(textbook)}$. In practice, only a few exceptions do exist: Buslaev
and Grecchi [9], for example, found a rather exceptional anharmonic-oscillator
quantum system in which the OT strategy and the active use of both of the
formally equivalent representation operators $H_{({BG})}$ and
$\mathfrak{h}_{({BG})}$ have been found feasible and proved equally easy. In
such a case, naturally, the ultimate choice between the two operators may be
dictated by some complementary phenomenological criteria [17, 18].

  * •

In the case of the strategy based on the reconstruction of the necessary
amended physical Hilbert-space metric $\Theta$ one can encounter multiple non-
trivial and not always expected technical obstacles. Their source is that
whenever one decides to treat Eq. (6) as an implicit definition of
$\Theta=\Theta(H)$, one reveals that such a definition is not only ambiguous
[19] but also, mostly, yielding just an approximate form of the metric: A
fairly long list of these technical MA-related obstacles may be found
presented, e.g., in review paper [12].

The main idea of our present methodical innovation can be traced back to the
Dyson’s physics-rooted papers [1] in which the essence of the success (i.e.,
the sufficiently persuasive reason for the preference of the use of the non-
Hermitian Hamiltonian model $H_{(Dyson)}$) lied in the access to some
additional information about the properties (i.e., typically, correlations [7]
or clusterings [20]) of the system in question. Naturally, such a
supplementary knowledge facilitated the specification of the “good” forms of
the mapping $\Omega$.

Without the latter advantage the probability of success (i.e., of the
simplifications achieved as a compensation of the loss of the Hermiticity)
would be lower. Still, we believe that the “user-friendliness” of the mapping
$\Omega$ and/or of the metric $\Theta(H)$ (which is, for the success, crucial)
could be also enhanced, in the generic situation, by a tentative factorization
of the operator $\Omega$ itself. In particular, we propose the use of the two-
term ansatz

$\Omega=\Omega_{M}\,\Omega_{H}\,.$ (11)

Under this assumption, the generic Dyson map can be reinterpreted as a
sequence of the two sub-maps. Subsequently, also the operator-transformation
of the Hamiltonian can be made sequential. In such a setting the emerging
“interpolative” step

$H\ \to\ H_{H}=\Omega_{H}\,H\,\Omega^{-1}_{H}\neq H_{H}^{\dagger}\,$ (12)

might precede the ultimate OT Hermitization

$H_{H}\ \to\
\mathfrak{h}=\Omega_{M}\,H_{H}\,\Omega^{-1}_{M}=\Omega\,H\,\Omega^{-1}=\mathfrak{h}^{\dagger}\,.$
(13)

This is, in nuce, the main message of our present paper.

One can almost immediately deduce that the “last step” (13) is in fact not
unavoidable. Alternatively one may finalize the Hermitization in the MA and
Hilbert-space-amendment spirit. Technically, such a methodical alternative
would simply mean that one keeps the new non-Hermitian Hamiltonian $H_{H}$
unchanged. Thus, one recalls only the Hermiticity of $\mathfrak{h}$ and re-
expresses this property as the quasi-Hermiticity property of $H_{H}$. In this
manner, the Hermiticity rule $\mathfrak{h}=\mathfrak{h}^{\dagger}$ becomes
replaced by the formally equivalent relation

$H_{H}^{\dagger}\,\Theta_{M}=\Theta_{M}\,H_{H}\,,\ \ \ \ \
\Theta_{M}=\Omega_{M}^{\dagger}\,\Omega_{M}\,$ (14)

in which one just has to confirm that the definition

$\Theta_{M}=\Omega_{M}^{\dagger}\Omega_{M}$ (15)

of the new metric operator is correct and consistent with all of the formal
QHQM postulates.

Fortunately, the latter confirmation requires just an elementary algebra so
that it may be left to the readers. What one obtains is a hybrid-form
Hermitization (HF Hermitization) of $H$. What is, at the same time, most
important is the question about the potential usefulness of the new HF-type
Hermitization (12) + (15) in applications. In the next section, we will try to
persuade the readers about the existence of the specific merits of the HF
strategy via an implementation of the recipe to a schematic illustrative two-
level quantum system.

##  5 Illustration

For an explicit illustration of our abstract HF representation of the hiddenly
unitary quantum evolution let us pick up just a maximally schematic two-by-two
real-matrix model. For the sake of simplicity let us also assume that the
Hermitian textbook Hamiltonian (with the real and non-degenerate spectrum,
say, $E_{0}=1$ and $E_{1}=2$) is, for our methodical purposes, diagonalized,

$\mathfrak{h}=\left[\begin{array}[]{cc}1&0\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr 0&2\end{array}\right]\,.$ (16)

Also, let us choose, in (11), the two Dyson-map factors (and their product) as
follows,

$\Omega_{M}=\left[\begin{array}[]{cc}1&0\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr s&1\end{array}\right]\,,\ \ \ \ \
\Omega_{H}=\left[\begin{array}[]{cc}1&t\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr 0&1\end{array}\right]\,,\ \ \ \ \
\Omega=\left[\begin{array}[]{cc}1&t\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
s&st+1\end{array}\right]\,.$ (17)

In the light of Eq. (2) we may replace our $\mathfrak{h}$ of Eq. (16) by its
manifestly non-Hermitian isospectral alternative

$H=\left[\begin{array}[]{cc}-st+1&-\left(st+1\right)t\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr s&st+2\end{array}\right]\,.$ (18)

Obviously, in a sharp contrast to trivial Eq. (16), the new Hamiltonian-
sampling matrix $H$ looks “complicated”. For our present methodical purposes
it is important that the latter observation can be also reinterpreted as a
statement that the contrast of complexities will survive in any other basis,
with an optimal representation relocated, including the opposite extreme case
in which the matrix form of $H$ would look decisively simpler than the matrix
form of $\mathfrak{h}$.

For the sake of simplicity let us just keep in mind the latter methodical
argument while using the illustrative matrices, due to their maximal
pedagogical appeal, in their fixed and given form. This means that we may
still speak about a generic nature of the sharp contrast between the
triviality of the secular equation for one of the models (here, for
$\mathfrak{h}$) and an apparently more complicated form of its Dyson-mapping-
mediated analogue.

In our particular example, therefore, one is able to estimate the complexity
of the matrix model only after some explicit algebraic manipulations. These
manipulations also enable us to reveal and demonstrate that the apparently
complicated and $s-$ and $t-$dependent secular polynomial $\det(H-E)$
associated with $H$ is in fact also elementary and $s-$ and $t-$independent
and easily factorizable, $\det(H-E)={E}^{2}-3\,E+2=(E-1)\,(E-2)$.

Another apparent contrast in complexity between $\mathfrak{h}$ and $H$ emerges
during the verification of the validity of the MA-related quasi-Hermiticity
relation (6). Indeed, after insertion we obtain the most complicated matrix
result

$H^{\dagger}\Theta=\Theta\,H=\left[\begin{array}[]{cc}1+2\,{s}^{2}&t+2\,t{s}^{2}+2\,s\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
t+2\,t{s}^{2}+2\,s&{t}^{2}+2\,{t}^{2}{s}^{2}+4\,ts+2\end{array}\right]\,.$
(19)

Indeed, it takes time to see that this matrix is Hermitian and positive
definite at any positive values of the real parameters $s$ a $t$:
Incidentally, in our recent paper [21] the operator products
$H^{\dagger}\Theta=\Theta\,H$ were denoted by a dedicated symbol $Y$ and
studied per se. Incidentally, such a study was well motivated since these
products prove highly relevant in the context of the random matrix theories
[22]. Naturally, in such an applied physics context their complexity or
simplicity would be an important characteristics, indeed.

In any physical and QHQM context, after all, one finally has to turn attention
to the explicit form of the MA-mediating Hermitizing metric. In our model we
have

$\Theta=\left[\begin{array}[]{cc}1+{s}^{2}&\left(1+{s}^{2}\right)t+s\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr
t+\left(ts+1\right)s&\left(t+\left(ts+1\right)s\right)t+ts+1\end{array}\right]$
(20)

Indeed, one might think that the necessary (and,, quite often, difficult [23])
proofs of its positive definiteness can still be done, quite routinely, via
the computer-assisted symbolic manipulations. In this sense we encountered a
technical surprise because in spite of a virtual triviality of the underlying
two-by-two-matrix algebra the explicit evaluation of the spectrum
$\\{\theta_{\pm}\\}$ of the metric as provided by the routines of MAPLE [24]
appeared complicated,

$\theta_{\pm}=1/2\,{t}^{2}+1/2\,{t}^{2}{s}^{2}+ts+1+1/2\,{s}^{2}\pm
1/2\,\sqrt{D}$ (21)

containing, in addition, an unexpectedly long polynomial

$D={2\,{t}^{4}{s}^{2}+4\,{t}^{3}s+4\,{t}^{3}{s}^{3}+{t}^{4}{s}^{4}+{s}^{4}+{t}^{4}+4\,{s}^{2}+8\,ts+4\,{t}^{2}+10\,{t}^{2}{s}^{2}+2\,{t}^{2}{s}^{4}+4\,t{s}^{3}}\,.$
(22)

Still, one arrives at a nice result.

###### Lemma 1

The matrix of metric (20) is positive definite.

Proof. The proof of the positivity of the metric remained, fortunately, easy
and straightforward because after the check of the calculation which was still
feasible to perform by hand the helpful key simplification

$D={\left(4+{t}^{2}{s}^{2}+({s}+{t})^{2}\right)\left({t}^{2}{s}^{2}+({s}+{t})^{2}\right)}$
(23)

of the long expression (22) by factorization was immediately visible.
$\square$

We believe that the latter set of formulae persuaded the readers that the
technical obstacles encountered during the pure-form OT or MA Hermitizations
need not be inessential. We believe that the hybrid approach offering, via Eq.
(12), the next-to elementary Hamiltonian

$H_{H}=\left[\begin{array}[]{cc}1&0\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
s&2\end{array}\right]\,$ (24)

in combination with the next-to elementary inner-product metric of Eqs. (14)
and (15), viz.,

$\Theta_{M}=\left[\begin{array}[]{cc}1+{s}^{2}&s\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr s&1\end{array}\right]$ (25)

delivers a fairly persuasive argument in favor of the optimality of our newly
proposed third, HF Hermitization strategy. Pars pro toto  it is worth noting
that in contrast to the MA-related Lemma 1 the analogous HF-validating proof
of the positive definiteness of metric (25) at all $s$ proceeds via the
positivity of its much more elementary eigenvalues $\ \sim
2+s^{2}\pm\sqrt{(2+s^{2})^{2}-4}$.

In our final argument supporting the innovative HF Hermitization strategy we
may say that the single-parameter dependence of the HF-representing pair of
operators $H_{H}$ and $\Theta_{M}$ symbolizes and underlines a half-way formal
character of the HF approach. We see that also by the criteria of complexity
of formulae it lies in between the two (viz., OT and MA) extremes, one of
which is always, in our schematic illustration, two-parametric.

##  6 Conclusions

We may summarize our present proposal of an innovative HF reformulation of
quantum mechanics as a composition of the partial OT mapping $H\to H_{H}$ of
Eq. (12) with the partial MA quasi-Hermiticity constraint (14) which is
formulated in terms of the reduced HF metric $\Theta_{M}$. In conclusion let
us now add a few complementary comments on such a form of the model-building
strategy.

First of all, let us underline the importance of the stationarity assumption
(5). It represents a key to the practical applicability and feasibility of the
present HF Hermitization. In the language of mathematics the stationarity is
really the constraint which enabled us to make the full use of our fundamental
Dyson-map-factorization ansatz (11). Now, we just have to add that in the
nearest future we plan to generalize and develop the HF theory also towards
its non-stationary extensions. In fact, the first steps in this direction were
already made, in 2008, in our three-Hilbert-space description of the unitary
quantum evolution as proposed in paper [25] and as reviewed, later, in [15].
On these grounds we were able to formulate a non-Hermitian (or, better,
hiddenly Hermitian) generalization of the standard unitary quantum mechanics
in interaction picture [26] (see also [27]) and, in the special cases, not
only in Heisenberg picture [28] (see also the parallel developments in [29])
but also in the generalized Heisenberg picture (see [30] and also [27, 31]).

In all of the latter papers the Dyson maps were assumed manifestly non-
stationary, $\Omega=\Omega(t)$. Interested readers should certainly consult
the most recent forms of the non-stationary QHQM theory, say, in the
comprehensive reviews [32, 33]. Naturally, due to the non-stationarity of
these innovations there exists no direct comparison with our present,
stationary version of the HF approach. Nevertheless, it is worth adding that
the authors of review [33] proposed to call the non-stationary Dyson maps
“generalized vielbeins”. This is a truly remarkable detail because it combines
the apparently purely terminological decision with a truly deep insight in the
possible applicability of the non-stationary Dyson maps in quantum cosmology.
Unfortunately, such an observation and remark already go fairly beyond the
scope of our present letter. Still, interested readers are recommended to have
a look at the corresponding newest considerations in [34] or in the very
recent preprint [35].

In our second, independent and last concluding comment on our present HF
method let us point out that it is, in some sense, surprising that in the
newest literature on the non-Hermitian quantum Hamiltonians (cf., e.g., [13,
36]) the “mainstream” attention is slowly moving from the closed-system theory
to the studies of the open and nonlinear quantum systems. In this case,
naturally, the unitarity ceases to be an issue. One of the possible
explanations of the shift could be sought in the fact that in spite of the
initially enthusiastic acceptance of the Hermitization ideas, their subsequent
more detailed analysis and implementations encountered multiple (and also not
quite expected) conceptual as well as technical complications [3, 37, 38]. Not
enough attention has been paid to the quantum systems (sampled, e.g., by the
Buslaev’s and Grecchi’s paper [9]) in which a way out of the difficulties
could be sought in a decomposition of the Dyson map into a sequence of its
simpler factors. In this sense, our present paper on Hermitizations based on
the composite Dyson map ansatz (11) can be read as an encouragement of a
renewal of interest in unitarity and of a start of a new active research in
the apparently old-fashioned closed-system direction.

## References

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