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

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 Hfriendly is required to coexist with the unitarity of system in another, ad hoc Hilbert space Hphysical . Thus, quantum mechanics in its non-Hermitian interaction picture (NIP) representation is to be used. We must construct the time-dependent states (say, ψ(t)) as well as the time-dependent observables (say, Λ(t)). Their evolution in time is generated by the operators denoted, here, by the respective symbols G(t) (a Schrödinger-equation generator) and Σ(t) (a Heisenberg-equation generator, a.k.a. quantum Coriolis force). The unitarity of evolution in Hphysical is then guaranteed by the reality of spectrum of the energy observable alias Hamiltonian H(t) = G(t) + Σ(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 Θ(N=2)(t), for the related Dyson maps Ω(N=2)(t), for the Coriolis force Σ(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).


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 η−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 H physical ) with mathematics (set in another space 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 H physical = H unphysical , cf. also several more rigorous and updated reviews in [11]) one always has to keep in mind that any operator Λ representing an observable is, by definition, simultaneously self-adjoint in the hypothetical "correct space" H physical and non-Hermitian in the friendlier, "unphysical" but preferred construction space H mathematical .
In practice it appeared convenient to work just in H unphysical = H.Using the common notation convention one stays in H, writes Λ = Λ † and introduces the so called inner-product metric Θ in order to guarantee the observability status of Λ via the Dieudonné's [7] quasi-Hermiticity postulate Λ † Θ = Θ Λ (cf.also [1] for details).In such a setting one treats H physical as represented in H unphysical while one only amends the inner product, 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, Λ (N SP ) = Λ (N SP ) (t).An ultimate remedy and clarification of the misunder-standing 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 Θ N IP = Θ N IP (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.

Stationary cases and NSP physical Hilbert spaces
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, 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, nonequivalent 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 H physical ).The other, non-equivalent Hilbert space (i.e., say, H mathematical = H f riendly ) 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 H f riendly = H unphysical and H physical = H unf riendly .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 H unf riendly appeared to be comparatively easily represented in the mathematically more suitable working space H f riendly [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 L, viz., from the equation of textbooks to its alternative living in The price to pay for the change is that the new version of Hamiltonian (which is self-adjoint in H unf riendly ) appears non-Hermitian in the mathematical representation space.

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 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 Due to the emergence of a non-vanishing Coriolis-force operator (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 Eq. ( 1) into its NSP partner (2) but rather into its modified, NIP partner 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 h(t) (or, more precisely, of its non-Hermitian isospectral image H(t) defined as acting in H f riendly ) does not still enable us to write down Schrödinger Eq. ( 5) and, via its solution, to reconstruct the evolution of ket-vectors |ψ(t) ∈ H f riendly .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 h(t) in L via the time-dependent generalization of the Dieudonné's quasi-Hermiticity property of its isospectral observable-Hamiltonian avatar 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 H f riendly ), 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 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 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 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 nonstationary metric, manifestly time-dependent form of Hilbert space H physical = 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 was strongly encouraged by the dynamical input knowledge status of the boundary-value matrix elements z(t) ∈ C of our potentially observable time-dependent quantum Hamiltonian.
Our mathematically motivated choice of such an N by N matrix model with N < ∞ will also decisively facilitate the physics-oriented construction of the predictions of measurements (7).

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] and/or by its numerically motivated difference-equation equidistant-lattice analogue [21] 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 Hermiticityviolating two-parametric (i.e., Robin-boundary-condition) alternatives (in (9), with two free real parameters α , β ∈ R) or (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, The latter, innocent-looking generalization leads to a number of nontrivial constructive tasks.
In the forthcoming, methodically sufficiently instructive analysis only the difference Schrödingerequation model will be considered.

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 → 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 − iα h).
Once we rewrote the corresponding stationary Schrödinger equation in its equivalent N by N matrix form in H f riendly , (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 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, HP T = P T H. 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.).

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 → 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 ∈ C gets replaced by a more specific real variable.An amended insight emerged when we replaced the value of z by its redefinition i √ 1 − r 2 using a real variable r and yielding A key merit of such a (still, stationary) reduction appeared to lie not only in the extremely elementary form of spectrum E (2) ± = 2 ± 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 remains restricted to r = r (EP ) = 0.Alternatively, the latter matrix becomes tractable also as the special diagonal self-adjoint textbook Hamiltonian in 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 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 (6) n seems to be a purely numerical task because the secular determinant det(H − E) is equal to polynomial 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 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.

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.

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 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 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 then all of the eligible (i.e., invertible and positive definite [1]) inner-product metrics form an N−parametric family, Different physics becomes represented by the different choices of parameters κ n (t) > 0. In what follows, for the sake of simplicity, we will work just with the trivial choice of κ (t) = 1 and we will speak about a "special" alias "standard" choice of Θ ( 1) (t) = Θ S (t).

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 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 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 where the matrix of eigenvalues h S (t) is diagonal.Relation ( 21) can be interpreted as a specific realization of correspondence between Hilbert spaces L ↔ H f riendly as defined in Eq. ( 3).In this sense the set of all of the ketket eigenvectors |ξ n (t) of H † (t) (cf.Eq. ( 19)) can be re-visualized as the set of separate columns of matrix Ω † S (t).As a consequence, we can put Θ S (t) = Ω † S (t) Ω 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 κ(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) − Σ(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 Σ(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.

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 ϕ 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 we got not only the above-mentioned spectrum E ± = 2 + sin ϕ (where one can simulate ± by sign ϕ 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, Immediately, this yields the metric, 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 θ ± = 2 ± 2 cos ϕ as well as by its diagonality and proportionality to a unit operator in the Hermitian-Hamiltonian limit of cos ϕ → 0.

The second task: closed-form Coriolis force
By the direct computation we get matrices as well as Coriolis force, This result leads us to an important observation that both of the eigenvalues σ ± of Σ S are always complex, Moreover, these two eigenvalues do not even form a complex conjugate doublet.This means that there exists no operator of parity P which could make the Coriolis force (i.e., the Heisenberg-equation generator alias "Heisenberg Hamiltonian") PT −symmetric.

The ultimate task: closed-form Schrödinger equation
Once we abbreviate D = φ(t)/(2 sin ϕ(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) − Σ S (t), Its eigenvalues are available in compact form g ± = 2 − D sin ϕ + w ± with In the "almost stationary" dynamical regime with small φ such that D 2 < sin 2 ϕ we get yielding the two strictly non-real eigenvalues of G S which are even not mutually conjugate, In the opposite, "strongly non-stationary" case with small sin 2 ϕ < D 2 we get so that the whole correction reflecting the influence of the Coriolis force (i.e., of operator (24) proportional to φ = 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 PT −symmetric.

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.

Square-well quantum models
The idea of the more or less revolutionary replacement of conventional self-adjoint Hamiltonians (say, h) by their less usual non-Hermitian isospectral but, presumably, user-friendlier partners H = Ω −1 h Ω 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 h = h † , 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 nonstationary extension of the model of paper [17] can be perceived as opening broad new horizons.

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 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 dynamicalinput 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), 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.

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 Σ(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" Σ(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].

Figure 1 :
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) = i √ 1 − r 2 .