.

Systematics of quasi-Hermitian representations of non-Hermitian quantum models

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 the currently quickly growing area of applications of non-Hermitian
Hamiltonians $H\neq H^{\dagger}$ which are quasi-Hermitian (i.e., such that
$H^{\dagger}\,\Theta=\Theta\,H$, with a suitable inner-product metric
$\Theta\neq I$), the correct probabilistic interpretation of the model is
needed. Achieved either in the Buslaev-Grecchi-inspired spirit (BGI; one
factorizes $\Theta=\Omega^{\dagger}\,\Omega$ and reconstructs the conventional
Hermitian Hamiltonian,
$H\to\mathfrak{h}=\Omega\,H\,\Omega^{-1}=\mathfrak{h}^{\dagger}$) or in the
much user-friendlier, Dyson-inspired spirit (DI; one eliminates the
“difficult” reference to $\mathfrak{h}$ via the quasi-Hermiticity rule, i.e.,
via an “easier” reconstruction of $\Theta$). Here, the two model-building BGI
and DI recipes are identified as the two extreme special cases of a general
correct-interpretation-providing strategy. We show that at any preselected
integer $N$ one may make a choice between the BGI extreme and an $N-$plet of
the other, consistent and physical DI-type representations of the system in
which a specific, partially modified Hamiltonian is constructed as quasi-
Hermitian with respect to a specific, simplified inner-product metric. In
applications, any one of these $N+1$ options may prove optimal for a given
$H$: A schematic three-state quantum system is discussed as an illustrative
example.

## Keywords

.

non-Hermitian quantum mechanics of unitary systems;

isospectral preconditionings of Hamiltonians;

physical Hilbert spaces with ad hoc inner-product metrics;

alternative reconstructions of hidden Hermiticity;

##  1 Introduction

In the conventional formulation of quantum mechanics in Schrödinger picture
[1] one has to solve Schrödinger equation

${\rm
i}\,\frac{d}{dt}\,|\psi\\!\\!\succ\,\,=\mathfrak{h}\,|\psi\\!\\!\succ\,\,\,,\
\ \ \ \ \mathfrak{h}=\mathfrak{h}^{\dagger}\,,\ \ \ \ \
|\psi\\!\\!\succ\,\,\in{\cal H}_{textbook}\,.$ (1)

This process is often facilitated by a unitary isospectral preconditioning of
the Hamiltonian,

$\mathfrak{h}\,\to\,\mathfrak{h}^{\prime}={\cal
U}^{\dagger}\,\mathfrak{h}\,{\cal U}\,,\ \ \ \ \
\mathfrak{h}^{\prime}=\left(\mathfrak{h}^{\prime}\right)^{\dagger}\,.$

In 1956 F. Dyson revealed, during his study of ferromagnetism [2], that the
efficiency of the preconditioning can decisively be enhanced when one omits
the formally redundant condition of the unitarity of the transformation. Then,
one generalizes

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

and speaks about the Dyson-inspired (DI) “non-Hermitian Hamiltonian”
reformulation of the conventional quantum mechanics (for its compact outline
see Appendix A below).

In the latter framework the original Schrödinger Eq. (1) for
$|\psi\\!\\!\succ\,\,=\Omega\,|\psi\rangle$ is merely replaced by its
mathematically equivalent non-Hermitian alternative

${\rm i}\,\frac{d}{dt}\,|\psi\rangle=H\,|\psi\rangle\,,\ \ \ \ \
|\psi\rangle\in{\cal H}_{math}\,$ (3)

which is, presumably, easier to solve. Naturally, the latter assumption is
essential and nontrivial because the amended representation $H$ of the
Hamiltonian becomes less standard, viz., quasi-Hermitian [3, 4],

$H^{\dagger}\,\Theta=\Theta\,H\,,\ \ \ \ \ \Theta=\Omega^{\dagger}\Omega\,.$
(4)

In practice, nevertheless, the generalized DI preconditioning (2) alias
“Dyson map” alias  “quasi-Hermitian quantum mechanics” (QHQM) found several
successful applications, say, in nuclear physics (see, e.g., [5, 6] and also
[7]).

On these grounds, Scholtz et al [4] proposed, in 1992, a closely related
modification of the conventional model-building process. The essence of their
innovative proposal lied in a reversal of the preconditioning,

$H\to\mathfrak{h}=\Omega\,H\,\Omega^{-1}=\mathfrak{h}^{\dagger}\,.$ (5)

In place of the traditional approach in which one recalls the principle of
correspondence and in which one preselects a self-adjoint candidate
$\mathfrak{h}$ for the Hamiltonian, an alternative, textbooks-complementing
upgrade of QHQM in Schrödinger picture has been described and promoted. In
this approach the non-Hermitian but still user-friendly “input” Hamiltonian
$H$ is expected to be given in advance.

Table 1: The DI-Hermitization pattern ${\cal H}_{math}\to{\cal H}_{phys}$. input: | aim: | construction: | tool: | result:
---|---|---|---|---
Hamiltonian | unitarity | find the metric | amend the products | Hermiticity in ${\cal H}_{phys}$
$H\neq H^{\dagger}$ in ${\cal H}_{math}$ | (in ${\cal H}_{phys}$) | $\Theta=\Theta(H)$ | $\langle\psi|\psi^{\prime}\rangle\to\langle\psi|\Theta|\psi^{\prime}\rangle$ | $H=\Theta^{-1}\,H^{\dagger}\,\Theta\equiv H^{\sharp}$

Naturally, the preselected operator $H$ must be assigned the obligatory
probabilistic physical interpretation [8]. The purpose can be served either by
a direct backward reconstruction of $\mathfrak{h}$ (via Eq. (5)) or by the
direct reference to the quasi-Hermiticity of $H$. In the latter, DI setting
(see its compact outline in Table 1) the construction is usually more
straightforward because one merely has to solve Eq. (4) in order to obtain the
so called metric operator $\Theta=\Theta(H)$ (the technical details may be
found outlined, e.g., in reviews [4, 9]). Another reason of the preference of
the DI strategy in applications lies in its conceptual simplicity because the
initial non-Hermitian Hamiltonian $H$ may be kept unchanged. In many
applications it is not even necessary to know or reconstruct the Dyson map.

For a sample of applicability of the former, Hamiltonian-amending approach one
may recall the realistic anharmonic-oscillator example as described, in 1993,
by Buslaev and Grecchi [10]. This result (see its account in Appendix B below)
constituted, in fact, also a key source of inspiration of our present study.
Nevertheless, one has to admit that in comparison with the DI quasi-
Hermitization, the straightforward Buslaev-Grecchi-inspired (BGI)
Hermitization (5) (where one must obligatorily re-factorize the metric
$\Theta=\Omega^{\dagger}\Omega$, see Table 2) has to be perceived as
technically less friendly.

Table 2: BGI Hermitization ${\cal H}_{math}\to{\cal H}_{textbook}$. input: | aim: | prepartory step: | main step: | result:
---|---|---|---|---
Hamiltonian | unitarity | find | decompose | Hermiticity in ${\cal H}_{textbook}$
$H\neq H^{\dagger}$ in ${\cal H}_{math}$ | (in ${\cal H}_{textbook}$) | $\Theta=\Theta(H)$ | $\Theta=\Omega^{\dagger}\Omega$ | $\mathfrak{h}=\Omega\,H\,\Omega^{-1}=\mathfrak{h}^{\dagger}$

For compensation, it makes sense to point out that the Buslaev’s and Grecchi’s
constructive Hermitization (5) was rendered possible by a hidden symmetry of
their non-Hermitian model $H^{(BG)}$ with respect to a family of partial Dyson
sub-maps $\Omega_{j}$. This opens the way towards an innovative $N-$term
factorization of the global Dyson-map operator itself,

$\Omega=\Omega_{N}\Omega_{N-1}\ldots\Omega_{1}\,.$ (6)

One of the main goals of our present analysis will be, therefore, a model-
independent, abstract clarification of the role and of the consequences of
such a factorization.

Having the factorization (6) at their disposal, Buslaev and Grecchi were able
to realize the Hermitization (5) via its $N-$step decomposition,

$H\,\to\,H_{01}\,\to\,H_{02}\,\to\,\ldots\,\to\,H_{0N-1}\,\to\,\mathfrak{h}\,.$
(7)

In their specific model these authors obtained formula (7) in which
$H=H^{(BG)}\equiv H_{00}$ represented a maximally computation-friendly and, in
some sense, maximally non-Hermitian Hamiltonian operator. The strictly
Hermitian final item $\mathfrak{h}=\mathfrak{h}^{(BG)}\equiv H_{0N}$ proved
reconstructed exactly, via auxiliary isospectral-operator elements
$H_{0j}=H_{0j}^{(BG)}$ of the sequence with $0<j<N$. In their model they used
a not too small value of $N=7$ [10].

In a model-independent setting the existence of sequence (7) is in fact a
direct consequence of the factorization (6) of the Dyson map. In what follows
we will draw just multiple further consequences out of these ansatzs. The
underlying factorization (6) will be given here a deeper algebraic
interpretation. The recurrent $N-$step BGI process (7) of a constructive
guarantee of the unitarity of the evolution will be characterized as a mere
special case of a much broader new class of eligible re-Hermitizations of
Schrödinger equation (3).

##  2 Two forms of re-Hermitization at $N=1$

The current enormous growth of popularity of the QHQM studies only took place
after the development of certain robust DI Hermitization of $H$ (see a concise
summary of the history in [8]). A key characteristics of the process lies in
the amendment of the inner product in Hilbert space (see, e.g., comprehensive
reviews [9, 11]). One just keeps the Hamiltonian $H$ unchanged and guarantees
the unitarity of evolution in a new, physical Hilbert space ${\cal H}_{phys}$
obtained by an amendment of the inner product in the initial Hilbert space
${\cal H}_{math}$.

Given a candidate for the Hamiltonian $H\neq H^{\dagger}$ one cannot get rid
of the feeling that the BGI Hermitization (5) looks more natural. In what
follows we intend to strengthen the feeling. A constructive analysis will be
provided in which the standard BGI- and DI-path Hermitizations will be shown
competitive. Moreover, we will complement these options by a multitude of
other Hermitizations. We will show that in the QHQM theory with non-Hermitian
Hamiltonians $H$ the optimal representation of the Hermitizable Hamiltonians
may be selected out of a much richer menu.

Formally, the equivalence between the two alternative processes of the
Hermitization can be characterized by the diagram of review [12],

$\begin{array}[]{c}\begin{array}[]{|c|}\hline\cr{\cal H}_{math}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm DI\
path}}}{{\longrightarrow}}\ \begin{array}[]{|c|}\hline\cr{\cal H}_{phys}\\\
\hline\cr\end{array}\\\ \ \ \ \ \ \ \ \ \ \ \ \stackrel{{\scriptstyle{\rm BGI\
path}}}{{}}\ \searrow\ \ \ \ \ \nearrow\\!\swarrow\
\stackrel{{\scriptstyle{\rm equivalence\,.}\ }}{{}}\\\ \ \
\begin{array}[]{|c|}\hline\cr\ \ {\cal H}_{textbook}\\\
\hline\cr\end{array}\\\ \end{array}$ (8)

In spite of the fact that the Hermiticity of $H$ only enters the scene ”in
disguise”, the calculations may be perceivably facilitated when performed in
${\cal H}_{math}$ and interpreted along the DI or BGI path. In comparison, a
specific merit of the technically more difficult BGI Hermitization pattern
lies in the trivial inner product in the target space ${\cal H}_{textbook}$,
rendering the ultimate physical probabilistic interpretation of the system in
question maximally transparent.

The first step towards our goal (i.e., to establishing a deeper connection
between the BGI and DI Hermitizations) has been performed in our recent study
[13]. We complemented there the standard DI methods of the one-step
construction of the physical-Hilbert-space metric $\Theta$ by an innovation
based on an auxiliary $N-$term factorization of the operator,

$\Theta=\Theta_{N}=Z_{N}Z_{N-1}\ldots Z_{2}Z_{1}\,.$ (9)

Such an $N>1$ DI approach represented the quantum system in question in an
innovated non-Hermitian Schrödinger picture. From our present perspective the
latter results can be perceived as a preparatory step. Its scope remained
interpreted as an amendment of the DI approach in a way which is more
thoroughly discussed, at $N=2$ and at $N=3$, in Appendix C.

In the present continuation of the project we will point out that one of the
unnoticed but still most important consequences of the metric-factorization
postulate (9) has to be seen in the parallel induction of the factorization of
the related Dyson map (6). This is our present main idea. Its use will enable
us to develop a systematic extension of both of the DI and BGI Hermitization
patterns beyond their existing realizations.

The core of our message will lie in the shift of attention from the factorized
metric to the factorized Dyson map. At $N=1$ this is closely connected with a
change of interpretation of the three vertices in diagram (8) and of the
Dyson-map-mediated relations

$|\psi_{n}\\!\\!\succ\,\,=\Omega\,|\psi_{n}\rangle\,$

which connect the ket vectors $|\psi_{n}\\!\\!\succ\,\,\in{\cal H}_{textbook}$
and $|\psi_{n}\rangle\in{\cal H}_{math}$. This means that diagram (8) may be
re-displayed in its more explicit version

$\begin{array}[]{c}\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\\\ {\rm in\ }{\cal H}_{math}\\\ \hline\cr\end{array}\
\stackrel{{\scriptstyle{\rm metric\
}\Omega^{\dagger}\Omega}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,\Omega^{\dagger}\Omega\\\ {\rm in\ }{\cal H}_{phys}\\\
\hline\cr\end{array}\\\ \ \ \ \ \ \ \stackrel{{\scriptstyle{\rm Dyson\
map}}}{{}}\ \searrow\ \ \ \ \ \ \ \ \ \ \ \ \ \ \nearrow\\!\swarrow\
\stackrel{{\scriptstyle{\rm equivalence\,.}\ }}{{}}\\\
\begin{array}[]{|c|}\hline\cr\Omega\,|\psi_{m}\rangle,\
\langle\psi_{n}|\,\Omega^{\dagger}\\\ {\rm in\ }{\cal H}_{textbook}\\\
\hline\cr\end{array}\\\ \end{array}$ (10)

Both the ket and bra vectors are written down here in their preferred
representation in ${\cal H}_{math}$. Occasionally, the “correct and physical”
Hilbert space ${\cal H}_{phys}$ will be re-denoted as ${\cal R}_{0}^{(0)}$.
Similarly, we will work with ${\cal H}_{math}$ alias  ${\cal R}_{N}^{(0)}$ and
with ${\cal H}_{textbook}$ alias  ${\cal R}_{0}^{(N)}$. Such an amendment of
the notation conventions will facilitate transition to the QHQM formalism
using arbitrary $N$ in (6).

##  3 The $N=2$ BGI Hermitization and its two shortcuts

In the studies of various realistic non-Hermitian models the references to the
$N>1$ BGI-path option (7) remain rare (see, e.g., [14, 15]). One of the
reasons is that in contrast to the DI path where one needs just the inner-
product metric $\Theta$, the BGI-path option requires also the knowledge of
its “square root” Dyson map $\Omega$. Another problem is that in some cases
the methodical gains provided by the Hermitization $H\to\mathfrak{h}$ may
happen to be, due the collateral technical complications, weakened or even
lost completely. In such a case one simply has to return to the traditional
calculations in ${\cal H}_{textbook}$. Definitely, a more detailed exploration
of the BGI theory is needed.

###  3.1 The Hamiltonian-preserving DI shortcut

In our preceding paper [13] it has been shown that the decomposition (9) of
the metric into a product of $N$ factors $Z_{j}$ can be associated with a
sequence of certain specific, auxiliary Hilbert spaces. This has been shown to
yield a specific pattern of a sequential DI-type quasi-Hermitization of $H$.
In particular, after the first nontrivial choice of $N=2$ one has to consider
a triplet of spaces

${\cal H}_{math}\to{\cal H}_{int}\to{\cal H}_{phys}\,.$ (11)

This means that the three-Hilbert-space BGI-DI diagram (8) had to be replaced
by the following four-Hilbert-space upgrade

$\begin{array}[]{c}\ \ \ \ \begin{array}[]{|c|}\hline\cr{\cal H}_{math}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm DI\ path,\ step\
a}}}{{\longrightarrow}}\ \begin{array}[]{|c|}\hline\cr{\cal H}_{int}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm DI\ path,\ step\
b}}}{{\longrightarrow}}\ \begin{array}[]{|c|}\hline\cr{\cal H}_{phys}\\\
\hline\cr\end{array}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\stackrel{{\scriptstyle{\rm BGI\ path}}}{{}}\searrow\ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\nearrow\\!\swarrow\stackrel{{\scriptstyle{\rm equivalence\,.}}}{{}}\\\ \ \ \
\ \ \begin{array}[]{|c|}\hline\cr\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\cal
H}_{textbook}\\\ \hline\cr\end{array}\\\ \end{array}$ (12)

In the upper line the diagram contains the two-step DI-path Hermitization
during which one does not change the Hamiltonian $H$ (nor the other
observables). One just has to keep the trivial metric assigned to ${\cal
H}_{math}$ and the full factorized metric (9) assigned to ${\cal H}_{phys}$.

From the point of view of unitary quantum physics it appeared useful to endow
the new, intermediate Hilbert space ${\cal H}_{int}$ with a new and rather
unusual inner-product metric $Z_{2}$. Although the latter metric has not been
assigned any immediate physical meaning, its introduction was well motivated
by mathematics. For the purpose (see also letter [16] for a more detailed
explanatory commentary) it was also recommended to assume the Hermiticity and
positivity of the candidate for the metric in ${\cal H}_{int}$. Thus, the
following, Dyson-like product decomposition of the auxiliary metric has been
postulated,

$Z_{2}=\Omega_{2}^{\dagger}\,\Omega_{2}\,.$ (13)

Nevertheless, in contrast to our forthcoming considerations, the existence of
such a re-factorization was just based on a purely formal analogy between the
choices of $N=2$ and $N=1$.

###  3.2 The Hamiltonian-amending alternative shortcut

In our present paper, relation (13) will be reinterpreted as a reason for a
split of the three-term sequence (11) into the two independent and elementary
step-1 and step-2 subsequences,

${\cal H}_{math}\to{\cal H}_{int}\,,\ \ \ \ {\cal H}_{int}\to{\cal
H}_{phys}\,.$ (14)

In the left, step-1 doublet of spaces the target space ${\cal H}_{int}$ can be
perceived as a strict analogue of the $N=1$ target space ${\cal H}_{phys}$ in
diagram (8). Similarly, the initial-space item ${\cal H}_{int}$ in the right
pair of spaces in (14) can equally well be interpreted as mimicking the
initial $N=1$ space ${\cal H}_{math}$ of diagram (10).

This is our basic idea. We will treat the two sub-steps (14) separately.
Indeed, in both of these cases there exists no full-fledged analogue of the
textbook space ${\cal H}_{textbook}$ of Eq. (10). This means, strictly
speaking, that neither one of the $N=2$ analogues of the textbook Hilbert
space ${\cal H}_{textbook}$ could immediately be assigned the status of a
physical Hilbert space. Still, in spite of this obstacle, it is possible to
preserve at least some of the parallels with the $N=1$ scenario.

In the context of step 1, ${\cal H}_{math}\to{\cal H}_{int}$, it is now easy
to introduce a manifestly unphysical formal analogue of the ${\cal
H}_{textbook}-$space component of diagram (10) and denote it, say, by symbol
${\cal K}_{math}$. The related step-1/DI-path sub-Hermitization process may be
then reinterpreted in terms of the following three-Hilbert-space subdiagram

$\begin{array}[]{c}\ \ \ \ \ \ \
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\ {\rm}\langle\psi_{n}|\\\
{\rm in\ }{\cal H}_{math}\\\ \hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm
sub\\!-\\!Hermitization}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,\Omega^{\dagger}_{2}\Omega_{2}\\\ {\rm in\ }{\cal
H}_{int}\\\ \hline\cr\end{array}\\\ \ \ \ \ \ \ \ \
\stackrel{{\scriptstyle{\rm sub\\!-\\!map}\ \Omega_{2}}}{{}}\ \searrow\ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \nearrow\\!\swarrow\ \stackrel{{\scriptstyle{\rm
sub\\!-\\!equivalence.}\ }}{{}}\\\ \ \
\begin{array}[]{|c|}\hline\cr\Omega_{2}\,|\psi_{m}\rangle,\
\langle\psi_{n}|\,\Omega^{\dagger}_{2}\\\ {\rm in\ }{\cal K}_{math}\\\
\hline\cr\end{array}\\\ \end{array}$ (15)

This is a perfect analogue of diagram (10). The innovation operates with the
Dyson-map sub-factor $\Omega_{2}$ as defined in Eq. (13). What is more
important is that we need also a parallel transformation of the Hamiltonian as
well as of all of the other operators of observables. The initial non-
Hermitian Hamiltonian operator $H$ acting in ${\cal H}_{math}$ and in ${\cal
H}_{int}$ as well as in ${\cal H}_{phys}$ has to be replaced, in ${\cal
K}_{math}$, by the new isospectral non-Hermitian Hamiltonian operator
$\Omega_{2}\,H\,\Omega_{2}^{-1}$ denoted, say, as $\mathfrak{h}_{2}$. This
enables us to finalize the ultimate upgrade of diagram (15),

$\begin{array}[]{c}\ \ \ \ \ \ \
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\ {\rm}\langle\psi_{n}|\\\
\ {\rm and\ }H\\\ {\rm in\ }{\cal H}_{math}\\\ \hline\cr\end{array}\
\stackrel{{\scriptstyle{\rm sub\\!-\\!Hermitization}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,\Omega^{\dagger}_{2}\Omega_{2}\\\ \ {\rm and\ }H\\\
{\rm in\ }{\cal H}_{int}\\\ \hline\cr\end{array}\\\ \ \ \ \ \ \ \ \
\stackrel{{\scriptstyle{\rm sub\\!-\\!map}\ \Omega_{2}}}{{}}\ \searrow\ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \nearrow\\!\swarrow\ \stackrel{{\scriptstyle{\rm
sub\\!-\\!equivalence.}\ }}{{}}\\\ \ \
\begin{array}[]{|c|}\hline\cr\Omega_{2}\,|\psi_{m}\rangle,\
\langle\psi_{n}|\,\Omega^{\dagger}_{2}\\\ \ {\rm and\
}\mathfrak{h}_{2}=\Omega_{2}\,H\,\Omega_{2}^{-1}\\\ {\rm in\ }{\cal
K}_{math}\\\ \hline\cr\end{array}\\\ \end{array}$ (16)

The isospectrality relation connects now the initial Hamiltonian $H$ with its
new partner $\mathfrak{h}_{2}$. The latter operator enters the respective new
hypothetical auxiliary Schrödinger equation of course.

A straightforward nature of the derivation of Eq. (16) seems to encourage the
use of the same factorization of our positive definite “generalized charge”
$Z_{1}$. Unfortunately, such an attempt fails to work for the reasons
clarified in Lemma Nr. 1 of paper [13]. The correct and consistent analogue of
relation (13) had to have a more sophisticated inner-product-dependent form

$Z_{1}=\Omega_{1}^{\ddagger(1)}\,\Omega_{1}\,,\ \ \ \
\Omega_{1}^{\ddagger(1)}=Z_{2}^{-1}\Omega_{1}^{\dagger}\,Z_{2}\,.$ (17)

After the incorporation of this inner-product-adapted factorization relation
the theory becomes consistent. One reveals an analogous three-Hilbert-space
substructure in the other, step-2/DI-path sub-Hermitization process in which
the ket- and bra-vectors as well as Hamiltonians may be given the explicit
form in their representation in the universal and unique working space ${\cal
H}_{math}$,

$\begin{array}[]{c}\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|Z_{2}\\\ \ {\rm and\ }H\\\ {\rm in\ }{\cal H}_{int}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm
sub\\!-\\!Hermitization}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,\Omega^{\dagger}_{1}Z_{2}\Omega_{1}\\\ \ {\rm and\
}H\\\ {\rm in\ }{\cal H}_{phys}\\\ \hline\cr\end{array}\\\ \ \ \ \
\stackrel{{\scriptstyle{\rm sub\\!-\\!map}\ \Omega_{1}}}{{}}\ \searrow\ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \nearrow\\!\swarrow\ \stackrel{{\scriptstyle{\rm
sub\\!-\\!equivalence.}\ }}{{}}\\\
\begin{array}[]{|c|}\hline\cr\Omega_{1}\,|\psi_{m}\rangle,\
\langle\psi_{n}|\,\Omega^{\dagger}_{1}Z_{2}\\\ \ {\rm and\
}\mathfrak{h}_{1}=\Omega_{1}\,H\,\Omega_{1}^{-1}\\\ {\rm in\ }{\cal
K}_{phys}\\\ \hline\cr\end{array}\\\ \end{array}$ (18)

This also offers an explicit definition of the respective operations of
Hermitian conjugation in ${\cal K}_{phys}$. The corresponding Hamiltonian
operator acquires its new and final “textbook-compatible” eligible, BGI-
shortcut form defined by formula
$\mathfrak{h}_{1}=\Omega_{1}\,H\,\Omega_{1}^{-1}$. This operator remains
manifestly non-Hermitian in our unique manipulation Hilbert space ${\cal
H}_{math}$ but, simultaneously, it can be perceived as safely self-adjoint in
the new physical Hilbert space ${\cal K}_{phys}$. In this sense one can
conclude that a new version of a quasi-Hermitian representation of the quantum
system in question is obtained via the partial, $\Omega_{2}-$mediated
amendment of the metric, $I\to Z_{2}$, accompanied by the partial,
$\Omega_{1}-$mediated amendment of the Hamiltonian itself,
$H\to\mathfrak{h}_{1}$.

A qualitatively new mathematical feature of diagram (18) is that in its upper-
left corner the operation of Hermitian conjugation is manifestly
$Z_{2}$-dependent. The sub-equivalence arrows then extend the physical status
from ${\cal H}_{phys}$ to ${\cal K}_{phys}$. In the lower half of Eq. (18)
containing the representation of the new kets and bras in ${\cal K}_{phys}$
one may just add the third eligible physical Hamiltonian operator
$\mathfrak{h}_{1}$. The description of physics in the new Hilbert space of
states ${\cal K}_{phys}$ becomes equivalent to the one described in its
partners ${\cal H}_{phys}$ and ${\cal H}_{textbook}$.

The resulting extension and completion of the considerations of Ref. [13] is
new. By the replacement ${\cal K}_{math}\to{\cal K}_{phys}$ we managed to
complement the standard doublet of the physical Hilbert spaces ${\cal
H}_{textbook}$ and ${\cal H}_{phys}$ by the third physical Hilbert space
${\cal K}_{phys}$. In this space we simultaneously  amended both  the inner
product metric (via the operator replacement $I\to Z_{2}$) and  the
Hamiltonian (via the transformation $H\to\mathfrak{h}_{1}$). Now, in only
remains for us to add that our analysis is also opening, as its methodical
byproduct, a new path towards a further, still more sophisticated realization
of the Hermitization programme.

###  3.3 The ultimate six-Hilbert-space classification diagram at $N=2$

At the moment, the realization of the programme is incomplete even at $N=2$.
What remains to be done is the search for the missing link between the three
Hilbert spaces ${\cal K}_{math}$, ${\cal K}_{phys}$ and ${\cal H}_{textbook}$.
The search may use the same technique as above. The initial, purely
mathematical Hilbert space ${\cal H}_{math}$ finds its analogue, in (16), in
the unphysical Hilbert space ${\cal K}_{math}$. Similarly, the old, $N=1$
physical Hilbert space ${\cal H}_{textbook}$ finds, in (18), its formal $N=2$
analogue in the new physical Hilbert space ${\cal K}_{phys}$. Thus, the
specification of the connection between ${\cal K}_{math}$ and ${\cal
K}_{phys}$ may still proceed by analogy. Due to the subtlety of the inclusion
of the upper-left-space Hermiticity, the completion of the analogy requires
more care. Still, its result is obvious and has the form of diagram

$\begin{array}[]{c}\begin{array}[]{|c|}\hline\cr{\rm}\Omega_{2}\,|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,\Omega_{2}^{\dagger}\\\ {\rm in\ }{\cal K}_{math}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm
amendment}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}\Omega_{1}\,|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,\Omega_{1}^{\dagger}\,\Omega^{\dagger}_{2}\Omega_{2}\\\
{\rm in\ }{\cal K}_{phys}\\\ \hline\cr\end{array}\\\
\stackrel{{\scriptstyle{\rm}\Omega_{21}}}{{}}\ \searrow\ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \nearrow\\!\swarrow\\\
\begin{array}[]{|c|}\hline\cr{\rm}\Omega_{2}\,\Omega_{1}\,|\psi_{m}\rangle,\
\langle\psi_{n}|\,\Omega_{1}^{\dagger}\,\Omega_{2}^{\dagger}\\\ {\rm in\
}{\cal H}_{textbook}\\\ \hline\cr\end{array}\\\ \end{array}$ (19)

where $\Omega_{21}=\Omega_{2}\,\Omega_{1}\,\Omega_{2}^{-1}$. We can now
concatenate all of the three sub-diagrams (15), (18) and (19) into a single
one. This leads to our final triangular six-space scheme at $N=2$,

$\begin{array}[]{c}\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\\\ \ {\rm and\ }H\\\ {\rm in\ }{\cal H}_{math}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm
amendment}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,Z_{2}\\\ \ {\rm and\ }H\\\ {\rm in\ }{\cal H}_{int}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm
amendment}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,Z_{2}\,Z_{1}\\\ \ {\rm and\ }H\\\ {\rm in\ }{\cal
H}_{phys}\\\ \hline\cr\end{array}\\\
\stackrel{{\scriptstyle{\rm}\Omega_{2}}}{{}}\ \searrow\ \ \ \ \ \ \ \ \ \ \ \
\nearrow\\!\swarrow\ \stackrel{{\scriptstyle{\rm}}}{{}}\ \ \ \ \ \ \ \ \
\stackrel{{\scriptstyle{\rm}\Omega_{1}}}{{}}\ \searrow\ \ \ \ \ \ \ \ \ \ \ \
\ \nearrow\\!\swarrow\\\
\begin{array}[]{|c|}\hline\cr{\rm}\Omega_{2}\,|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,\Omega_{2}^{\dagger}\\\ \ {\rm and\
}\mathfrak{h}_{2}=\Omega_{2}\,H\,\Omega_{2}^{-1}\\\ {\rm in\ }{\cal
K}_{math}\\\ \hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm
amendment}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}\Omega_{1}\,|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,\Omega_{1}^{\dagger}\,Z_{2}\\\ \ {\rm and\
}\mathfrak{h}_{1}=\Omega_{1}\,H\,\Omega_{1}^{-1}\\\ {\rm in\ }{\cal
K}_{phys}\\\ \hline\cr\end{array}\\\
\stackrel{{\scriptstyle{\rm}\Omega_{21}}}{{}}\ \searrow\ \ \ \ \ \ \ \ \ \ \
\nearrow\\!\swarrow\\\
\begin{array}[]{|c|}\hline\cr{\rm}\Omega_{2}\,\Omega_{1}\,|\psi_{m}\rangle,\
\langle\psi_{n}|\,\Omega_{1}^{\dagger}\,\Omega_{2}^{\dagger}\\\ \ {\rm and\
}\mathfrak{h}=\Omega\,H\,\Omega^{-1}\\\ {\rm in\ }{\cal H}_{textbook}\\\
\hline\cr\end{array}\\\ \end{array}$ (20)

The diagram lists all of the relevant Dyson mappings, kets, bras and the
Hamiltonians represented in the single Hilbert space ${\cal H}_{math}$ alias
${\cal R}_{N}={\cal R}_{N}^{(0)}$.

Optionally, one could also expand here
$Z_{2}=\Omega_{2}^{\dagger}\,\Omega_{2}$ and, in the light of Eq. (17),
$Z_{2}\,Z_{1}=\Omega^{\dagger}\,\Omega$ with $\Omega=\Omega_{2}\,\Omega_{1}$.
Such a correspondence between the two-term factorization of the metric and the
two-term factorization of the Dyson map was already noticed in paper [13].
Nevertheless, this observation was only interpreted there as a mathematical
curiosity because the subject of the paper, viz., the DI Hermitization of $H$
did not require the construction of the Dyson maps at all.

From our present perspective the latter formulae look much more important
because the knowledge of the factorized Dyson maps is needed in (20). In an
amended notation with ${\cal K}_{phys}={\cal R}_{0}^{(1)}$ and ${\cal
K}_{math}={\cal R}_{1}^{(1)}$ the resulting menu of the four eligible
Hermitization strategies may be now summarized in the more compact $N=2$
flowchart

$\begin{array}[]{c}\ \ \ \ \begin{array}[]{|c|}\hline\cr{\cal R}_{2}^{(0)}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm paths\
a,b}}}{{\longrightarrow}}\ \begin{array}[]{|c|}\hline\cr{\cal R}_{1}^{(0)}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm path\
a}}}{{\longrightarrow}}\ \begin{array}[]{|c|}\hline\cr{\cal R}_{0}^{(0)}\\\
\hline\cr\end{array}\\\ \ \ \ \ \ \ \ \ \ \ \stackrel{{\scriptstyle{\rm paths\
c,d\ }}}{{}}\searrow\ \ \ \ \ \ \ \ \ \ \stackrel{{\scriptstyle{\rm path\ b\
}}}{{}}\searrow\\\ \ \ \ \ \begin{array}[]{|c|}\hline\cr{\cal R}_{1}^{(1)}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm path\
c}}}{{\longrightarrow}}\ \begin{array}[]{|c|}\hline\cr{\cal R}_{0}^{(1)}\\\
\hline\cr\end{array}\\\ \ \stackrel{{\scriptstyle{\rm path\ d\
}}}{{}}\searrow\\\ \ \ \ \begin{array}[]{|c|}\hline\cr\ \ {\cal
R}_{0}^{(2)}\\\ \hline\cr\end{array}\\\ \end{array}$ (21)

For any preselected initial Hamiltonian $H$ this diagram offers the choice
among the four, formally equally eligible options a, b, c and d.

Table 3: The three optional choices of the model in QHQM at $N=2$. Hermitization | BGI | new | DI
---|---|---|---
(cf. diagram (21)) | (path d) | (paths b or c) | (path a)
physical metric | $I$ | $Z_{2}$ | $Z_{2}Z_{1}$
Hamiltonian | $\mathfrak{h}$ | $\mathfrak{h}_{1}$ | $H$
Hilbert space | ${\cal R}_{0}^{(2)}$ | ${\cal R}_{0}^{(1)}$ | ${\cal R}_{0}^{(0)}$

The list of the optional $N=2$ Hermitizations is now complete. Presumably, an
“optimal” element selected out of the four-path menu (see also its
rearrangement in Table 3) will be, in any realistic application of the theory,
characterized by the minimalized costs of the possible one- or two-step change
of the inner product (not needed along the BGI path d) and/or of the auxiliary
transformation(s) of the Hamiltonian (absent only along the DI path a).

The benefits of the separate options will vary with the input operator $H$ as
well as with the feasibility of the separate Hamiltonian-dependent Dyson sub-
mappings. A new feature of the $N=2$ scheme is that the choice of the path
ending at ${\cal K}_{phys}$ is ambiguous. Its realization may proceed along
the two alternative Hilbert-space triplets,

${\cal H}_{math}\to{\cal H}_{int}\to{\cal K}_{phys}\,,\,\ \ \ \ \ {\cal
H}_{math}\to{\cal K}_{math}\to{\cal K}_{phys}\,.$

Both of these paths are, from the point of view of Hermitization, equivalent.
In practice, therefore, one could often speak just about the three distinct
Hermitizations as listed in Table 3.

##  4 The formalism using the factorizations with $N=3$

Without factorization (i.e., at $N=1$) one can distinguish just between the
BGI Hermnitization (working with the trivial inner product and changing $H$)
and the DI $\Theta-$quasi-Hermnitization (keeping the initially non-Hermitian
Hamiltonian unchanged; see diagram (8)). In section 3 we choose $N=2$ and we
revealed the existence of another, non-DI quasi-Hermnitization in which a new,
non-BGI, partial amendment
$H\to\mathfrak{h}_{1}=\Omega_{1}\,H\,\Omega_{1}^{-1}$ of the Hamiltonian had
to be accompanied by a partial simplification of the physical inner product,
$\Theta\ (\equiv Z_{2}\,Z_{1})\,\to Z^{2}$.

In the $N=3$ diagram

$\begin{array}[]{c}\\\ \begin{array}[]{|c|}\hline\cr{\cal R}_{3}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle\rm DI\
path}}{{\longrightarrow}}\ \begin{array}[]{|c|}\hline\cr{\cal R}_{2}\\\
\hline\cr\end{array}\ \ \stackrel{{\scriptstyle\rm DI\
path}}{{\longrightarrow}}\ \begin{array}[]{|c|}\hline\cr{\cal R}_{1}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle\rm DI\
path}}{{\longrightarrow}}\ \begin{array}[]{|c|}\hline\cr{\cal R}_{0}\\\
\hline\cr\end{array}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \stackrel{{\scriptstyle{\rm
BGI\ path}}}{{}}\searrow\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \nearrow\\!\swarrow\stackrel{{\scriptstyle{\rm
equivalence\,}}}{{}}\\\ \begin{array}[]{|c|}\hline\cr\ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ {\cal H}_{textbook}\\\ \hline\cr\end{array}\\\ \end{array}$ (22)

the meaning of the symbols ${\cal R}_{k}$ is obvious. We only have to add here
that in full parallel to its $N=2$ predecessor in (12), the DI-type upper-line
Hermitization pattern is exceptional in the sense that the Hamiltonian remains
the same along the whole sequence of Hilbert spaces. The related technical
subtleties then concern just the step-by-step amendment of the metric.

Now we are going to offer a systematic description of the non-DI-type options.
The new Hermitization paths will be found branched in a way which has not been
noticed in our previous papers [13, 16]. The basic idea behind the “branching”
will be shown again related to the factorization of the metric. Still, its
description will require a modification and extension of the paradigm.

The necessary change of the paradigm will be counterintuitive, based on
introduction of further auxiliary inner-product spaces. For these reasons, it
makes sense to start its description from its first nontrivial $N=3$ special
case again. In such a case one merely has to replace the two $N=2$ doublets of
spaces in Eq. (14) by the three doublets,

${\cal R}_{3}^{(0)}\to{\cal R}_{2}^{(0)}\,,\ \ \ \ {\cal R}_{2}^{(0)}\to{\cal
R}_{1}^{(0)}\,,\ \ \ \ {\cal R}_{1}^{(0)}\to{\cal R}_{0}^{(0)}\,.$ (23)

After the three-term factorization (9) of the correct metric making the
ultimate Hilbert space ${\cal H}_{phys}={\cal R}_{0}^{(0)}$ physical we will
assume again that all of the three factors $Z_{1}$, $Z_{2}$ and $Z_{3}$ are
positive definite. This opens the possibility of a generalization of the
Dyson-map-resembling sub-factorization of these factors. We are now able to
require not only that we can always decompose

$Z_{3}=\Omega^{\dagger}_{3}\,\Omega_{3}$ (24)

(relation reflecting the fact that operator $Z_{3}$ gets factorized using a
Dyson sub-map $\Omega_{3}$) but also that we can always decompose

$Z_{3}\,Z_{2}=\widetilde{Z_{2}}=\widetilde{\Omega_{2}}^{\dagger}\,\widetilde{\Omega_{2}}\,,\
\ \ \ \widetilde{\Omega_{2}}=\Omega_{3}\,\Omega_{2}$ (25)

(this decomposition determines the untilded Dyson sub-map
$\Omega_{2}=\Omega_{3}^{-1}\,\widetilde{\Omega_{2}}$) and that we can always
decompose

$Z_{3}\,Z_{2}\,Z_{1}=\widetilde{Z_{1}}=\widetilde{\Omega_{1}}^{\dagger}\,\widetilde{\Omega_{1}}\,,\
\ \ \ \widetilde{\Omega_{1}}=\Omega_{3}\,\Omega_{2}\,\Omega_{1}$ (26)

(this factorization defines, finally, the third untilded Dyson sub-map
$\Omega_{1}=\widetilde{\Omega_{2}}^{-1}\,\widetilde{\Omega_{1}}$).

The most unexpected consequence of the factorizations (24), (25) and (26) is,
according to Lemma Nr. 2 of paper [13], that they specify, explicitly, the
composite nature of the conventional Dyson map of Eq. (6),

$\Omega=\widetilde{\Omega_{1}}=\Omega_{3}\,\Omega_{2}\,\Omega_{1}\,.$ (27)

Such a map connects the initial, unphysical representation space ${\cal
R}_{3}^{(0)}$ with the conventional textbook physical Hilbert space ${\cal
H}_{textbook}$.

Now we have to search for all of the possible shortcuts of the $N-$step BGI
Hermitizations, i.e., for the multiple direct and indirect analogues of the
ultimate DI path. Once we return to the $N=2$ cases for inspiration, and once
we recall the two triangular sub-diagrams (15) and (18) we get encouraged to
use the same mathematics yielding the $N=3$ triplet of the following
concatenated triangles,

$\begin{array}[]{c}\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\\\ {\rm in\ }{\cal R}_{3}^{(0)}\\\ \hline\cr\end{array}\
\stackrel{{\scriptstyle{\rm}Z_{3}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,Z_{3}\\\ {\rm in\ }{\cal R}_{2}^{(0)}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm}Z_{2}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,Z_{3}\,Z_{2}\\\ {\rm in\ }{\cal R}_{1}^{(0)}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm}Z_{1}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,Z_{3}\,Z_{2}\,Z_{1}\\\ {\rm in\ }{\cal R}_{0}^{(0)}\\\
\hline\cr\end{array}\\\ \stackrel{{\scriptstyle{\rm}\Omega_{3}}}{{}}\
\searrow\ \ \ \ \ \ \ \ \ \nearrow\\!\swarrow\
\stackrel{{\scriptstyle{\rm}}}{{}}\ \ \ \
\stackrel{{\scriptstyle{\rm}\Omega_{2}}}{{}}\ \searrow\ \ \ \ \ \ \ \ \ \
\nearrow\\!\swarrow\ \ \ \ \ \ \ \ \ \
\stackrel{{\scriptstyle{\rm}\Omega_{1}}}{{}}\ \searrow\ \ \ \ \ \ \ \ \ \ \
\nearrow\\!\swarrow\\\
\begin{array}[]{|c|}\hline\cr\Omega_{3}\,|\psi_{m}\rangle,\
\langle\psi_{n}|\,\Omega_{3}^{\dagger}\\\ {\rm in\ }{\cal J}_{2}\\\
\hline\cr\end{array}\ \ \ \ \
\begin{array}[]{|c|}\hline\cr\Omega_{2}\,|\psi_{m}\rangle,\
\langle\psi_{n}|\,\Omega_{2}^{\dagger}\,Z_{3}\\\ {\rm in\ }{\cal J}_{1}\\\
\hline\cr\end{array}\ \ \ \ \
\begin{array}[]{|c|}\hline\cr\Omega_{1}\,|\psi_{m}\rangle,\
\langle\psi_{n}|\,\Omega_{1}^{\dagger}\,\,Z_{3}\,Z_{2}\\\ {\rm in\ }{\cal
J}_{0}\\\ \hline\cr\end{array}\,.\end{array}$ (28)

In this diagram the arrows at $\Omega_{k}$ mark the transitions from the
initial spaces ${\cal R}_{k}^{(0)}$ to a new triplet ${\cal J}_{k-1}$ with
$k=3,2$ and $1$. These changes of Hilbert spaces are accompanied by the
respective replacements of $H$ by the respective new isospectral Hamiltonian
operators $\mathfrak{h}_{k}=\Omega_{k}\,H\,\Omega_{k}^{-1}$. In a formal
parallel with Eq. (18) (where we had $N=2$) we now arrive at the ultimate
physical $N=3$ DI-path Hilbert space ${\cal R}_{0}^{(0)}$ as well as at its
new, manifestly physical partner ${\cal J}_{0}$.

In Eq. (28) the unphysical intermediate Hamiltonians $\mathfrak{h}_{2}$ and
$\mathfrak{h}_{3}$ may happen to be available and tractable as easily as the
original input Hamiltonian $H$. As we already mentioned, it may make sense to
use of such a knowledge and to study the unusual Hermitizations passing
through these two less usual intermediates. Nevertheless, as long as we have
$N=3$, there still exists a methodically challenging missing-line gap between
the triplet of spaces ${\cal J}_{k}$ in (28) and the ultimate unique textbook
Hilbert space ${\cal H}_{textbook}$.

We already explained above how such a gap has to be filled. Leaving,
therefore, the details of the construction to the readers let us merely
formulate the final result. It states that the role of the $N=3$ analogue of
the six-space $N=2$ triangle (20) is now played by the following ten-space
triangular lattice

$\begin{array}[]{c}\\\ \begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\\\ {\rm in\ }{\cal R}_{3}^{(0)}\\\ \hline\cr\end{array}\
\stackrel{{\scriptstyle{\rm amendment}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,Z_{3}\\\ {\rm in\ }{\cal R}_{2}^{(0)}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm
amendment}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,Z_{3}\,Z_{2}\\\ {\rm in\ }{\cal R}_{1}^{(0)}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm
amendment}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,\Theta\\\ {\rm in\ }{\cal R}_{0}^{(0)}\\\
\hline\cr\end{array}\\\ \stackrel{{\scriptstyle{\rm}\Omega_{3}}}{{}}\
\searrow\ \ \ \ \ \ \ \ \ \ \ \nearrow\\!\swarrow\
\stackrel{{\scriptstyle{\rm}}}{{}}\ \ \ \ \ \ \ \ \
\stackrel{{\scriptstyle{\rm}\Omega_{2}}}{{}}\ \searrow\ \ \ \ \ \ \ \ \ \ \ \
\ \nearrow\\!\swarrow\ \ \ \ \ \ \ \ \ \ \
\stackrel{{\scriptstyle{\rm}\Omega_{1}}}{{}}\ \searrow\ \ \ \ \ \ \ \ \ \ \ \
\ \nearrow\\!\swarrow\\\
\begin{array}[]{|c|}\hline\cr\Omega_{3}\,|\psi_{m}\rangle,\
\langle\psi_{n}|\,\Omega_{3}^{\dagger}\\\ {\rm in\ }{\cal J}_{2}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm
amendment}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr\Omega_{2}\,|\psi_{m}\rangle,\
\langle\psi_{n}|\,\Omega_{2}^{\dagger}\,Z_{3}\\\ {\rm in\ }{\cal J}_{1}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm
amendment}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr\Omega_{1}\,|\psi_{m}\rangle,\
\langle\psi_{n}|\,\Omega_{1}^{\dagger}\,Z_{3}\,Z_{2}\\\ {\rm in\ }{\cal
J}_{0}\\\ \hline\cr\end{array}\\\
\stackrel{{\scriptstyle{\rm}\Omega_{32}}}{{}}\ \searrow\ \ \ \ \ \ \ \ \ \ \ \
\nearrow\\!\swarrow\ \stackrel{{\scriptstyle{\rm}}}{{}}\ \ \ \ \ \ \ \ \ \ \ \
\ \ \stackrel{{\scriptstyle{\rm}\Omega_{21}}}{{}}\ \searrow\ \ \ \ \ \ \ \ \ \
\ \ \ \nearrow\\!\swarrow\\\
\begin{array}[]{|c|}\hline\cr\Omega_{3}\,\Omega_{2}\,|\psi_{m}\rangle,\
\langle\psi_{n}|\,\Omega_{2}^{\dagger}\,\Omega_{3}^{\dagger}\\\ {\rm in\
}{\cal K}_{1}\\\ \hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm
amendment}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr\Omega_{2}\,\Omega_{1}\,|\psi_{m}\rangle,\
\langle\psi_{n}|\,\Omega_{1}^{\dagger}\,\Omega_{2}^{\dagger}\,Z_{3}\\\ {\rm
in\ }{\cal K}_{0}\\\ \hline\cr\end{array}\\\
\stackrel{{\scriptstyle{\rm}\Omega_{321}}}{{}}\ \searrow\ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \nearrow\\!\swarrow\\\
\begin{array}[]{|c|}\hline\cr\Omega_{3}\,\Omega_{2}\,\Omega_{1}\,|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,\Omega_{1}^{\dagger}\,\Omega_{2}^{\dagger}\,\Omega_{3}^{\dagger}\\\
{\rm in\ }{\cal H}_{textbook}\\\ \hline\cr\end{array}\\\ \end{array}$ (29)

The abbreviations $\Omega_{32}=\Omega_{3}\,\Omega_{2}\,\Omega_{3}^{-1}$,
$\Omega_{21}=\Omega_{2}\,\Omega_{1}\,\Omega_{2}^{-1}$ and
$\Omega_{321}=\Omega_{3}\,\Omega_{21}\,\Omega_{3}^{-1}$ have a recurrent
structure. This reflects the fact that under the three-term factorization
assumption (27) we had to introduce the two new Hilbert spaces denoted as
${\cal K}_{1}$ and ${\cal K}_{0}$. In these spaces the Hamiltonians may easily
be found to have the respective forms
$\mathfrak{h}_{32}=\Omega_{3}\,\Omega_{2}\,H\,\Omega_{2}^{-1}\,\Omega_{3}^{-1}$
(which is still non-Hermitian in ${\cal K}_{1}$) and
$\mathfrak{h}_{21}=\Omega_{2}\,\Omega_{1}\,H\,\Omega_{1}^{-1}\,\Omega_{2}^{-1}$
(which is quasi-Hermitian in ${\cal K}_{0}$).

Table 4: The BGI Hermitization and its three quasi-Hermitian shortcuts at $N=3$. The definitions of $\mathfrak{h}_{j}=\Omega_{j}\,H\,\Omega_{j}^{-1}$, $\mathfrak{h}_{jj-1}=\Omega_{jj-1}\,\mathfrak{h}_{j}\,\Omega_{jj-1}^{-1}$ and of $\mathfrak{h}_{jj-1j-2}=\Omega_{jj-1j-2}\,\mathfrak{h}_{jj-1}\,\Omega_{jj-1j-2}^{-1}$ are recurrent. BGI | quasi-Hermiticity shortcuts
---|---
| the $1^{\rm st}$ | the $2^{\rm nd}$ | the $3^{\rm rd}$ (DI)
non-Hermitian mathematical input
$H$ |  |  |
$\mathfrak{h}_{3}$ | $H$ |  |
$\mathfrak{h}_{32}$ | $\mathfrak{h}_{2}$ | $H$ |
physical versions of Hamiltonians
$\mathfrak{h}_{321}$ | $\mathfrak{h}_{21}$ | $\mathfrak{h}_{1}$ | $H$
physical Hilbert-space metrics
$I$ | $Z_{3}$ | $Z_{3}Z_{2}$ | $Z_{3}Z_{2}Z_{1}$

We see that the $N=3$ version of the formalism may be characterized by the
menu of as many as four alternative physical scenarios realized in as many as
four physical Hilbert spaces ${\cal R}_{0}^{(0)}$, ${\cal J}_{0}$ (alias
${\cal R}_{0}^{(1)}$), ${\cal K}_{0}$ (alias  ${\cal R}_{0}^{(2)}$), and
${\cal H}_{textbook}$ (alias , ${\cal R}_{0}^{(3)}$). In a different format
this observation is also summarized in Table 4. In practice, certainly, the
choice out of this menu will critically depend on the user-friendliness (i.e.,
on the complexity) of the respective Schrödinger equations and/or on the
accessibility of the desirable phenomenological predictions, i.e., on the
feasibility of the evaluation of certain relevant matrix elements.

##  5 The classification of the quasi-Hermitian reformulations of QHQM at any
$N$

Our detailed derivation of the latter, $N=3$ diagram (29) can be read as a
methodical guide. At arbitrary $N>3$ the process of its generalization may
start again from the Hamiltonian-preserving, metric-mediated DI-type
Hermitization which proceeds through an $(N+1)-$plet of Hilbert spaces

${\cal R}_{N}^{(0)}\to{\cal R}_{N-1}^{(0)}\to\ldots\to{\cal
R}_{1}^{(0)}\to{\cal R}_{0}^{(0)}\,.$ (30)

The leftmost symbol denotes here again the initial, preferred representation
space ${\cal R}_{N}^{(0)}={\cal H}_{math}$ with the inner-product metric equal
to identity. At the right-hand side of the set we have the ultimate physical
space ${\cal R}_{0}^{(0)}={\cal H}_{phys}$ in which the Hamiltonian $H$
becomes self-adjoint, the quantum measurements become correctly interpreted as
probabilistic, and the unitarity of the evolution becomes guaranteed by the
Stone theorem.

In this direction we already made the first steps in [13] where we managed to
augment, in the language of diagrams, the two-line DI pattern (22) from $N=3$
to any $N$,

$\begin{array}[]{c}\\\ \begin{array}[]{|c|}\hline\cr{\cal R}_{N}^{(0)}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle\rm DI\ step\
a}}{{\longrightarrow}}\ \begin{array}[]{|c|}\hline\cr{\cal R}_{N-1}^{(0)}\\\
\hline\cr\end{array}\ \ \stackrel{{\scriptstyle\rm DI\ step\
b}}{{\longrightarrow}}\ \ldots\ \stackrel{{\scriptstyle\rm DI\ step\
y}}{{\longrightarrow}}\ \begin{array}[]{|c|}\hline\cr{\cal R}_{1}^{(0)}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle\rm DI\ step\
z}}{{\longrightarrow}}\ \begin{array}[]{|c|}\hline\cr{\cal R}_{0}^{(0)}\\\
\hline\cr\end{array}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \stackrel{{\scriptstyle{\rm
BGI\ path}}}{{}}\searrow\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\nearrow\\!\swarrow\stackrel{{\scriptstyle{\rm equivalence\,.}}}{{}}\\\
\begin{array}[]{|c|}\hline\cr\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ {\cal H}_{textbook}\\\ \hline\cr\end{array}\\\ \end{array}$ (31)

In place of the single intermediate Hilbert space ${\cal H}_{int}$ of Eq. (12)
at $N=2$, the general scheme comprises as many as $N-1$ different, zero-
superscripted intermediate Hilbert spaces. In this sense the main result of
our preceding paper [13] may be characterized as a discovery of a one-to-one
correspondence between the separate Hilbert-space elements of the $(N+1)-$plet
(30) and the individual metric-operator factors $Z_{j}$ (cf. our present Eq.
(9) above and the Tables Nr. 1 and 2 and the equation Nr. (16) in loc. cit).

In our present paper we just promised to generalize the latter scheme. In this
spirit we started from the triangular three-space diagram (8) at $N=1$, and we
obtained the six-space triangular diagram (20) at $N=2$ and the analogous ten-
space diagram (29) at $N=3$. Thus, the climax of the story may be expected to
provide, at any $N$, a triangular diagram which will comprise as many as
$\\!\\!\begin{array}[]{c}\\!\\!\left(\\!\\!\begin{array}[]{c}N+2\\\
2\end{array}\\!\\!\right)\end{array}\\!\\!\\!$ Hilbert spaces.

Our derivation of the extrapolation may proceed quickly. We may start again
from the known DI-type $N-$step recipe of paper [13]. With the ultimate
physical metric $\Theta$ given by formula (9) we may characterize the starting
step by the diagram

$\begin{array}[]{c}\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\langle\psi_{n}|\\\
{\rm for}\ {\cal R}_{N}^{(0)}\\\ \hline\cr\end{array}\
\stackrel{{\scriptstyle\rm step\ 1}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\langle\psi_{n}|Z_{N}\\\
{\rm for}\ {\cal R}_{N-1}^{(0)}\\\ \hline\cr\end{array}\
\stackrel{{\scriptstyle\rm step\ 2}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\langle\psi_{n}|Z_{N}Z_{N-1}\\\
{\rm for}\ {\cal R}_{N-2}^{(0)}\\\ \hline\cr\end{array}\
\stackrel{{\scriptstyle\rm step\ 3}}{{\longrightarrow}}\ \ldots\
\stackrel{{\scriptstyle\rm step\ N}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\langle\psi_{n}|\Theta\\\
{\rm for}\ {\cal R}_{0}^{(0)}\\\ \hline\cr\end{array}\,.\end{array}$

The mathematics behind such a structure has been fully developed elsewhere
[13, 17]: The diagram just generalizes its two special cases as listed in
Appendix C (cf. Eqs. (51) and (54) below). Thus, we may proceed along the same
lines as above. The general construction will be fully analogous to the
preceding special cases so that the description of the details may be left to
the readers. Let us display just the final result which generalizes diagram
(21) beyond $N=2$,

$\begin{array}[]{c}\ \ \ \ \begin{array}[]{|c|}\hline\cr{\cal R}_{N}^{(0)}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm 2^{N-1}\
paths}}}{{\longrightarrow}}\ \begin{array}[]{|c|}\hline\cr{\cal
R}_{N-1}^{(0)}\\\ \hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm 2^{N-2}\
paths}}}{{\longrightarrow}}\ \begin{array}[]{|c|}\hline\cr{\cal
R}_{N-2}^{(0)}\\\ \hline\cr\end{array}\
\stackrel{{\scriptstyle{\rm}}}{{\longrightarrow}}\ \ldots\
\stackrel{{\scriptstyle{\rm 2\ paths}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\cal R}_{1}^{(0)}\\\ \hline\cr\end{array}\
\stackrel{{\scriptstyle{\rm 1\ path}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\cal R}_{0}^{(0)}\\\ \hline\cr\end{array}\\\
\stackrel{{\scriptstyle{\rm 2^{N-1}\ paths}}}{{}}\searrow\ \ \ \ \ \ \ \ \ \
\stackrel{{\scriptstyle{\rm 2^{N-2}\ paths}}}{{}}\searrow\ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \stackrel{{\scriptstyle{\rm}}}{{}}\searrow\ \ \ \ \ \ \ \ \ \
\ldots\ \ \ \ \ \ \ \ \ \ \stackrel{{\scriptstyle{\rm 1\
path}}}{{}}\searrow\\\ \ \ \ \ \ \begin{array}[]{|c|}\hline\cr{\cal
R}_{N-1}^{(1)}\\\ \hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm 2^{N-2}\
paths}}}{{\longrightarrow}}\ \begin{array}[]{|c|}\hline\cr{\cal
R}_{N-2}^{(1)}\\\ \hline\cr\end{array}\
\stackrel{{\scriptstyle{\rm}}}{{\longrightarrow}}\ \ \ \ldots\ \ \
\stackrel{{\scriptstyle{\rm}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\cal R}_{1}^{(1)}\\\ \hline\cr\end{array}\
\stackrel{{\scriptstyle{\rm N-1\ paths}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\cal R}_{0}^{(1)}\\\ \hline\cr\end{array}\\\
\stackrel{{\scriptstyle{\rm 2^{N-2}\ paths}}}{{}}\searrow\ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \stackrel{{\scriptstyle{\rm}}}{{}}\searrow\ \ \ \ \ \ \ \ \ \
\ \ldots\ \ \ \ \ \ \ \ \ \ \ \ \stackrel{{\scriptstyle{\rm}}}{{}}\searrow\\\
\ldots\ \ \ \ \ \ \ldots\ \ \ \ \ \ \ldots\ \ \ \ \ \ \ldots\ \ \ \ \ \
\ldots\ \ \ \ \ \ \ldots\ \ \ \ \ \ \ldots\\\ \ \stackrel{{\scriptstyle{\rm 2\
paths}}}{{}}\searrow\ \ \ \ \ \ \stackrel{{\scriptstyle{\rm N-1\
paths}}}{{}}\searrow\\\ \begin{array}[]{|c|}\hline\cr{\cal R}_{1}^{(N-1)}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle{\rm 1\
path}}}{{\longrightarrow}}\ \begin{array}[]{|c|}\hline\cr{\cal
R}_{0}^{(N-1)}\\\ \hline\cr\end{array}\\\ \ \stackrel{{\scriptstyle{\rm 1\
path}}}{{}}\searrow\\\ \ \ \ \begin{array}[]{|c|}\hline\cr\ \ {\cal
R}_{0}^{(N)}\\\ \hline\cr\end{array}\,.\\\ \end{array}$ (32)

For the enumeration of all of the paths of the eligible Hermitizations, is is
now sufficient to fix one of integers $k=0,1,\ldots,N$ and the Hilbert-space
sequence

${\cal R}_{N}^{(0)}\,\to\,{\cal R}_{N-1}^{(k_{1})}\,\to\,\ldots\,\to\,{\cal
R}_{1}^{(k_{N-1})}\,\to\,{\cal R}_{0}^{(k)}$ (33)

in which the superscripts are variable.

###### Lemma 1

Every eligible Hermitization path (33) is determined by its set of
superscripts such that

$0=k_{0}\leq k_{1}\leq k_{2}\leq\ldots\leq k_{N-1}\leq k_{N}=k\,.$ (34)

The number of all of the eligible Hermitization paths is large, equal to
$2^{N}$. Nevertheless, only the Hilbert spaces ${\cal R}_{0}^{(k)}$ with
$k=0,1,\ldots,N$ offer a consistent Hermitian or quasi-Hermitian picture of
physics. This enables us to compactify our ultimate classification of the
(quasi-)Hermitizations in the form of Table 5. .

Table 5: The complete list of possible Hermitizations of a given non-Hermitian Hamiltonian $H$. | non-Hermitian Hamiltonians | physical | the amended metric | the $H\to H_{kN}$
---|---|---|---|---
$k$ | (auxiliary sequence) | Hamiltonian | in Hilbert space | transformation
0 | $H$ | $H_{01}$ | $H_{02}$ | $\ldots$ | $H_{0N-1}$ | $\mathfrak{h}$ | $I$ | in $N$ steps (BGI)
1 |  | $\ H$ | $H_{12}$ | $\ldots$ | $H_{1N-1}$ | $H_{1N}$ | $Z_{N}$ | in $N-1$ steps
$\vdots$ |  |  | $\ddots$ | $\ddots$ | $\ \ \ \vdots$ | $\ \ \ \vdots$ | $\ \ \vdots$ | $\vdots$
$N-2$ |  |  |  | $H$ | $H_{N-2N-1}$ | $H_{N-2N}$ | $Z_{N}Z_{N-1}\ldots Z_{3}$ | in two steps
$N-1$ |  |  |  |  | $\ \ \ H$ | $H_{N-1N}$ | $Z_{N}Z_{N-1}\ldots Z_{3}Z_{2}$ | in one step
$N$ |  |  |  |  |  | $H_{NN}\equiv H$ | $Z_{N}Z_{N-1}\ldots Z_{3}Z_{2}Z_{1}$ | no change (DI)

According to this ultimate classification the role of the correct physical
Hamiltonian can be played, at any $N$, either by the Hermitian operator
$\mathfrak{h}=H_{0N}$ or, at the $k-$th non-trivial inner-product metric, by
one of the quasi-Hermitian operators $H_{kN}\,$ with $k=1,2,\ldots,N$. What is
not displayed in Table 5 are the explicit formulae which would specify the
Hamiltonians at all of the admissible values of the subscripts. The derivation
of these formulae is easy and, moreover, it may be further facilitated by the
straightforward extrapolation of the $N=2$ formulae of section 3 (cf. also
Table 3) and of their $N=3$ descendants in section 4 (cf. Table 4). Hence, the
task is left to the readers.

##  6 The path-dependence of complexity in a schematic three-level quantum
model

During the future use of our formalism in quantum physics of (hiddenly)
unitary systems, one of the key decisions to be made will concern the choice
of the physical space ${\cal R}_{0}^{(k)}$, i.e., the selection of the $k-$th-
line-strategy out of Table 5. The decision may be expected to vary with the
dynamical information encoded in the input operator $H$. For this reason there
probably exists no universally valid recommendation of one of the
alternatives. Still, let us now return, once more, to the “first nontrivial”
$N=3$ scenario, and let us try to use such a special case for a more concrete
illustration and a sampling of at least some of the meaningful criteria.

At $N=3$ the general menu of Table 5 degenerates to the mere quadruple choice
as offered by Table 4. The compatibility between the notation used in the
respective diagrams (32) and (29) can be achieved via the above-mentioned
identifications like ${\cal J}_{0}\equiv{\cal R}_{0}^{(1)}$, ${\cal
K}_{0}\equiv{\cal R}_{0}^{(2)}$, ${\cal H}_{textbook}\equiv{\cal R}_{0}^{(3)}$
and ${\cal H}_{phys}\equiv{\cal R}_{0}\equiv{\cal R}_{0}^{(0)}$. Next, we
believe that for the present methodical purposes, some persuasive results of
the comparison of the separate strategies can already be provided by the study
of the real toy-model matrices of dimension three. After all, even such “first
nontrivial” matrices could be interpreted as already forming fairly reasonable
and realistic three-state quantum models.

The comparisons may then start from the Dyson-map-factorization formula (6).
Thus, let us make the following one-parametric illustrative choice of

$\Omega_{3}=\left[\begin{array}[]{ccc}1&0&0\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr r&1&r\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&0&1\end{array}\right]\,,\ \ \ \ \ \
\Omega_{2}=\left[\begin{array}[]{ccc}1&0&0\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr s&1&0\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&s&1\end{array}\right]\,,\ \ \ \ \ \
\Omega_{1}=\left[\begin{array}[]{ccc}1&t&0\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr 0&1&t\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&0&1\end{array}\right]\,.$ (35)

This is a schematic ansatz which already yields a more complicated but still
just tridiagonal three-parametric Dyson map product (6),

$\Omega=\Omega_{321}=\left[\begin{array}[]{ccc}1&t&0\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr r+s&\left(r+s\right)t+1+sr&\left(1+sr\right)t+r\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr 0&s&ts+1\end{array}\right]$ (36)

as well as the two two-parametric auxiliary Dyson sub-maps used in diagram
(29),

$\Omega_{32}=\left[\begin{array}[]{ccc}1&0&0\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr r+s&1+sr&r\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&s&1\end{array}\right]\,,\ \ \ \
\Omega_{21}=\left[\begin{array}[]{ccc}1-ts&t&0\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr 0&1&t\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
t{s}^{3}&-t{s}^{2}&ts+1\end{array}\right]\,.$ (37)

Obviously, using the knowledge of the Dyson-map factors (35) we may
immediately evaluate the inner-product metrics entering the last line of Table
4,

$Z_{3}=\left[\begin{array}[]{lll}1+{r}^{2}&r&{r}^{2}\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr r&1&r\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr{r}^{2}&r&1+{r}^{2}\end{array}\right]\,,\ \ \
Z_{3}Z_{2}=\left[\begin{array}[]{lll}1+({r}+{s})^{2}&(r+s)(1+sr)&\left(r+s\right)r\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr(r+s)(1+sr)&(1+sr)^{2}+{s}^{2}&r+s{r}^{2}+s\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr\left(r+s\right)r&r+s{r}^{2}+s&1+{r}^{2}\end{array}\right]$ (38)

and

$Z_{3}Z_{2}Z_{1}=\left[\begin{array}[]{lll}1+({r}+{s})^{2}&\ldots&\ldots\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
t\,(1+({r}+{s})^{2})+(r+s)(1+sr)&\ldots&\ldots\\\ \vskip 6.0pt plus 2.0pt
minus
2.0pt\cr\left(r+s\right)\left(t+rts+r\right)&\ldots&\ldots\end{array}\right]\,.$
(39)

In the last item we only displayed the first column of the matrix because the
other matrix elements would already require more than one line.

We are prepared to finalize the comparison. Our message (that the preferences
may be expected to depend on the structure of $H$) may be accompanied by the
reference to paper [10] (showing that the first-column BGI approach appeared
best suited for the very specific Buslaev’s and Grecchi’s non-Hermitian but
analytic model (44) of Appendix B) and also by the reference to papers [2, 4]
where the authors advocated the last-column DI option which proved optimal for
an efficient numerical tractability of a number of many-body quantum systems.

An important supplementary simplification of our argumentation will be based
on the idea that for any finite-dimensional matrix model the concept of the
complexity of an operator depends on the choice of the basis. Really, one can
always find such an (in general, biorthonormal) basis in which $H$ (or
$\mathfrak{h}_{1}$, etc) is diagonalized. In this sense, we decided to make
our present, model-based demonstration of the existence of the “differences in
the complexity” between the separate physical versions of the eligible
Hamiltonians in Table 4 more transparent by using the particular basis in
which one postulates the diagonality of the leftmost matrix
$\mathfrak{h}_{321}$.

In the latter case, indeed, the correct and physical inner-product metric is
most elementary and, in fact, trivial, $\Theta(\mathfrak{h}_{321})=I$. This
enables us to simplify our comparison of complexities by starting from a
Hermitian matrix $\mathfrak{h}_{321}$ chosen in its simplest, diagonal and
zero-parametric form. Once we choose it in the following special version
possessing equidistant spectrum,

$\mathfrak{h}_{321}=\left[\begin{array}[]{ccc}1&0&0\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr 0&3&0\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&0&5\end{array}\right]$ (40)

we may assign it, in the BGI scenario, a phenomenologically particularly
appealing truncated-harmonic-oscillator interpretation.

In the opposite, maximal-shortcut DI-based representation our choice of the
example then leads to the maximally complicated three-parametric result of the
Dyson map (2),

$H=\left[\begin{array}[]{lll}-2\,r{t}^{2}s-2\,tr-2\,{t}^{2}{s}^{2}-2\,ts+1&-2\,r{t}^{3}s-2\,{t}^{2}r-2\,{t}^{3}{s}^{2}-2\,t+2\,{t}^{2}{s}^{2}r+2\,rts&\ldots\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
2\,rts+2\,r+2\,t{s}^{2}+2\,s&2\,r{t}^{2}s+2\,tr+2\,{t}^{2}{s}^{2}-2\,{s}^{2}rt+3-2\,sr&\ldots\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr-2\,\left(r+s\right)s&-2\,rts-2\,t{s}^{2}+2\,s+2\,{s}^{2}r&\ldots\end{array}\right]\,.$

This matrix (in which we omitted the last column to fit the page) is, in some
sense, “maximally” non-Hermitian. Any other isospectral and physical quasi-
Hermitian Hamiltonian of Table 4 may be expected to be more user-friendly.

The latter expectations are manifestly confirmed by the lengthy but
straightforward linear-algebraic calculations yielding the two-parametric
isospectral partner Hamiltonian

$\mathfrak{h}_{1}=\mathfrak{h}_{1}(r,s)=\left[\begin{array}[]{ccc}1&0&0\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr 2\,r+2\,s&3-2\,sr&-2\,r\\\ \vskip 6.0pt
plus 2.0pt minus
2.0pt\cr-2\,\left(r+s\right)s&2\,s+2\,{s}^{2}r&2\,sr+5\end{array}\right]$ (41)

(which already contains two vanishing matrix elements) and, last but not
least, the one-parametric isospectral partner Hamiltonian

$\mathfrak{h}_{21}=\mathfrak{h}_{21}(r)=\left[\begin{array}[]{ccc}1&0&0\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr 2\,r&3&-2\,r\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr 0&0&5\end{array}\right]$ (42)

with as many as four vanishing matrix elements.

Table 6: A sample test of the benefits of alternative Hermitizations.  $N=3$ | BGI | three shortcuts
---|---|---
Hamiltonian | $\mathfrak{h}_{321}$ | $\mathfrak{h}_{21}$ | $\mathfrak{h}_{1}$ | $H$
${\cal N}_{par}$ | zero | one | two | three
${\cal N}_{zero}$ | 6 | 4 | 2 | 0

The results of all of these observations are collected in Table 6. The growth
of the complexity of the model is mimicked there by the growth of the number
${\cal N}_{par}$ of its independently variable parameters. The desirable
sparsity of the Hamiltonian matrix is measured by the number ${\cal N}_{zero}$
of its vanishing matrix elements. We may conclude that in spite of a highly
schematic nature of our illustrative example, even the use of rather
elementary criteria indicates the marked differences between the alternatives.
Moreover, both of our criteria lead to the same ordering of the recommended
preferences, with an optimal Hermitization being found in the BGI approach. In
other words, the test of benefits is consistent in having both the physical
Hamiltonians and the corresponding inner-product metrics ordered, in our
methodically simplest illustration, from the trivial BGI scenario up to the
opposite, least-friendly DI extreme.

For the sake of brevity we decided to describe, in detail, just the scenario
in which our toy model appeared most complicated in the DI approach of the
last column of Table 4. In a sharp contrast the model is trivial in the other,
BGI extreme of the trivial-metric BGI Hermitization as listed in the first
column of the same Table.

In a way, our present model can be then perceived as a simpler parallel of the
rather exceptional Buslaev’s and Grecchi’s quantum system in which the BGI
choice also appeared to be optimal. In our present matrix model, nevertheless,
it would be possible to rotate the Hilbert-space basis in a way reaching and
simulating the preference of any other Hermitization scenario. Naturally, in
the less elementary unitary quantum systems with non-Hermitian Hamiltonians,
the determination of a computationally optimal Hermitization path will be less
easy, in a way strongly dependent on the properties of the dynamical
information as carried by the input operator $H$.

Obviously, we can summarize that our unitary but still just hiddenly Hermitian
three-level quantum system is, for our present purposes, sufficiently
realistic. In parallel, in the methodical context, the model has been found
sufficient to illustrate the possibility of the sharp contrasts between the
extent of the user-friendliness (or, if you wish, of the calculation
complexities) of the separate Hermitization strategies. The model samples and
reflects, sufficiently distinctly, not only the qualitative features of the DI
and BGI extremes (with the latter one being deliberately constructed as the
most user-friendly case) but also the sufficiently persuasive differences
between the intermediate items in the Hermitization menu. This seems to
indicate that in applications it truly might make sense to make use of the
whole, exhaustive list of all of the partial-Dyson-map-mediated Hermitizations
of the form as summarized in Table 5.

##  7 Discussion

From the perspective of practical applicability of the present reformulation
of quantum theory our motivation had two sources. The first one may be found
in the multiple existing difficulties of our current understanding of the
quantization of gravity [18] and in their potential tractability by the
mathematical tools of QGQM [9]. The second, related one lied in the study of
the concept of the so called minimal length. For example, the authors of paper
[19] claimed that in such a context the role of the quasi-Hermiticity of the
operators of observables might be decisive.

For illustration they decided to treat the position operator $X$ and the
momentum operator $P$ as the dynamical variables which are non-Hermitian but
Hermitizable. Subsequently, having used, in our terminology, the DI-type
version of the Hermitization, their basic conjecture (concerning the
possibility of a simultaneous  Hermitization of both  of the candidates $X$
and $P$ for an observable) appeared to be unfounded. The main conclusion of
our subsequent critical comments in [20] or [21] was that a guarantee of a
consistent Hermitization of several  non-Hermitian candidates $\Lambda_{j}$
for an observable has to be understood as a truly difficult open theoretical
problem.

What followed was a very recent partial resolution of the problem in [17]. In
that paper our message and construction were based on the $N-$term
factorization of the physical inner product metric $\Theta$. Our main result
was a constructive recipe by which a multiplet of the observables
$\Lambda_{j}$ appeared obtainable in terms of the metric factors $Z_{n}$. Such
an innovative DI-type result offered in fact an additional, physics-based
encouragement of our present study.

One of the related and currently open questions is the search for a BGI-type
complement of the latter DI-type construction of the observables. This is,
obviously, directly connected with the applicability of our present universal
diagram (32). One only has to keep in mind that even the BGI extreme option
did not find applications in too many realistic models. In this sense the
results obtained for our toy model of section 6 are encouraging. Indicating
that the new horizons seem open by the exhaustive classification of
(quasi-)Hermitizations via Eqs. (33) and (34). As long as the gap between the
DI and BGI methodical extremes is now fully covered, one might now return to
the older BGI-type results (cf., e.g., [14, 15]) and re-analyze the
possibility of their further simplification in terms of the new shortcuts of
the BGI Hermitization.

The specification of an optimal choice of one of the present upgrades of the
general QHQM theory is in fact a difficult problem. Even the merits of our
illustrative three-state model of section 6 would quickly be lost after
transition to larger finite-dimensional matrices. It is not clear in which way
the necessary simplification of the more realistic matrix models should
proceed. The general criteria of the preference of a particular Hermitization
pattern are not yet known at present. In the future, the enumeration of the
merits of the choice will certainly require a deeper insight in the favorable
properties of $H$.

New sources of difficulties will also emerge for the infinite-dimensional
Hilbert spaces in which, for a given $H$, the inner-product metric
$\Theta=\Theta(H)$ need not exist at all [3, 4, 21, 22]. New ways towards the
resolution of the related mathematical problems might start from the
observation that in the light of Eq. (4) the symbol $\Theta$ is just an
abbreviation for a product of the better-defined Dyson-map operators. Besides
their deeper physical meaning (cf. Eq. (5)), even the mathematical conditions
of their existence may be perceivably less restrictive (see Ref. [10] for
illustration once more).

From a purely pragmatic perspective people should still search for the
criteria of optimality. One might feel encouraged by the fact that even in the
BGI extreme based on the direct use and construction of the fully factorized
Dyson map (6) the area of applicability is non-empty. One may return, once
more, to Table 2 summarizing such a trivial-inner-product strategy. Now, the
same Table can be re-read as admitting the choice of any $N>1$, with $k=N$ in
Lemma 1 and in Eq. (34).

Table 7: Intermediate Hermitizations ${\cal R}_{N}^{(0)}\to{\cal R}_{0}^{(i)}$ with $i=1,2,\ldots,N-1$ . input: | aim: | specification: | needed: | calculated:
---|---|---|---|---
Hamiltonian | unitarity | path in diagram (32) | product in ${\cal R}_{0}^{(i)}$ | Hamiltonian in ${\cal R}_{0}^{(i)}$
$H\neq H^{\dagger}$ in ${\cal R}_{N}^{(0)}$ | (in ${\cal R}_{0}^{(i)}$) | (cf. Eqs. (33), (34)) | (cf., e.g., Eq. (29)) | (cf., e.g., Table 4)

In this light the main innovation and result of our present paper may be seen
in the completion of systematics. We described some basic features of a broad
family of the new, “intermediate” Hermitizations (summarized in Table 7)
guaranteeing, equally well, the unitarity of the evolution in the overall QHQM
context. On this background the message delivered by our present paper is
threefold. Firstly, we reconfirmed the usefulness as well as a not quite
expected methodical productivity of the general $N-$term factorization of the
correct physical Hilbert-space metric $\Theta$ (note that at $N=2$ one
reobtains, along this line, even the by far most popular parity times charge
ansatz for the metric).

Secondly, via a deeper analysis of the closely related composite-Dyson-map
algebraic structures we managed to offer a deeper interpretation of the
Buslaev’s and Grecchi’s “exceptional” anharmonic-oscillator example. In a way
concerning the feasibility and usefulness of the similar systematic non-DI
Hermitization constructions, and in a way opposing the existing BGI-related
scepticism we also proposed and described another, exactly solvable finite-
matrix model for which the simplest Hermitization appeared to be provided
strictly in its BGI extreme.

Thirdly, last but not least, we promoted the idea that the factorization
ansatzs may make the general QHQM formulation of the theory more user-friendly
and efficient even between the two DI and BGI methodical extremes. The
availability of the widened spectrum of Hermitizations will enable one to
combine an optimal partial amendment of the physical inner product with a
suitable complementary partial amendment of the Hamiltonian and/or of the
other relevant observables in practice.

Concluding the discussion we believe that the present completion of the list
of optional Hermitizations made the QHQM form of unitary quantum theory, via
an explicit and exhaustive enumeration of all of the alternative Hermitization
patterns at a preselected fixed $N$, more flexible and open not only to its
further methodical analyses but also to its various ambitious phenomenological
applications – for the freshmost illustration see the discussion of the
potential relevance of the QHQM language in quantum cosmology in [23].

##  8 Summary

In our present paper we felt challenged, in the context of the use of non-
Hermitian Hamiltonians $H\neq H^{\dagger}$ in quantum mechanics of unitary
systems, by the dominance and prevailing preferences of the most conventional
$N=1$ DI-Hermitization philosophy (see its compact summary in Table 1). We
also felt encouraged by our preceding preliminary results concerning the
possible $N>1$ metric-factorization options: In [13] we described the most
straightforward amendment of the special, Hamiltonian-preserving DI quasi-
Hermitization based on an $N-$term factorization of the physical Hilbert-space
inner-product metric $\Theta$.

In our present continuation of the analysis we were inspired by the Buslaev’s
and Grecchi’s paper [10]. We recalled their anharmonic-oscillator Hamiltonian
$H=H^{(BG)}$ as a methodical guide. We emphasized that the similar models are
characterized by the $N-$term factorization (6) of the underlyi8ng global
Dyson map $\Omega$. We imagined that the existence of such a factorization may
admit not only the brute-force constructive BGI proof of the isospectrality
between $H^{(BG)}$ and its Hermitian avatar $\mathfrak{h}^{(BG)}$ but also a
new family of alternative Hermitizations. In a constructive and model-
independent manner we proposed and described an exhaustive classification of
all of such $k-$step Hermitizations with $0\leq k\leq N\,$ in Lemma 1.

In the language of diagrams we re-classified the $(N+1)-$plet of Hilbert
spaces ${\cal R}_{j}^{(0)}$ which appears in diagram (31) as the mere upper-
line component of the more adequate, much richer and, in some sense,
exhaustive triangular classification diagram (32). In the diagram the single-
path DI Hermitization (30) only emerges as an extreme, $k=0$ special case. The
gap in the methodical considerations of paper [13] is filled. New perspectives
are opened in the analysis of models using non-Hermitian observables. A number
of problems may now find new solutions mediated by the conjugate $N-$term
factorizations of the physical inner-product metrics and of the related Dyson
maps.

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## Appendix A. Hidden forms of Hermiticity

The numerical diagonalization of many realistic self-adjoint Hamiltonians
$\mathfrak{h}=\mathfrak{h}^{\dagger}$ can be accelerated by their judicious
isospectral non-unitary preconditioning $\mathfrak{h}\to H$ (see Eq. (2)).
More than 60 years ago this has been revealed by Freeman Dyson [2]. One only
has to keep in mind that the manifest non-Hermiticity $H\neq H^{\dagger}$ of
$H$ is inessential because the Hermiticity of
$\mathfrak{h}=\mathfrak{h}^{\dagger}$ can be perceived as equivalent to the
quasi-Hermiticity of $H$ [3]. In 1992, the arrow in (2) has been inverted by
Scholtz et al [4] (cf. Eq. (5)). A given, manifestly non-Hermitian $H$ with
real spectrum has been declared, under certain not too restrictive technical
conditions, an acceptable Hamiltonian tractable as an alternative
representation of $\mathfrak{h}$.

Even though the evolution of the wave functions then appeared controlled by a
manifestly non-Hermitian generator $H$ (cf. Eq. (3)), the standard
probabilistic interpretation of the theory remained, due to the Dyson-map
equivalence $\Omega$ and due to the Hermiticity of the partner $\mathfrak{h}$,
unchanged. At present, the original quasi-Hermitian formulation of quantum
mechanics (QHQM, [4]) based on relation (4) is considered, nevertheless,
difficult to implement (see, e.g., the arguments summarized on p. 1216 of
newer review [9]). For this reason the “Dyson-inspired” (DI) Hermitization (5)
has been modified and made, more than twenty years ago, technically more user-
friendly. A decisive simplification of the formalism has been achieved within
a slightly narrower framework of the ${\cal PCT}-$symmetric quantum mechanics
(see its standard review [11]).

One of the characteristic features of the latter approach is an ad hoc  two-
term factorization of the metric: see our comment on this feature in [17].
Still, even after the simplification, every user of a non-Hermitian but
Hermitizable Hamiltonian $H$ had to ask the questions about the compatibility
of the non-Hermitian $H$ with Stone theorem [24]. At present, it is already
widely accepted that the Hermitizations can be constructed as proceeding
either via a “BGI-path” transformation of the Hamiltonian or via the mere “DI-
path” amendment of the inner product in ${\cal H}_{math}$.

In both cases, operator $\mathfrak{h}$ is much less user-friendly than $H$.
This explains why the vast majority of the successful QHQM calculations are
performed in ${\cal H}_{math}$. In the former case, nevertheless, the
Hermitization seems more difficult, tractable in a narrower class of the
solvable or at least sufficiently elementary models [10, 14]. Still, the
picture reconstructed in the textbook space ${\cal H}_{textbook}$ may appear
useful, say, in semi-classical considerations [25].

In the latter, more robust case of the DI Hermitization the user-friendly
operator $H$ is kept unmodified. The information about the Hermiticity of
$\mathfrak{h}$ in ${\cal H}_{textbook}$ only has to be transferred to the
working space ${\cal H}_{math}$ which is, by definition, manifestly
unphysical. Hence, the transfer leads to the quasi-Hermiticity constraint (4).
This property of $H$ keeps the operator tractable, under certain additional
technical assumptions [4], as hiddenly Hermitian. Moreover, condition (4) can
be interpreted as inducing an ad hoc  change of inner product,

$\langle\psi_{a}|\psi_{b}\rangle\ \to\
\langle\psi_{a}|\Theta|\psi_{b}\rangle\,.$ (43)

Such a change can be read as a replacement of the initial, manifestly
unphysical manipulation space ${\cal H}_{math}$ by its standard physical
partner space of states ${\cal H}_{phys}$. In it, the amendment of the
Hermitian conjugation renders the initial Hamiltonian $H$ self-adjoint in
${\cal H}_{phys}$ [3, 4].

## Appendix B. The Buslaev’s and Grecchi’s oscillator

In the specific illustrative model of paper [10] (using, incidentally, a not
too small $N=N^{(BG)}=7$ in (6) and (7)), all of the recurrent isospectrality-
guaranteeing transformations $H_{0j}^{(BG)}\to H_{0j+1}^{(BG)}$ were entirely
elementary, mediated by a non-numerical Fourier transformation or even by the
mere change of variables. In this sense, the results were exceptional: Their
authors paid attention just to a special non-Hermitian Hamiltonian

$H=H_{\eta}({\rm
i}g,j)=-\frac{d^{2}}{dx^{2}}+\frac{1}{4}V^{(BG)}_{(j,g,\eta)}(x)\neq
H^{\dagger}\,,\ \ \ \ \ x\in\mathbb{R}\,$ (44)

containing an unusual, complex and asymptotically wrong-sign anharmonic-
oscillator potential

$V^{(BG)}_{(j,g,\eta)}(x)=\frac{j^{2}-1}{r^{2}_{\eta}(x)}+{r^{2}_{\eta}(x)}{}-{g^{2}\,r^{4}_{\eta}(x)}{}\,,\
\ \ \ \ r_{\eta}(x)=x-{\rm i}{\eta}\,,\ \ \ \ \eta>0\,,\ \ \ j>0\,$ (45)

(cf. equation Nr. 16 in loc. cit.). These authors identified the model as
${\cal PT}-$symmetric (parity-time-symmetric) and proved that its spectrum is
discrete, real and bounded from below. The main idea of the proof lied in the
observation that Hamiltonian $H_{\eta}({\rm i}g,j)$ (which is non-Hermitian in
the conventional Hilbert space $L^{2}(\mathbb{R})\,\equiv\,{\cal H}_{math}$)
is equivalent (i.e., isospectral) to another operator

$Q(g,j)=-\frac{d^{2}}{dx^{2}}+\mathfrak{v}^{(BG)}_{(j,g)}(x)=Q^{\ddagger}(g,j)\,,\
\ \ \ \ x\in\mathbb{R}\,$ (46)

with a real and confining double-well potential

$\mathfrak{v}^{(BG)}_{(j,g)}(x)=(gx-1)^{2}\,x^{2}+j\,(1/2-gx)\,.$ (47)

The double well model was safely self-adjoint in the Hilbert space
$L^{2}(\mathbb{R})\,\equiv\,{\cal H}_{textbook}$ and admitted, therefore, the
standard unitary-evolution quantum-mechanical interpretation (cf. equation Nr.
17 and Theorem 5 in loc. cit.).

At its time, the result was treated as a mere mathematical curiosity [26]. The
situation has only changed when, a few years later, Bender with Milton [27],
Bender with Boettcher [28] and a few other teams of authors [29, 30] pointed
out that the similar non-Hermitian but ${\cal PT}-$symmetric quantum
Hamiltonians $H\neq H^{\dagger}$ with real spectra might be of an immediate
theoretical as well as phenomenological interest (see also reviews [9, 11, 12,
31]).

In this context the birth of the motivation of our present considerations
dates back to the year 2000 when Vincenzo Grecchi [26] recalled his
collaboration with Vladimir Buslaev and when he pointed out that at least some
of the topical open questions concerning the ${\cal PT}-$symmetric quantum
systems have already been answered in [10]. One of these answers appeared in
the proof of Theorem 5. In it, the authors used the factorization (7) and they
filled the gap between the respective initial and final operators
$H_{\eta}({\rm i}g,j)$ and $Q(g,j)$ by a set of auxiliary isospectral
operators,

$H_{\eta}({\rm i}g,j)\sim K^{4}_{\eta}({\rm i}g,j)\sim\ldots\sim
Q^{6}_{\eta}(g,j)\sim Q(g,j)\,.$ (48)

These operators formed a set of equivalent candidates for the Hamiltonian
defined by equations Nr. 19 - 24 in [10]. They were allowed to fail to be
Hermitian or ${\cal PT}-$symmetric. Their mutual isospectrality was
comparatively easily deduced from the underlying recurrent definitions of the
operator-transformation form $K^{4}_{\eta}({\rm i}g,j)={\cal
S}_{1}H_{\eta}({\rm i}g,j){\cal S}^{-1}_{1}$, etc. All of the elements of
multiplet (48) happened to be just ordinary linear differential operators
defined, in closed form, analytically.

## Appendix C. Factorized metrics

In the three-space diagram (10) the realization of both of the BGI and DI
Hermitizations needs just $N=1$ in Eq. (6). At the same time, the DI-path
strategy itself is already known in the generalized form using any $N$ [13,
17]. This is one of the reasons why the more flexible $N-$step DI-path
Hermitization might seem preferable in applications. Indeed, it requires just
the solution of Eq. (4) for metric $\Theta=\Theta(H)$ (see also a few related
comments on p. 1216 in review [9], and an outline of the construction methods
to be found, e.g., in section Nr. 4 of the same review).

### C. 1. $N=2$

A remarkable feature of the Hamiltonians $H$ which are quasi-Hermitian alias
“Hermitian in disguise” is that their popularity started to grow only after
the theory was complemented by the innocent-looking requirement of ${\cal
PT}-$symmetry (see also the similar comment in Appendix A). This led to a
decisive innovation of the search for the Hermitizing metric $\Theta$ which
has been found facilitated by its pre-factorization,

$\Theta=Z_{2}\,Z_{1}\,.$ (49)

The right factor was kept Hamiltonian-dependent and treated, initially at
least, as a charge, $Z_{1}=Z_{1}(H)=Z_{1}^{\dagger}={\cal C}$. The left one
was initially defined as the conventional parity, $Z_{2}=Z_{2}^{\dagger}={\cal
P}$. Some authors criticized ansatz (49) as not too well motivated (see, e.g.,
pp. 1198 and 1232 in review [9]). We oppose the criticism emphasizing that the
factorization contributed to the appeal of QHQM significantly.

We admit that the motivation of ansatz (49) seems mysterious and that it
deserves a clarification. For the purpose we reflected a part of the criticism
and we turned attention to a possible generalization of the ansatz. In [16],
for example, we reinterpreted the charge $Z_{1}$ as the mere independent non-
Hermitian but ${\cal PT}-$symmetric observable. In another, model-independent
study of the problem in [17] it has been reconfirmed that the apparent mystery
of ansatz (49) may be given a very natural clarification based on a less
specific choice of the factors $Z_{j}$. In this manner a better balance has
been established between the roles played by the factors $Z_{1}$ and $Z_{2}$.

The balance has been achieved at the expense of introduction of another,
third, “intermediate” inner-product space ${\cal H}_{int}$. In terms of
diagrams we just proposed the upgrade (12) of diagram (8). The presence of the
two factors in (49) and of the two steps in diagram (12) implies that the
metric $Z_{2}$ need not necessarily be positive definite. In our present
paper, nevertheless, only the simpler Hilbert-space scenario characterized by
the positive definite factors $Z_{1}$ and $Z_{2}$ will be considered for the
sake of brevity (see also the extensive discussion of such an option in [32]).

The two-term factorization of the metric (49) may be assigned a deeper
mathematical meaning. In [16] we emphasized that for consistency reasons it
would be sufficient when operator $Z_{1}$ is merely required quasi-Hermitian
with respect to another, auxiliary “partial metric” $Z_{2}$. This means that
the related modified, positively definite “anomalous observable charge”
$Z_{1}$ need not be Hermitian in ${\cal H}_{math}$, $Z_{1}\neq
Z_{1}^{\dagger}$. Due to the Hermiticity of the positive-definite operator
$Z_{2}=Z_{2}^{\dagger}>0$ the most fundamental condition of the Hermiticity of
the factorized metric (49) may be now most easily satisfied because it is
equivalent to the single condition

$Z_{1}^{\dagger}\,Z_{2}=Z_{2}\,Z_{1}\,$ (50)

representing the weakened requirement of the $Z_{2}-$mediated quasi-
Hermiticity of $Z_{1}$. Thus, one can conclude that it is possible to work
with the Hermitization diagram (12) which contains the inner-product space
${\cal H}_{int}$ endowed with the new auxiliary and positive definite metric
$Z_{2}$.

In the intermediate Hilbert space every ket vector $|\psi\rangle$ becomes
associated with an unusual form of the Hermitian-conjugate bra vector
$\langle\psi|\,Z_{2}$. This means that the information provided by the first
line of diagram (12) may be made more explicit, including the “local”, inner-
product-dependent bra-vector Hermitian conjugates,

$\begin{array}[]{c}\ \ \ \ \ \ \ \ \ \ \
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\ {\rm}\langle\psi_{n}|\\\
{\rm for}\ {\cal H}_{math}\\\ \hline\cr\end{array}\ \stackrel{{\scriptstyle\rm
step\ 1}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,Z_{2}\\\ {\rm for}\ {\cal H}_{int}\\\
\hline\cr\end{array}\ \stackrel{{\scriptstyle\rm step\ 2}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\
{\rm}\langle\psi_{n}|\,Z_{2}\,Z_{1}\\\ {\rm for}\ {\cal H}_{phys}\\\
\hline\cr\end{array}\ .\end{array}$ (51)

In such a two-step DI-path Hermitization scenario, the description of quantum
dynamics is still based on the solution of Schrödinger equation using the same
Hamiltonian operator $H$.

The idea leads to a reformulation of quantum mechanics in which the same
Hamiltonian $H$ is assigned the three different, inner-product-dependent
Hermitian conjugate forms. In [16] they were denoted by the three different
dedicated symbols, viz., as $H^{\dagger}$ in ${\cal H}_{math}$, as
$H^{\ddagger}$ in ${\cal H}_{int}$, and as $H^{\sharp}$ in ${\cal H}_{phys}$.
Later, all of the necessary formulae were represented in ${\cal H}_{math}$ so
that the introduction of the special dedicated superscript in $H^{\ddagger}$
or in $H^{\sharp}$ can be declared redundant. For this reason, the exclusive
representation of the system in ${\cal H}_{math}$ is also used in our present
paper.

### C. 2. $N=3$

In an immediate generalization of the above-outlined two-factor scheme let us
consider the three-term ansatz

$\Theta=Z_{3}\,Z_{2}\,Z_{1}\,.$ (52)

Its introduction merely leads to the replacement of the four-space
Hermitization pattern (12) by its five-space analogue (22). On the technical
level the Hermiticity property of parity is then transferred to another
operator, $Z_{3}=Z_{3}^{\dagger}$. The quasi-Hermiticity (50) has to be
upgraded yielding the doublet of the necessary and sufficient constraints,

$Z_{2}^{\dagger}\,Z_{3}=Z_{3}\,Z_{2}\,,\ \ \
Z_{1}^{\dagger}\,(Z_{3}\,Z_{2})=(Z_{3}\,Z_{2})\,Z_{1}\,.$ (53)

In practice, therefore, the “dynamical input” knowledge of the positive
definite operator $Z_{3}$ and of any positive definite Hermitian operator
$Y_{3}=Z_{3}\,Z_{2}$ enables us to reconstruct $Z_{2}$ in ${\cal H}_{math}$
while, similarly, another piece of the “dynamical input” knowledge of a
positive definite Hermitian operator $X_{3}=Z_{3}\,Z_{2}\,Z_{1}$ enables us to
reconstruct $Z_{1}$. Ultimately, we may deduce and prove the observability
status of the operators $\Lambda_{1}=Z_{1}$ and $\Lambda_{2}=Z_{2}\,Z_{1}$
[17]. The DI-path sequence of Eq. (51) can be replaced by the three-step
Hermitization pattern,

$\begin{array}[]{c}\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\langle\psi_{n}|\\\
{\rm for\ }{\cal R}_{3}^{(0)}\\\ \hline\cr\end{array}\
\stackrel{{\scriptstyle{\rm step\ 1}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\langle\psi_{n}|\,Z_{3}\\\
{\rm for\ }{\cal R}_{2}^{(0)}\\\ \hline\cr\end{array}\
\stackrel{{\scriptstyle{\rm step\ 2}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\langle\psi_{n}|\,Z_{3}\,Z_{2}\\\
{\rm for\ }{\cal R}_{1}^{(0)}\\\ \hline\cr\end{array}\
\stackrel{{\scriptstyle{\rm step\ 3}}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr{\rm}|\psi_{m}\rangle,\langle\psi_{n}|\,Z_{3}\,Z_{2}\,Z_{1}\\\
{\rm for\ }{\cal R}_{0}^{(0)}\\\ \hline\cr\end{array}\,.\end{array}$ (54)

Both of the two- and three-step DI-path Hermitizations remain conceptually the
same. The transition to the general $N-$step DI-path Hermitization is
immediate and may be found described, thoroughly, in [13].