Discrete-coordinate crypto-Hermitian quantum system controlled by time-
dependent Robin boundary conditions

Miloslav Znojila,b,c

a The Czech Academy of Sciences, Nuclear Physics Institute, Hlavní 130, 250 68
Řež,

Czech Republic, e-mail<EMAIL_ADDRESS>
b Department of Physics, Faculty of Science, University of Hradec Králové,
Rokitanského 62,

50003 Hradec Králové, Czech Republic, e-mail<EMAIL_ADDRESS>
c Institute of System Science, Durban University of Technology, Durban, South
Africa

### Abstract

A family of exactly solvable quantum square wells with discrete coordinates
and with certain non-stationary  Hermiticity-violating Robin boundary
conditions is proposed and studied. Manifest non-Hermiticity of the model in
conventional Hilbert space ${\cal H}_{friendly}$ is required to coexist with
the unitarity of system in another, ad hoc  Hilbert space ${\cal
H}_{physical}$. Thus, quantum mechanics in its non-Hermitian interaction
picture (NIP) representation is to be used. We must construct the time-
dependent states (say, $\psi(t)$) as well as the time-dependent observables
(say, $\Lambda(t)$). Their evolution in time is generated by the operators
denoted, here, by the respective symbols $G(t)$ (a Schrödinger-equation
generator) and $\Sigma(t)$ (a Heisenberg-equation generator, a.k.a. quantum
Coriolis force). The unitarity of evolution in ${\cal H}_{physical}$ is then
guaranteed by the reality of spectrum of the energy observable alias
Hamiltonian $H(t)=G(t)+\Sigma(t)$. The applicability of these ideas is
illustrated via an $N$ by $N$ matrix model. At $N=2$, closed formulae are
presented not only for the measurable instantaneous energy spectrum but also
for all of the eligible time-dependent physical inner-product metrics
$\Theta_{(N=2)}(t)$, for the related Dyson maps $\Omega_{(N=2)}(t)$, for the
Coriolis force $\Sigma_{(N=2)}(t)$ as well as, in the very ultimate step of
the construction, for the truly nontrivial Schrödinger-equation generator
$G_{(N=2)}(t)$.

### Keywords

quantum theory of unitary systems; non-Hermitian interaction representation;
non-stationary physical inner products; solvable discrete square well;

##  1 Introduction

Among applications of quantum mechanics working with stationary observables
which are non-Hermitian but quasi-Hermitian [1], a comparatively exceptional
position is taken by the theories in which the information about dynamics is
merely carried by boundary conditions [2, 3, 4, 5]. One of the simplest,
square-well examples of such a type can be found discussed in [6]. In the
context of scattering the boundary conditions of a manifestly non-Hermitian
form have been assigned there a fully conventional physical interpretation of
a perfect-transmission constraint. A less artificial-looking and, currently,
more widely accepted physical bound-state treatment of all of the analogous
non-Hermitian but quasi-Hermitian (i.e., hiddenly Hermitian) models and
Hamiltonians can be found described in a number of reviews in which one finds
the very general mathematical concept of quasi-Hermitian operators [7]
narrowed to the operators which are quasi-Hermitian but bounded [1] or merely
$\eta-$pseudo-Hermitian alias  pseudo-Hermitian [8] or even just parity-
pseudo-Hermitian alias  parity-time symmetric [9].

In the formulation of stationary quantum theory called non-Hermitian
Schrödinger picture (NSP) one has to combine physics (requiring the unitarity
of evolution which is just guaranteed in a certain hypothetical, user-
unfriendly Hilbert space ${\cal H}_{physical}$) with mathematics (set in
another space ${\cal H}_{mathematical}$ and needed during the practical
implementation of the theory).

In the conventional stationary quantum mechanics of textbooks [10], both of
the latter two Hilbert spaces coincide. In the upgraded NSP version of the
theory (in which one admits that ${\cal H}_{physical}\neq{\cal
H}_{unphysical}$, cf. also several more rigorous and updated reviews in [11])
one always has to keep in mind that any operator $\Lambda$ representing an
observable is, by definition, simultaneously self-adjoint in the hypothetical
“correct space” ${\cal H}_{physical}$ and non-Hermitian in the friendlier,
“unphysical” but preferred construction space ${\cal H}_{mathematical}$.

In practice it appeared convenient to work just in ${\cal
H}_{unphysical}={\cal H}$. Using the common notation convention one stays in
${\cal H}$, writes $\Lambda\neq\Lambda^{\dagger}$ and introduces the so called
inner-product metric $\Theta$ in order to guarantee the observability status
of $\Lambda$ via the Dieudonné’s [7] quasi-Hermiticity postulate
$\Lambda^{\dagger}\Theta=\Theta\,\Lambda$ (cf. also [1] for details). In such
a setting one treats ${\cal H}_{physical}$ as represented in ${\cal
H}_{unphysical}$ while one only amends the inner product,

$\langle\psi_{1}|\psi_{2}\rangle_{physical}=\langle\psi_{1}|\Theta|\psi_{2}\rangle_{unphysical}\,.$

In the second item the subscript unphysical can and will be omitted as
superfluous.

During the early years of development and applications of the NSP formalism
people believed that for unitary systems (of our present interest) “the inner
product of the physical Hilbert space cannot depend on time unless one defines
the dynamics of the quantum system by an operator that is not observable” (cf.
Theorem 2 and subsequent comments on p. 1272 in [8]).

Later, it became clear that the applicability of the latter “no-go” Theorem is
restricted just to the mere NSP framework in which, generically, the non-
Hermitian observables remain stationary, time-independent,
$\Lambda^{(NSP)}\neq\Lambda^{(NSP)}(t)$. An ultimate remedy and clarification
of the misunderstanding has been found, in 2008, in a non-stationary extension
of the quasi-Hermitian formulation of quantum mechanics (to be called, in what
follows, non-Hermitian interaction picture, NIP, cf. its introduction in [12,
13]).

A few years later, the situation has been re-analyzed [14] and the adequacy of
the use of time-dependent metric operators has been reconfirmed [15] (in this
respect cf. also the recent comprehensive NIP review [16]). In the NIP
framework of our present interest (where one admits the non-stationarity
$\Theta_{NIP}=\Theta_{NIP}(t)$) the necessary mathematics has been found
perceivably more complicated. For this reason, in a way inspired by the recent
NSP description of the role of non-Hermitian but stationary boundary
conditions in a discrete Schrödinger equation [17] we are now going to
describe a discrete but still unitary quantum system in which the boundary
conditions would be not only non-Hermitian but also non-stationary.

A detailed formulation of the problem as well as a constructive sample of its
solution will be given. We will outline the basic features of the properly
amended NIP alternative to the more traditional stationary NSP formulation of
quantum theory. After a concise exposition of necessary mathematics we will
redirect emphasis to physics. We will explain how the “input” knowledge of the
dynamics-determining non-Hermitian and time-dependent boundary conditions can
be consequently converted into a consistent theoretical scheme yielding the
“output” predictions of the results of measurements of observable
characteristics of a non-stationary but stable, hiddenly unitary physical
system.

The presentation of our results will be arranged as follows. First, in section
2 we will outline the basic features of the formalism. Then, section 3 will be
devoted to the introduction of our specific boundary-interaction model. This
will be followed by section 4 in which our attention will be turned to the
construction and properties of the physical Hilbert space. In subsequent
section 5 we will display the explicit illustrative formulae while in our
final sections 6 and 7 we will add a few final remarks and conclusions.

##  2 Hiddenly Hermitian quantum mechanics in nuce

###  2.1 Stationary cases and NSP physical Hilbert spaces

Given an arbitrary non-Hermitian Hamiltonian $H\neq H^{\dagger}$, the first
question to ask concerns the consistent probabilistic interpretation of the
model. The answer was formulated, in 1992, by Scholtz et al [1]. These authors
explained how the conventional requirement of self-adjointness of a
Hamiltonian can be weakened. In particular, they emphasized that in the
analysis of many realistic systems the uniqueness of a conventional textbook
Hilbert space of states (say, ${\cal L}$, which has to be, simultaneously,
user-friendly and physical [10]) may happen to be over-restrictive.

They proposed to split the roles and to work, simultaneously, with the two
separate, non-equivalent Hilbert spaces. Naturally, a preselected Hamiltonian
$H$ (or any other observable $\Lambda$) can only be self-adjoint in one of
them (i.e., say, in ${\cal H}_{physical}$). The other, non-equivalent Hilbert
space (i.e., say, ${\cal H}_{mathematical}={\cal H}_{friendly}$) may be
preferred, as the  representation space in calculations. At the same time, the
latter space must necessarily be perceived as manifestly unphysical since in
this space one has $H\neq H^{\dagger}$.

For our present purposes the two Hilbert spaces may be interpreted as
complementary since ${\cal H}_{friendly}={\cal H}_{unphysical}$ and ${\cal
H}_{physical}={\cal H}_{unfriendly}$. The main motivation of the split is that
for many quantum systems of practical interest the innovated formulation of
quantum mechanics might be more calculation-friendly. Moreover, it was of
paramount importance to imagine that the correct physical Hilbert space ${\cal
H}_{unfriendly}$ appeared to be comparatively easily represented in the
mathematically more suitable working space ${\cal H}_{friendly}$ [1]. What
appeared sufficient was the mere amendment of inner product (the related
technical details may be also found discussed in reviews [8, 9, 11]).

In applications (and, in particular, in applications in which the operators of
observables are stationary, time-independent), it is very natural to expect
that all of the necessary calculations will be simpler after transition from
the conventional Schrödinger equation living in ${\cal L}$, viz., from the
equation of textbooks

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

to its alternative living in ${\cal H}_{friendly}$,

${\rm i}\,\frac{d}{dt}\,|\psi(t)\rangle=H\,|\psi(t)\rangle\,,\ \ \ \
|\psi(t)\rangle\in{\cal H}_{friendly}\,,\ \ \ \ H\neq H^{\dagger}\,.$ (2)

The price to pay for the change is that the new version of Hamiltonian (which
is self-adjoint in ${\cal H}_{unfriendly}$) appears non-Hermitian in the
mathematical representation space.

###  2.2 Non-stationary NIP and physical Hilbert space

In the NSP formulation of quantum mechanics one of the most essential
assumption is that the one-to-one correspondence between the Hilbert spaces
and/or between the Schrödinger equations (realized, say, by an invertible
“Dyson map” operator $\Omega$ [18]) remains time-independent (cf., e.g.,
Theorem Nr. 2 in [8]). After a more or less straightforward non-stationary NIP
generalization of the theory, unfortunately, only too many changes did occur.

The first one was that in the specification of correspondence
$|\psi(t)\\!\succ\,\,\leftrightarrow|\psi(t)\rangle$ we had to keep in mind,
in general, the manifest time-dependence of all of the relevant operators.
Thus, using a time-dependent generalization $\Omega=\Omega(t)$ of the
invertible Dyson-map operator we postulate

$|\psi(t)\\!\succ\,\,=\Omega(t)\,|\psi(t)\rangle\,,\ \ \ \ \
\mathfrak{h}(t)=\Omega(t)\,H(t)\,\Omega^{-1}(t)\,.$ (3)

Due to the emergence of a non-vanishing Coriolis-force operator

$\Sigma(t)={\rm i}\,\Omega^{-1}(t)\,\dot{\Omega}(t)$ (4)

(where the dot over ${\Omega}(t)$ marks the differentiation with respect to
time) the insertion of the textbook ket
$|\psi(t)\\!\succ\,\,=\Omega(t)\,|\psi(t)\rangle$ does not convert the
textbook Schrödinger Eq. (1) into its NSP partner (2) but rather into its
modified, NIP partner

${\rm i}\,\frac{d}{dt}\,|\psi(t)\rangle=G(t)\,|\psi(t)\rangle\,,\ \ \ \
|\psi(t)\rangle\in{\cal H}_{friendly}\,,\ \ \ \ G(t)=H(t)-\Sigma(t)\,.$ (5)

In contrast to the stationary NSP models with vanishing $\Sigma(t)=0$, we now
have, in general, $G(t)\neq H(t)$. This means that the “dynamical information
input” knowledge of the textbook Hamiltonian $\mathfrak{h}(t)$ (or, more
precisely, of its non-Hermitian isospectral image $H(t)$ defined as acting in
${\cal H}_{friendly}$) does not still enable us to write down Schrödinger Eq.
(5) and, via its solution, to reconstruct the evolution of ket-vectors
$|\psi(t)\rangle\in{\cal H}_{friendly}$.

Fortunately, a hypothetical knowledge of the time-dependence of mapping
${\Omega}(t)$ and of the inner-product metric
$\Theta(t)=\Omega^{\dagger}(t)\,\Omega(t)$ enables us to re-express the self-
adjointness of $\mathfrak{h}(t)$ in ${\cal L}$ via the time-dependent
generalization of the Dieudonné’s quasi-Hermiticity property of its
isospectral observable-Hamiltonian avatar
$H(t)=\Omega^{-1}(t)\,\mathfrak{h}(t)\,\Omega(t)$ in ${\cal H}_{friendly}$
[7],

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

Hence, we may invert the flowchart and assume that given the Hamiltonian (with
real spectrum) in its non-Hermitian-representation version $H(t)$ (preselected
and defined as acting in ${\cal H}_{friendly}$), the necessary search for
inner-product metric $\Theta(t)$ can still be based on the solution of (6),
i.e., of the linear equation. We may conclude that in the non-stationary NIP
framework the construction of metric $\Theta(t)$ can proceed in full analogy
with the NSP recipes.

After the construction of $\Theta(t)$ one is also allowed to employ the
standard operator methods and to construct the “square root” $\Omega(t)$ of
metric as well as its inverse and the time derivative as needed in Eq. (4).
The necessity of construction of all of these “missing” components of Coriolis
force represents, in fact, a new and difficult technical challenge. Only its
satisfactory resolution may enable us to define, ultimately, the NIP
Schrödinger equation and to construct the ket vectors representing the states
(cf. also review [16] for details).

Only on this basis we would finally be able to restore the NIP-NSP parallels
and to predict the results of measurements. Typically, whenever one considers
an observable of interest (represented by an operator $\Lambda(t)$ with real
spectrum and such that
$\Lambda^{\dagger}(t)\,\Theta(t)=\Theta(t)\,\Lambda(t)$), the NIP predictions
will be based again on the evaluation of overlaps

$\langle\psi(t)|\,\Theta(t)\,\Lambda(t)|\psi(t)\rangle\,.$ (7)

The presence of the correct physical inner-product metric $\Theta(t)\neq I$
indicates that these overlaps only have their correct probabilistic
interpretation in the truly anomalous and, through non-stationary metric,
manifestly time-dependent form of Hilbert space ${\cal H}_{physical}={\cal
H}_{physical}(t)$.

##  3 Boundary-controlled square-well model

The existence of NIP-related “new and difficult technical challenges” as
mentioned in preceding section was one of the main sources of inspiration of
our forthcoming detailed and constructive study of the non-stationary version
of the discrete square-well model endowed with nontrivial, manifestly time-
dependent boundary conditions.

Before we start addressing the related technical challenges, let us briefly
mention that in a broader physical context, the studies of quantum systems
characterized by time-dependent boundary conditions are currently finding
phenomenological applications which range from mathematical and condensed-
matter physics to cosmology (for a concise reference let us just cite the
recent preprint [19]). In this setting, our choice of model

$H(t)=\left[\begin{array}[]{ccccc}2-z(t)&-1&0&\ldots&0\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr-1&2&-1&\ddots&\vdots\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr 0&-1&\ddots&\ddots&0\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\vdots&\ddots&\ddots&2&-1\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&\ldots&0&-1&2-z^{*}(t)\end{array}\right]\,$ (8)

was strongly encouraged by the dynamical input knowledge status of the
boundary-value matrix elements $z(t)\in\mathbb{C}$ of our potentially
observable time-dependent quantum Hamiltonian.

Our mathematically motivated choice of such an $N$ by $N$ matrix model with
$N<\infty$ will also decisively facilitate the physics-oriented construction
of the predictions of measurements (7).

###  3.1 Robin boundary conditions

Many salient features of bound-state problems of conventional textbooks [10]
are well illustrated by the exactly solvable ordinary differential square-well
Schrödinger equation [20]

$-\frac{d^{2}}{dx^{2}}\psi_{n}(x)=\varepsilon_{n}\psi_{n}(x)\,,\ \ \
\psi_{n}(0)=\psi_{n}(L)=0\,$ (9)

and/or by its numerically motivated difference-equation equidistant-lattice
analogue [21]

$-\psi_{n}(x_{k-1})+2\,\psi_{n}(x_{k})-\psi_{n}(x_{k+1})=E_{n}^{(N)}\psi_{n}(x_{k})\,,\
\ \ \psi_{n}(x_{0})=\psi_{n}(x_{N+1})=0\,$ (10)

where $n=0,1,\ldots$ and where either $x\in(0,L)$ or $k=1,2,\ldots,N$,
respectively. One of the important new methodical merits of both of the latter
two old toy models emerged, recently, in the framework of the so called non-
Hermitian reformulations of quantum mechanics: For our present purposes we may
recall, in this respect, either the older reviews [1, 8, 9] (and speak about a
stationary non-Hermitian Schrödinger picture, NSP) or paper [12] and newer
reviews [11, 16] (and speak about a non-stationary non-Hermitian interaction
picture, NIP).

We are returning to these questions with a new motivation provided by the
emergence of difficulties accompanying the growth of interest in certain non-
stationary  NIP models [15, 22]. During the formulation of our present project
we felt encouraged by the mutual relationship between the two square-well
Schrödinger equations (9) and (10). In parallel, a strictly phenomenological
source of our interest can be seen in a consequent restriction of the “input”
information about dynamics to the boundaries, i.e., in a certain “minimality”
of the non-Hermitian ingredients in these models.

We decided to replace the conventional Dirichlet boundary conditions by their
Hermiticity-violating two-parametric (i.e., Robin-boundary-condition)
alternatives

$\psi(0)=\frac{\rm i}{\alpha+{\rm i}\beta}\,\frac{d}{dx}\psi(0)\,,\ \ \ \ \ \
\psi(L)=\frac{\rm i}{\alpha-{\rm i}\beta}\,\frac{d}{dx}\psi(L)\,$ (11)

(in (9), with two free real parameters $\alpha\,,\,\beta\,\in\,\mathbb{R}$) or

$\psi_{n}(x_{0})=\frac{\rm i}{\alpha+{\rm
i}\beta}\,\left(\frac{\psi_{n}(x_{1})-\psi_{n}(x_{0})}{h}\right)\,,\ \ \ \ \ \
\psi_{n}(x_{N+1})=\frac{\rm i}{\alpha-{\rm
i}\beta}\,\left(\frac{\psi_{n}(x_{N+1})-\psi_{n}(x_{N})}{h}\right)\,$ (12)

(in (10), with a suitable lattice grid-point distance $h>0$), respectively.

A wealth of consequences may be expected to emerge. The most obvious one lies
in the necessity of an upgrade of the conventional formulation of quantum
mechanics. A key challenge emerges due to our innovated interpretation of
parameters in conditions (11) and (12) which will be allowed non-stationary,
time-dependent,

$\alpha=\alpha(t)\,,\ \ \ \ \beta=\beta(t)\,.$ (13)

The latter, innocent-looking generalization leads to a number of nontrivial
constructive tasks. In the forthcoming, methodically sufficiently instructive
analysis only the difference Schrödinger-equation model will be considered.

###  3.2 Condition number one: the reality of spectrum

The two stationary and manifestly non-Hermitian square-well bound-state
problems as mentioned in Introduction were thoroughly studied, in the NSP
framework, in [17]. We noticed there that the differential-equation problem
can be perceived as a specific (i.e., $h\to 0$ and $N\to\infty$) special-case
limit of its difference-equation partner. Thus, we just studied the
difference-equation bound-state problem with finite $N$ and with a single
complex parameter $z=1/(1-\beta\,h-i\alpha\,h)$.

Once we rewrote the corresponding stationary Schrödinger equation in its
equivalent $N$ by $N$ matrix form in ${\cal H}_{friendly}$,

$\left[\begin{array}[]{ccccc}2-z&-1&0&\ldots&0\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr-1&2&-1&\ddots&\vdots\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&-1&\ddots&\ddots&0\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\vdots&\ddots&\ddots&2&-1\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&\ldots&0&-1&2-z^{*}\end{array}\right]\,\left[\begin{array}[]{c}\phi_{1}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\phi_{2}\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr\vdots\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\phi_{N-1}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\phi_{N}\\\
\end{array}\right]=E_{n}^{(N)}\,\left[\begin{array}[]{c}\phi_{1}\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr\phi_{2}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\vdots\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\phi_{N-1}\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr\phi_{N}\\\ \end{array}\right]\,$ (14)

(cf. Eq. Nr. 13 in [17]) it was comparatively straightforward to reveal the
exact solvability of the problem. Indeed, after abbreviation
$2-E_{n}^{(N)}=2y$ our Schrödinger Eq. (14) acquired, strictly, the form of
recurrences satisfied by Chebyshev polynomials of the first and second kind
[23]. In other words, we could set

$\phi_{n}=A\,T_{n-1}(y)+B\,U_{n-1}(y)\,,\ \ \ \ n=1,2,\ldots,N\,.$ (15)

Subsequently, we could fix the values of the two complex parameters $A$ and
$B$ and of the energy via the normalization and boundary conditions, i.e., via
the first and last line of Eq. (14).

Precisely this has been done in [17]. Serendipitously we discovered there that
the Hamiltonian in (14) is PT-symmetric, $HPT=PTH$. Whenever this symmetry
proves spontaneously unbroken, the energies are all real, i.e., the evolution
in time remains unitary [9]. We proved that the existence of such a dynamical
regime is guaranteed in a non-empty complex vicinity of real $z=1$ (cf.
Proposition Nr. 1 in loc. cit.).

The quantum system in question has also been given the standard probabilistic
interpretation. Realized, in some cases, via explicit formulae determining a
suitable stationary NSP metric $\Theta$ at any $N$ (cf., e.g., Proposition Nr.
2 in loc. cit.).

###  3.3 Numerics and reparametrizations

In our present paper we decided to extend the latter analysis to the non-
stationary dynamical regime in which the complex dynamics-controlling
parameter becomes allowed to vary with time, $z=z(t)$. The motivation of such
a project was threefold. Firstly, we felt encouraged by the fact that the
introduction $z\to z(t)$ of the non-stationarity of dynamics leaves the formal
solvability of eigenvalue problem (14) via ansatz (15) unchanged. Secondly, we
imagined that the extremely elementary one-parametric nature of our $N$ by $N$
Hamiltonian $H(t)$ enhances the chances of the constructive considerations
being successful.

Thirdly, having performed a few preliminary tests at $N=2$ we revealed that in
a way paralleling a few stationary-theory observations as made in [17], the
insight in the structure and properties of bound states can significantly be
enhanced when parameter $z\in\mathbb{C}$ gets replaced by a more specific real
variable. An amended insight emerged when we replaced the value of $z$ by its
redefinition $\,i\sqrt{1-{r}^{2}}\,$ using a real variable $r$ and yielding

$H=\left[\begin{array}[]{cc}2-i\sqrt{1-{r}^{2}}&-1\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr-1&2+i\sqrt{1-{r}^{2}}\end{array}\right]\,.$ (16)

A key merit of such a (still, stationary) reduction appeared to lie not only
in the extremely elementary form of spectrum $E^{(2)}_{\pm}=2\pm r$ but also
in a serendipitous discovery that the matrix $H$ ceases to be diagonalizable
in the limit of vanishing $r\to 0$.

In this limit, in the language of mathematics, the model acquires the Kato’s
[24] exceptional-point (EP) singularity. Hence, the usual diagonal-matrix
representation of Hamiltonian

$\mathfrak{h}_{S}=\mathfrak{h}_{S}(r)=\left[\begin{array}[]{cc}r+2&0\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr 0&-r+2\end{array}\right]\,$ (17)

remains restricted to $r\neq r^{(EP)}=0$. Alternatively, the latter matrix
becomes tractable also as the special diagonal self-adjoint textbook
Hamiltonian in ${\cal L}$.

Even in the language of physics, the choice of $r=0$ must be excluded as not
compatible with the postulates of consistent quantum theory of closed systems.
At EP, matrix (17) has to be replaced by a canonical non-diagonal Jordan block

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

in a process which is discontinuous in $r$.

Figure 1: Graphical version of the closed-form representation (18) of the
strictly real bound-state energy spectrum of Schrödinger Eq. (14) at $N=6$
with specific $z=z(r)={\rm i}\sqrt{1-r^{2}}$.

The methodical message delivered by the latter analysis can easily be extended
to any $N$. For example, once we choose $N=6$, the evaluation of energies
$E_{n}^{(6)}$ seems to be a purely numerical task because the secular
determinant $\det(H-E)$ is equal to polynomial

${E}^{6}-12\,{E}^{5}+\left(56-{r}^{2}\right)E^{4}+\left(-128+8\,{r}^{2}\right){E}^{3}+\left(147-21\,{r}^{2}\right){E}^{2}+\left(-76+20\,{r}^{2}\right)E+12-5\,{r}^{2}$

of the sixth degree in $E$. Fortunately, this polynomial is just a linear
function of $r^{2}$. Thus, the $N=6$ spectrum can exactly  be defined by the
following closed implicit-function formula

$r_{\pm}=r_{\pm}(E)=\pm\sqrt{\frac{{E}^{4}-8\,{E}^{3}+20\,{E}^{2}-16\,E+3}{{E}^{4}-8\,{E}^{3}+21\,{E}^{2}-20\,E+5}}\left(E-2\right)\,.$
(18)

This is an analytic result which could be extended to any finite Hilbert-space
dimension $N$. Numerically it is complemented by Figure 1 where we can see
that the EP singularity (also known as “non-Hermitian degeneracy” [25]) at
$r=0$ manifests itself by the merger of two levels in the middle of the
spectrum.

##  4 Non-stationary inner-product metrics

In our present toy model the nontrivial (i.e., non-Hermitian and  time-
dependent) dynamics is introduced via boundary conditions. Partially, it can
simplify a consequent application of the theory.

###  4.1 Definition

Once we managed to confirm the reality of spectrum we have to move to the next
model-building task which lies, in both of the NSP and NIP contexts, in the
construction or selection of the physical inner product metric $\Theta$ which
would be compatible with quasi-Hermiticity constraint (6). At this stage of
development we may feel encouraged by the observation that for the same but
stationary, time-independent interaction (i.e., in the simpler NSP dynamical
regime), incidentally, a closed-form solution $\Theta(H)$ of Eq. (6) appeared
available [26].

One of the most universal construction strategies which might be also used in
the NIP setting has been described in [27]. After a restriction of attention
to the unitary quantum models living in the finite-dimensional Hilbert spaces
we showed there that whenever one manages to solve the conjugate version of
Schrödinger eigenvector problem or, in the notation of Ref. [13], of the
ketket-vector problem

$H^{\dagger}(t)\,|\xi_{n}(t)\rangle\\!\rangle=E_{n}\,|\xi_{n}(t)\rangle\\!\rangle\,,\
\ \ \ n=1,2,\ldots,N\,$ (19)

then all  of the eligible (i.e., invertible and positive definite [1]) inner-
product metrics form an $N-$parametric family,

$\Theta_{(\vec{\kappa}(t))}(t)=\sum_{n}\,|\xi_{n}(t)\rangle\\!\rangle\,\kappa_{n}(t)\,\langle\\!\langle\xi_{n}(t)|\,,\
\ \ \ \forall\ \kappa_{n}(t)>0\,.$ (20)

Different physics becomes represented by the different choices of parameters
$\kappa_{n}(t)>0$. In what follows, for the sake of simplicity, we will work
just with the trivial choice of $\kappa_{n}^{(constant)}(t)=1$ and we will
speak about a “special” alias  “standard” choice of
$\Theta_{(\vec{1})}(t)=\Theta_{S}(t)$.

###  4.2 Non-stationary Dyson maps

Let us now fully concentrate on the non-stationary NIP model-building
dynamical scenarios with the Hilbert-space metrics which vary with time.
Keeping in mind that this time-dependence is essential, contributing to an
enormous increase of the model-building difficulties caused by the transition
from the NSP formalism to its NIP generalization.

The first duty to fulfill is the Dyson-map factorization of the manifestly
non-statinary metric $\Theta(t)=\Omega^{\dagger}(t)\Omega(t)$. Often, this is
performed via taking a self-adjoint square root of $\Theta(t)$ followed by a
completion of the construction by considering also the unitary-matrix
ambiguity of eligible $\Omega(t)$s. As long as such a task seems enormously
complicated, its upgrade will be based here on a return to a diagonal
Hamiltonian matrix $\mathfrak{h}_{S}(t)$ (cf. its $N=2$ sample (17)). Within
such a project, definition (19) of the ketkets has to be re-read as a matrix
intertwining problem

$H^{\dagger}(t)\,\Omega_{S}^{\dagger}(t)=\Omega_{S}^{\dagger}(t)\,\mathfrak{h}_{S}(t)\,$
(21)

where the matrix of eigenvalues $\mathfrak{h}_{S}(t)$ is diagonal.

Relation (21) can be interpreted as a specific realization of correspondence
between Hilbert spaces ${\cal L}\leftrightarrow{\cal H}_{friendly}$ as defined
in Eq. (3). In this sense the set of all of the ketket eigenvectors
$|\xi_{n}(t)\rangle\\!\rangle$ of $H^{\dagger}(t)$ (cf. Eq. (19)) can be re-
visualized as the set of separate columns of matrix $\Omega_{S}^{\dagger}(t)$.
As a consequence, we can put

$\Theta_{S}(t)=\Omega_{S}^{\dagger}(t)\,\Omega_{S}(t)\,$ (22)

and speak about the metric in which the change of normalization of the
eigenketkets in (19) (which is admissible) can be perceived as equivalent to
the change of parameters $\vec{\kappa}(t)$ in (20).

##  5 Non-numerical results at $N=2$

Whenever we are given the non-stationary and non-Hermitian operator $H(t)$
representing the instantaneous real and observable bound-state energies, the
main obstacle on our way towards the tests (i.e., towards the predictions
represented by formula (7)) is the necessity of construction of the generator
$G(t)=H(t)-\Sigma(t)$ of evolution of the relevant ket vectors (cf. Eq. (5)).

In the literature, very often, the authors circumvent the problem and,
typically, complement the knowledge of $H(t)$ by some additional information
about $G(t)$ or $\Sigma(t)$. In our present paper we intend to argue that the
dynamical input knowledge of $H(t)$ can be, in some cases and sense,
sufficient.

The latter statement is strongly model-dependent. Even in the case of our
present specific and sufficiently elementary discrete square-well model (8),
an explicit and exhaustive description of its properties would be, in spite of
its solvability, complicated and not too illuminative, especially at the
larger Hilbert-space dimensions $N$. As long as we intend to provide here just
an overall qualitative support of the user-friendliness of applicability of
the NIP-based models, we will restrict our attention to the mere first
nontrivial version (16) of our model with $N=2$.

Surprisingly enough, the results of this study will be shown to be not only
encouraging but also compact and, in a way, persuasive and sufficiently
informative.

###  5.1 The first task: closed-form Dyson map

During our preliminary search for the closed-form solutions of the $N=2$
matrix version of Schrödinger Eq. (19) alias  (21) we tried to use several
computer-assisted symbolic-manipulation techniques, and we failed. The success
only came with a return to the paper-and-pencil techniques. In the model of
Eq. (16) they guided us to perform another change of variables setting
$r=\sin\varphi$ where $\varphi=\varphi(t)\neq\varphi^{(EP)}=0,\pm\pi,\ldots$.

We will drop, in some cases, the ubiquitous time-dependence-emphasizing
brackets $(t)$ as redundant. In particular, having turned attention to non-
stationary conjugate-Hamiltonian operator

$H^{\dagger}=\left[\begin{array}[]{cc}2+i\cos\varphi&-1\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr-1&2-i\cos\varphi\end{array}\right]$

we got not only the above-mentioned spectrum $E_{\pm}^{(2)}=2+\sin\varphi$
(where one can simulate $\pm$ by ${\rm sign\,}\varphi$ and omit the subscript)
but also, having solved Eq. (21), one of the most compact eligible non-
stationary forms of the respective conjugate and non-conjugate Dyson maps,

$\Omega_{S}^{\dagger}=\left[\begin{array}[]{cc}1&-{\rm i}\,\exp{\rm
i}\varphi\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr{\rm i}\,\exp{\rm
i}\varphi&1\end{array}\right]\,,\ \ \ \
\Omega_{S}=\left[\begin{array}[]{cc}1&-{\rm i}\,\exp(-{\rm i}\varphi)\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr{\rm i}\,\exp(-{\rm
i}\varphi)&1\end{array}\right]\,$

Immediately, this yields the metric,

$\Theta_{S}=\Omega_{S}^{\dagger}\,\Omega_{S}=\left[\begin{array}[]{cc}2&-2{\rm
i}\,\cos\varphi\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr 2{\rm
i}\,\cos\varphi&2\end{array}\right]\,.$ (23)

This formula coincides with the one constructed in [17] where we,
unfortunately, did not find the way towards its Dyson-map factorization. A
consistency of the latter metric proves also supported, off the EP
singularity, by the positivity of its two eigenvalues $\theta_{\pm}=2\pm
2\,\cos\varphi$ as well as by its diagonality and proportionality to a unit
operator in the Hermitian-Hamiltonian limit of $\cos\varphi\to 0$.

###  5.2 The second task: closed-form Coriolis force

By the direct computation we get matrices

$\Omega_{S}^{-1}=\frac{1}{1-\exp(-2{\rm
i}\varphi)}\,\left[\begin{array}[]{cc}1&{\rm i}\,\exp(-{\rm i}\varphi)\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr-{\rm i}\,\exp(-{\rm
i}\varphi)&1\end{array}\right]\,,\ \ \ \ \
\dot{\Omega}_{S}=\dot{\varphi}\cdot\left[\begin{array}[]{cc}0&-\exp(-{\rm
i}\varphi)\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\exp(-{\rm
i}\varphi)&0\end{array}\right]\,$

as well as Coriolis force,

$\Sigma_{S}={\rm
i}\Omega_{S}^{-1}\dot{\Omega}_{S}=\frac{\dot{\varphi}}{2\,\sin\varphi}\,\left[\begin{array}[]{cc}{\rm
i}\exp(-{\rm i}\varphi)&-1\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr 1&{\rm
i}\exp(-{\rm i}\varphi)\end{array}\right]\,.$ (24)

This result leads us to an important observation that both  of the eigenvalues
$\sigma_{\pm}$ of $\Sigma_{S}$ are always  complex,

$\sigma_{\pm}=\left(1+{\rm i}\,\frac{\cos\varphi\pm
1}{\sin\varphi}\right)\cdot{\dot{\varphi}}/{2}=\left\\{\begin{array}[]{c}\left(1+{\rm
i}\,{\cot\varphi/2}\right){\dot{\varphi}}/2\\\ \left(1-{\rm
i}\,{\tan\varphi/2}\right){\dot{\varphi}}/2\,.\end{array}\right.$

Moreover, these two eigenvalues do not even form a complex conjugate doublet.

This means that there exists no operator of parity ${\cal P}$ which could make
the Coriolis force (i.e., the Heisenberg-equation generator alias  “Heisenberg
Hamiltonian”) ${\cal PT}-$symmetric.

###  5.3 The ultimate task: closed-form Schrödinger equation

Once we abbreviate $D={\dot{\varphi}(t)}/({2\,\sin\varphi(t)})$ and set
$1-D=A=A(t)$ and $1+D=B=B(t)$, we may recall Schrödinger Eq. (5) and evaluate
the difference $G_{S}(t)=H(t)-\Sigma_{S}(t)$,

$G_{S}=\left[\begin{array}[]{cc}2-D\sin\varphi-{\rm i}B\cos\varphi&-A\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr-B&2-D\sin\varphi+{\rm
i}A\cos\varphi\end{array}\right]\,.$

Its eigenvalues are available in compact form $g_{\pm}=2-D\sin\varphi+w_{\pm}$
with

$w_{\pm}=-{\rm i}D\cos\varphi\pm\sqrt{\sin^{2}\varphi-D^{2}}\ .$

In the “almost stationary” dynamical regime with small ${\dot{\varphi}}$ such
that $D^{2}<\sin^{2}\varphi$ we get

$w_{\pm}=\pm\sin\varphi-{\rm i}D\cos\varphi+{\cal O}(D^{2})$

yielding the two strictly non-real eigenvalues of $G_{S}$ which are even not
mutually conjugate,

$g_{\pm}=2\pm\sin\varphi-D\sin\varphi-{\rm i}D\cos\varphi+{\cal O}(D^{2})\,.$

In the opposite, “strongly non-stationary” case with small
$\sin^{2}\varphi<D^{2}$ we get

$w_{\pm}={\rm
i}D\left[-\cos\varphi\pm({1-D^{-2}\sin^{2}\varphi})^{1/2}\right]\,$

so that the whole correction reflecting the influence of the Coriolis force
(i.e., of operator (24) proportional to ${\dot{\varphi}}\neq 0$) becomes
strictly purely imaginary.

Again, the two strictly non-real eigenvalues of $G_{S}$ are not mutually
conjugate. This implies that such a “Schrödinger Hamiltonian” alias  generator
of evolution of state vectors can never be required ${\cal PT}-$symmetric.

##  6 Discussion

In the overall context of our present paper it is worth inserting a
terminological remark that the widely used and popular word “non-Hermitian”
could be potentially misleading and deserves explanation: What the majority of
authors of the reviews and papers on the subject had in mind was a
mathematical formalism which just describes the conventional unitary alias
closed quantum systems in an innovative NSP or NIP representation.

###  6.1 Square-well quantum models

The idea of the more or less revolutionary replacement of conventional self-
adjoint Hamiltonians (say, $\mathfrak{h}$) by their less usual non-Hermitian
isospectral but, presumably, user-friendlier partners
$H=\Omega^{-1}\,\mathfrak{h}\,\Omega$ can be traced back to Dyson’s paper
[18]. He introduced the notion of an invertible preconditioning operator
$\Omega$ which has been allowed stationary but non-unitary. This made his
followers able to define a Hamiltonian-dependent inner-product Hilbert-space
metric [1], the knowledge of which enabled them to re-read the self-
adjointness of $\mathfrak{h}=\mathfrak{h}^{\dagger}$, formally at least, as
equivalent to the NSP (and, later, also to the NIP) quasi-Hermiticity of $H$.

In the related innovative model-building process, the simplicity of non-
Hermitian $H$ with real spectrum was essential. This was the reason why some
of the most user-friendly quasi-Hermitian generalizations of models (9) and/or
(10) were only modified “minimally”, by the mere change of boundary condition.
In this setting, the discrete-square-well dynamics controlled by certain non-
Hermitian but PT-symmetric and time-independent, stationary  boundary
conditions can be found described in our older NSP paper [17].

Among the methodically welcome features of this (i.e., still just stationary)
model we may mention its exact solvability. In certain intervals of parameters
the bound-state energies were real and given as roots of certain elementary
trigonometric expressions. Also the wave-functions were expressed, in loc.
cit., in closed form. The model has been rendered quasi-Hermitian by means of
an explicit construction of a nontrivial NSP inner-product metric $\Theta$.

From the purely methodical point of view the assumption of stationarity of the
model was essential because it enabled us to recall and apply just the
formulation of quantum mechanics of reviews [1, 8, 9]. In this context we were
able to reduce the analysis to the mere diagonalization of a non-Hermitian $N$
by $N$ matrix. The role and influence of boundary conditions were represented
by the single complex time-independent parameter. In this sense the present,
NIP-based non-stationary extension of the model of paper [17] can be perceived
as opening broad new horizons.

###  6.2 The problem of observables

In the majority of publications on hiddenly Hermitian models the authors are
accepting the assumption of stationarity of the Dyson’s map in Eq. (3). For
several good reasons: One of the most important ones is purely technical
because the assumption of stationarity implies the full formal equivalence
between “the old” Schrödinger equation (1) and “the new” NSP Schrödinger
equation (2). The self-adjointness of $\mathfrak{h}$ can be perceived as
equivalent to the quasi-Hermiticity of $H$. Thus, given a stationary non-
Hermitian $H$ with real spectrum, a key to the completion of the NSP theory
can be seen in the specification of such a self-adjoint and positive definite
operator $\Theta$ which would make the operators of such NSP observables
quasi-Hermitian [1].

A weak point of such a philosophy lies in the necessity of formulation of the
dynamical-input information in terms of the operators of observables which are
non-Hermitian. Usually, one succeeds in choosing a sufficiently interesting
non-Hermitian candidate $H$ for the energy-representing NSP Hamiltonian, say,
in its Klein-Gordon form [28], or in its Proca-field version [29, 30], etc.
Nevertheless, the resulting non-triviality of physical metric $\Theta\neq I$
becomes a source of difficulties. It implies that any other eligible
observable (say, $\Lambda$) must satisfy the hidden-Hermiticity relation with
the same  metric $\Theta=\Theta(H)$,

$\Lambda^{\dagger}\,\Theta(H)=\Theta(H)\,\Lambda\,.$ (25)

In this light it is obvious that in the future studies, more attention will
have to be paid to the most fundamental concept of the observable spatial
coordinate $\Lambda_{x}$. Indeed, the task of its construction becomes highly
nontrivial even for the stationary square-well potentials [31] or for various
even more elementary delta-function interactions [32]. Naturally, also in such
a context the present, NIP-based extension of the model-building philosophy to
the non-stationary-metric domain opens new methodical as well as
phenomenological challenges and questions. Pars pro toto, what becomes
particularly important is the role of the exact solvability as sampled by our
illustrative model, and as rendered possible by its simplified, boundary-
controlled dynamics.

##  7 Summary

In our present paper, non-stationary version of unitary quantum mechanics
formulated in non-Hermitian (or, more precisely, in hiddenly Hermitian)
interaction-picture representation was recalled and illustrated. The purpose
was served by an elementary $N$ by $N$ matrix Hamiltonian $H(t)$ mimicking a
1D-box system in which the physics is controlled by time-dependent boundary
conditions.

The model was presented as analytically solvable at $N=2$. Expressis verbis
this means that for both of the underlying Heisenberg and Schrödinger
evolution equations the generators (i.e., in our notation, the respective
operators $\Sigma(t)$ and $G(t)$) became available in closed form. In this
light, the key message delivered by our paper is that contrary to the
conventional beliefs and in spite of the unitarity of evolution of the system,
neither its “Heisenbergian Hamiltonian” $\Sigma(t)$ nor its “Schrödingerian
Hamiltonian” $G(t)$ possesses a real spectrum (or even some spectrum
containing the conjugate pairs of complex eigenvalues).

Such an observation can be perceived as being of paramount importance in the
quickly developing field of study of the role of non-self-adjoint operators in
quantum physics. One of the technically most relevant division lines
separates, in this context, the stationary from non-stationary models [11].
This observation motivated also our present paper. We imagined that at least
some of the existing solvable stationary models still wait for a non-
stationary extension.

In the older review paper [8] we read that the inner product metric “cannot
depend on time, unless …[operator $G(t)$] is not observable.” In fact, the
latter non-observability is easy to accept and, after all, fully compatible
with our present results as well as with the explicit theoretical description
of unitary quantum dynamics. Indeed, one can work, in a fully consistent
manner, with a broad class of non-observable operators $G(t)$ (cf. [12] and
also a few later confirmations and reconfirmations of this observation in
papers [33, 34, 35]).

Naturally, the transition to non-Hermitian and  time-dependent operators of
observables leads to multiple new - and not always expected - technical
obstacles. This is, in fact, the main weakness of our present non-stationary
amendment of the more common stationary models. For this reason the early
attention of researchers turned to the non-stationary models which were
exactly solvable [13, 36, 37, 38]. Only recently, the progress in our
understanding of various technical subtleties led to the more systematic
analyses and to the less schematic methodical considerations [15, 39]. Still,
the exactly solvable models keep playing a dominant role.

In parallel, suitable approximative techniques had to be developed [40, 41,
42]. Several new directions of applicability of the NIP constructive
philosophy and of its innovative modifications emerged [43, 44, 45]. Among the
most recent ones let we would like to recall paper [5] by Fring and Taira in
which the authors were able to study the time-dependent boundary conditions in
an implementation to the well known Swanson’s “benchmark” non-Hermitian
Hamiltonian [46, 47].

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