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

Given an arbitrary non-Hermitian Hamiltonian H ≠ H^† 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, ${ \mathcal L }$, 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 Λ) can only be self-adjoint in one of them (i.e. say, in ${{ \mathcal H }}_{{physical}}$). The other, non-equivalent Hilbert space (i.e. say, ${{ \mathcal H }}_{{mathematical}}={{ \mathcal H }}_{{friendly}}$) 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 ≠ H^†.

For our present purposes the two Hilbert spaces may be interpreted as complementary since ${{ \mathcal H }}_{{friendly}}={{ \mathcal H }}_{{unphysical}}$ and ${{ \mathcal H }}_{{physical}}={{ \mathcal H }}_{{unfriendly}}$. 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 ${{ \mathcal H }}_{{unfriendly}}$ appeared to be comparatively easily represented in the mathematically more suitable working space ${{ \mathcal H }}_{{friendly}}$ [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 ${ \mathcal L }$, viz., from the equation of textbooks

$$\begin{eqnarray}&&{\rm{i}}\,\displaystyle \frac{d}{dt}\,|\psi (t)\succ ={\mathfrak{h}}\,|\psi (t)\succ ,\text{}\text{}\text{}|\psi (t)\succ \in { \mathcal L },\text{}\text{}\text{}{\mathfrak{h}}={{\mathfrak{h}}}^{\dagger }\end{eqnarray} \tag{ 1 }$$

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

$$\begin{eqnarray}&&{\rm{i}}\,\displaystyle \frac{d}{dt}\,|\psi (t)\rangle =H\,|\psi (t)\rangle ,\text{}\text{}\text{}|\psi (t)\rangle \in {{ \mathcal H }}_{friendly},\text{}\text{}\text{}H\ne {H}^{\dagger }.\end{eqnarray} \tag{ 2 }$$

The price to pay for the change is that the new version of Hamiltonian (which is self-adjoint in ${{ \mathcal H }}_{{unfriendly}}$) appears non-Hermitian in the mathematical representation 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 Ω [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 ∣ψ(t) ≻ ↔ ∣ψ(t)〉 we had to keep in mind, in general, the manifest time-dependence of all of the relevant operators. Thus, using a time-dependent generalization Ω = Ω(t) of the invertible Dyson-map operator we postulate

$$\begin{eqnarray}&&| {\psi }^{}(t)\succ ={\rm{\Omega }}(t)| {\psi }^{}(t)\rangle ,\ \ \ \ {\mathfrak{h}}(t)={\rm{\Omega }}(t)H(t){{\rm{\Omega }}}^{-1}(t).\end{eqnarray} \tag{ 3 }$$

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

$$\begin{eqnarray}&&{\rm{\Sigma }}(t)={\rm{i}}\,{{\rm{\Omega }}}^{-1}(t)\dot{{\rm{\Omega }}}(t)\end{eqnarray} \tag{ 4 }$$

(where the dot over Ω(t) marks the differentiation with respect to time) the insertion of the textbook ket ∣ψ(t) ≻ = Ω(t)∣ψ(t)〉 does not convert the textbook Schrödinger equation (1) into its NSP partner (2) but rather into its modified, NIP partner

$$\begin{eqnarray}&&{\rm{i}}\,\displaystyle \frac{d}{dt}\,|\psi (t)\rangle =G(t)|\psi (t)\rangle ,\text{}\text{}\text{}|\psi (t)\rangle \in {{ \mathcal H }}_{friendly},\text{}\text{}\text{}G(t)=H(t)-{\rm{\Sigma }}(t).\end{eqnarray} \tag{ 5 }$$

In contrast to the stationary NSP models with vanishing Σ(t) = 0, we now have, in general, G(t) ≠ 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 ${{ \mathcal H }}_{{friendly}}$) does not still enable us to write down Schrödinger equation (5) and, via its solution, to reconstruct the evolution of ket-vectors $| {\psi }^{}(t)\rangle \in {{ \mathcal H }}_{{friendly}}$.

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

$$\begin{eqnarray}&&{H}^{\dagger }(t){\rm{\Theta }}(t)={\rm{\Theta }}(t)H(t).\end{eqnarray} \tag{ 6 }$$

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 ${{ \mathcal H }}_{{friendly}}$), the necessary search for inner-product metric Θ(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 Θ(t) can proceed in full analogy with the NSP recipes.

After the construction of Θ(t) one is also allowed to employ the standard operator methods and to construct the 'square root' Ω(t) of metric as well as its inverse and the time derivative as needed in equation (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 Λ(t) with real spectrum and such that Λ^†(t)Θ(t) = Θ(t)Λ(t)), the NIP predictions will be based again on the evaluation of overlaps

$$\begin{eqnarray}&&\langle \psi (t)|\,{\rm{\Theta }}(t)\,{\rm{\Lambda }}(t)|\psi (t)\rangle .\end{eqnarray} \tag{ 7 }$$

The presence of the correct physical inner-product metric Θ(t) ≠ 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 ${{ \mathcal H }}_{{physical}}={{ \mathcal H }}_{{physical}}(t)$.

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]

$$\begin{eqnarray}&&-\displaystyle \frac{{d}^{2}}{{{dx}}^{2}}{\psi }_{n}(x)={\varepsilon }_{n}{\psi }_{n}(x),\ \ {\psi }_{n}(0)={\psi }_{n}(L)=0\,\end{eqnarray} \tag{ 9 }$$

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

$$\begin{eqnarray}&&-{\psi }_{n}({x}_{k-1})+2\,{\psi }_{n}({x}_{k})-{\psi }_{n}({x}_{k+1})={E}_{n}^{(N)}{\psi }_{n}({x}_{k}),\ \ {\psi }_{n}({x}_{0})={\psi }_{n}({x}_{N+1})=0\,\end{eqnarray} \tag{ 10 }$$

where n = 0, 1, ... and where either x ∈ (0, L) or k = 1, 2,...,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

$$\begin{eqnarray}&&\psi (0)=\displaystyle \frac{{\rm{i}}\ }{\alpha +{\rm{i}}\beta }\,\displaystyle \frac{d}{{dx}}\psi (0),\ \ \ \ \ \psi (L)=\displaystyle \frac{{\rm{i}}\ }{\alpha -{\rm{i}}\beta }\,\displaystyle \frac{d}{{dx}}\psi (L)\end{eqnarray} \tag{ 11 }$$

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

$$\begin{eqnarray}&&{\psi }_{n}({x}_{0})=\displaystyle \frac{{\rm{i}}\ }{\alpha +{\rm{i}}\beta }\left(\displaystyle \frac{{\psi }_{n}({x}_{1})-{\psi }_{n}({x}_{0})}{h}\right),\ \ \ \ \ {\psi }_{n}({x}_{N+1})=\displaystyle \frac{{\rm{i}}\ }{\alpha -{\rm{i}}\beta }\left(\displaystyle \frac{{\psi }_{n}({x}_{N+1})-{\psi }_{n}({x}_{N})}{h}\right)\end{eqnarray} \tag{ 12 }$$

(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,

$$\begin{eqnarray}&&\alpha =\alpha (t),\ \ \ \beta =\beta (t).\end{eqnarray} \tag{ 13 }$$

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.

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 → 0 and N → ∞ ) 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 − β h − i α h).

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

$$\begin{eqnarray}\left[\begin{array}{ccccc}2-z & -1 & 0 & \ldots & 0\\ -1 & 2 & -1 & \ddots & \vdots \\ 0 & -1 & \ddots & \ddots & 0\\ \vdots & \ddots & \ddots & 2 & -1\\ 0 & \ldots & 0 & -1 & 2-{z}^{* }\end{array}\right]\left[\begin{array}{c}{\phi }_{1}\\ {\phi }_{2}\\ \vdots \\ {\phi }_{N-1}\\ {\phi }_{N}\end{array}\right]={E}_{n}^{(N)}\left[\begin{array}{c}{\phi }_{1}\\ {\phi }_{2}\\ \vdots \\ {\phi }_{N-1}\\ {\phi }_{N}\end{array}\right]\end{eqnarray} \tag{ 14 }$$

(cf equation 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 equation (14) acquired, strictly, the form of recurrences satisfied by Chebyshev polynomials of the first and second kind [23]. In other words, we could set

$$\begin{eqnarray}&&{\phi }_{n}=A\,{T}_{n-1}(y)+B\,{U}_{n-1}(y),\ \ \ n\,=\,1,2,\ldots ,N.\end{eqnarray} \tag{ 15 }$$

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 equation (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 Θ at any N (cf e.g. Proposition Nr. 2 in loc. cit.).

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 → 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

$$\begin{eqnarray}H=\left[\begin{array}{cc}2-i\sqrt{1-{r}^{2}} & -1\\ -1 & 2+i\sqrt{1-{r}^{2}}\end{array}\right].\end{eqnarray} \tag{ 16 }$$

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

$$\begin{eqnarray}{{\mathfrak{h}}}_{S}={{\mathfrak{h}}}_{S}(r)=\left[\begin{array}{cc}r+2 & 0\\ 0 & -r+2\end{array}\right]\,\end{eqnarray} \tag{ 17 }$$

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

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

$$\begin{eqnarray*}{{\mathfrak{h}}}_{S}(0)=\left[\begin{array}{cc}2 & 1\\ 0 & 2\end{array}\right].\end{eqnarray*}$$

in a process which is discontinuous in r.

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

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

of the sixth degree in E. Fortunately, this polynomial is just a linear function of r². Thus, the N = 6 spectrum can exactly be defined by the following closed implicit-function formula

$$\begin{eqnarray}&&{r}_{\pm }={r}_{\pm }(E)=\pm \sqrt{\displaystyle \frac{{E}^{4}-8\,{E}^{3}+20\,{E}^{2}-16\,E+3}{{E}^{4}-8\,{E}^{3}+21\,{E}^{2}-20\,E\,+\,5}}\left(E-2\right).\end{eqnarray} \tag{ 18 }$$

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.

Zoom In Zoom Out Reset image size

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 Θ 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 Θ(H) of equation (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 [13], of the ketket-vector problem

$$\begin{eqnarray}&&{H}^{\dagger }(t)|{\xi }_{n}(t)\rangle \rangle ={E}_{n}\,|{\xi }_{n}(t)\rangle \rangle ,\text{}\text{}\text{}n\,=\,1,2,\ldots ,N\,\end{eqnarray} \tag{ 19 }$$

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

$$\begin{eqnarray}&&{{\rm{\Theta }}}_{\left(\overrightarrow{\kappa }\left(t\right)\right)}(t)=\displaystyle \sum _{n}\,|{\xi }_{n}(t)\rangle \rangle \,{\kappa }_{n}(t)\langle \langle {\xi }_{n}(t)|,\,\,\,{\rm{\forall }}{\kappa }_{n}(t)\gt 0.\end{eqnarray} \tag{ 20 }$$

Different physics becomes represented by the different choices of parameters ${\kappa }_{n}^{}(t)\gt 0$. 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 ${{\rm{\Theta }}}_{(\vec{1})}(t)={{\rm{\Theta }}}_{S}(t)$.

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 Θ(t) = Ω^†(t)Ω(t). Often, this is performed via taking a self-adjoint square root of Θ(t) followed by a completion of the construction by considering also the unitary-matrix ambiguity of eligible Ω(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

$$\begin{eqnarray}&&{H}^{\dagger }(t){{\rm{\Omega }}}_{S}^{\dagger }(t)={{\rm{\Omega }}}_{S}^{\dagger }(t){{\mathfrak{h}}}_{S}(t)\end{eqnarray} \tag{ 21 }$$

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 ${ \mathcal L }\leftrightarrow {{ \mathcal H }}_{{friendly}}$ as defined in equation (3). In this sense the set of all of the ketket eigenvectors ∣ξ_n (t)〉〉 of H^†(t) (cf equation (19)) can be re-visualized as the set of separate columns of matrix ${{\rm{\Omega }}}_{S}^{\dagger }(t)$. As a consequence, we can put

$$\begin{eqnarray}&&{{\rm{\Theta }}}_{S}(t)={{\rm{\Omega }}}_{S}^{\dagger }(t){{\rm{\Omega }}}_{S}(t)\end{eqnarray} \tag{ 22 }$$

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).

During our preliminary search for the closed-form solutions of the N = 2 matrix version of Schrödinger equation (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 equation (16) they guided us to perform another change of variables setting $r=\sin \varphi$ where φ = φ(t) ≠ φ^(EP) = 0, ± π, ....

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

$$\begin{eqnarray*}\begin{array}{c}{H}^{\dagger }=\left[\begin{array}{cc}2+i\,\cos \varphi & -1\\ -1 & 2-i\,\cos \varphi \end{array}\right]\end{array}\end{eqnarray*}$$

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

$$\begin{eqnarray*}{{\rm{\Omega }}}_{S}^{\dagger }=\left[\begin{array}{cc}1 & -{\rm{i}}\,\exp \,{\rm{i}}\varphi \\ {\rm{i}}\,\exp \,{\rm{i}}\varphi & 1\end{array}\right],\ \ \ {{\rm{\Omega }}}_{S}=\left[\begin{array}{cc}1 & -{\rm{i}}\,\exp (-{\rm{i}}\varphi )\\ {\rm{i}}\,\exp (-{\rm{i}}\varphi ) & 1\end{array}\right]\,\end{eqnarray*}$$

Immediately, this yields the metric,

$$\begin{eqnarray}{{\rm{\Theta }}}_{S}={{\rm{\Omega }}}_{S}^{\dagger }\,{{\rm{\Omega }}}_{S}=\left[\begin{array}{cc}2 & -2{\rm{i}}\,\cos \varphi \\ 2{\rm{i}}\,\cos \varphi & 2\end{array}\right].\end{eqnarray} \tag{ 23 }$$

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 $\cos \varphi \to 0$.

By the direct computation we get matrices

$$\begin{eqnarray*}{{\rm{\Omega }}}_{S}^{-1}=\displaystyle \frac{1}{1-\exp (-2{\rm{i}}\varphi )}\left[\begin{array}{cc}1 & {\rm{i}}\,\exp (-{\rm{i}}\varphi )\\ -{\rm{i}}\,\exp (-{\rm{i}}\varphi ) & 1\end{array}\right],\ \ \ \ {\dot{{\rm{\Omega }}}}_{S}=\dot{\varphi }\cdot \left[\begin{array}{cc}0 & -\exp (-{\rm{i}}\varphi )\\ \exp (-{\rm{i}}\varphi ) & 0\end{array}\right]\,\end{eqnarray*}$$

as well as Coriolis force,

$$\begin{eqnarray}{{\rm{\Sigma }}}_{S}={\rm{i}}{{\rm{\Omega }}}_{S}^{-1}{\dot{{\rm{\Omega }}}}_{S}=\displaystyle \frac{\dot{\varphi }}{2\,\sin \varphi }\left[\begin{array}{cc}{\rm{i}}\,\exp (-{\rm{i}}\varphi ) & -1\\ 1 & {\rm{i}}\,\exp (-{\rm{i}}\varphi )\end{array}\right].\end{eqnarray} \tag{ 24 }$$

This result leads us to an important observation that both of the eigenvalues σ_± of Σ_S are always complex,

$$\begin{eqnarray*}&&{\sigma }_{\pm }=\left(1+{\rm{i}}\,\displaystyle \frac{\cos \varphi \pm 1}{\sin \varphi }\right)\cdot \dot{\varphi }/2=\left\{\begin{array}{c}\left(1+{\rm{i}}\,\cot \varphi /2\right)\dot{\varphi }/2\\ \left(1-{\rm{i}}\,\tan \varphi /2\right)\dot{\varphi }/2.\end{array}\right.\end{eqnarray*}$$

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

This means that there exists no operator of parity ${ \mathcal P }$ which could make the Coriolis force (i.e. the Heisenberg-equation generator alias 'Heisenberg Hamiltonian') ${ \mathcal P }T-$ symmetric.

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 equation (5) and evaluate the difference G_S (t) = H(t) − Σ_S (t),

$$\begin{eqnarray*}\begin{array}{c}{G}_{S}=\left[\begin{array}{cc}2-D\,\sin \varphi -{\rm{i}}\,B\,\cos \varphi & -A\\ -B & 2-D\,\sin \varphi +{\rm{i}}\,A\,\cos \varphi \end{array}\right].\end{array}\end{eqnarray*}$$

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

$$\begin{eqnarray*}&&{w}_{\pm }=-{\rm{i}}D\,\cos \varphi \pm \sqrt{{\sin }^{2}\varphi -{D}^{2}}.\end{eqnarray*}$$

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

$$\begin{eqnarray*}&&{w}_{\pm }=\pm \sin \varphi -{\rm{i}}D\,\cos \varphi +{ \mathcal O }({D}^{2})\end{eqnarray*}$$

yielding the two strictly non-real eigenvalues of G_S which are even not mutually conjugate,

$$\begin{eqnarray*}&&{g}_{\pm }=2\pm \sin \varphi -D\,\sin \varphi -{\rm{i}}D\,\cos \varphi +{ \mathcal O }({D}^{2}).\end{eqnarray*}$$

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

$$\begin{eqnarray*}&&{w}_{\pm }={\rm{i}}D\,\left[-\cos \varphi \pm {\left(1-{D}^{-2}{\sin }^{2}\varphi \right)}^{1/2}\right]\,\end{eqnarray*}$$

so that the whole correction reflecting the influence of the Coriolis force, i.e., of operator (24) proportional to $\dot{\varphi }\ne 0$ 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 ${ \mathcal P }T-$ symmetric.

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 $H={{\rm{\Omega }}}^{-1}\,{\mathfrak{h}}\,{\rm{\Omega }}$ can be traced back to Dyson's paper [18]. He introduced the notion of an invertible preconditioning operator Ω 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 Θ.

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.

In the majority of publications on hiddenly Hermitian models the authors are accepting the assumption of stationarity of the Dyson's map in equation (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 Θ 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 Θ ≠ I becomes a source of difficulties. It implies that any other eligible observable (say, Λ) must satisfy the hidden-Hermiticity relation with the same metric Θ = Θ(H),

$$\begin{eqnarray}&&{{\rm{\Lambda }}}^{\dagger }\,{\rm{\Theta }}(H)={\rm{\Theta }}(H){\rm{\Lambda }}.\end{eqnarray} \tag{ 25 }$$

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 Λ_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.