Emergence and Localization of Exceptional Points in an Exactly Solvable Toy Model

Abstract

In contrast to classical physics, there are not too many mathematical tools facilitating the study of singularities in quantum systems. One of the exceptions is Kato’s notion of exceptional points (EPs). Their emergence and localization are analyzed here via a family of schematic toy models.

1. Introduction

Evolutionary singularities emerging in a classical dynamical system are a phenomenon that found its appropriate mathematical clarification and qualitative classification in the framework of the popular theory of catastrophes [1]. In the majority of applications of the procedure of quantization, people have revealed that there is a deep conceptual difference between the emergence of singularities in classical and quantum systems. This has often led to the conclusion that there is no immediate quantum analog of the theory of catastrophes because the classical singularity seems always smeared out after quantization [2].

The latter belief found its particularly persuasive reconfirmation in quantum cosmology. In this field the significant progress achieved via loop quantum gravity [3] offered strong support of a replacement of the point-like Big Bang by its regularized version called the Big Bounce (cf., e.g., the comprehensive monographs [4,5] for details).

We plan to defend our belief that the situation became radically changed after the recent innovation of the formalism of quantum mechanics using non-Hermitian operators [6]. Widely, the innovated theory became known under the nicknames of quasi-Hermitian quantum mechanics [7] alias $\mathcal{PT}$-symmetric quantum mechanics [8] alias pseudo-Hermitian quantum mechanics [9] (see a compact outline of the basic ideas behind these approaches in Appendix A and, in particular, in Appendix A.1).

In the framework of the innovated theory, the apparently unavoidable nature of the regularization after quantization has been called into question [10]. It has been noticed that the disappearance or survival of singularities may be model-dependent. Thus, in particular, one has to conclude that a strictly quantum Big Bang can still be treated as a singularity-representing extreme of a conventional unitary quantum evolution [11].

In our present paper, the occurrence and role of some strictly quantum singularities will be discussed. They will be interpreted as an inseparable part of a remarkable non-Hermitian (or rather quasi-Hermitian) collapse or, in the opposite direction, of another specific process of a non-Hermitian singularity unfolding.

For the sake of definiteness, only a schematic model will be considered. It will be shown to exhibit a number of counterintuitive features. In particular, we will emphasize that a pair of some of its neighboring bound or resonant states may merge at a value of a parameter called the exceptional point (EP, cf. [12] or Appendix A.2 below).

In contrast to several rather sceptical conclusions about the model as reached in our recent contribution to conference proceedings [11], we will be able to report significant progress in our understanding of the underlying quantum dynamics. In particular, in several benchmark special cases we will be able to prove the existence of the EP singularities, which will appear localizable non-numerically.

2. The Model

The kinetic energy of a quantum particle that moves freely along an equidistant 1D lattice is represented by a discrete Laplacean [13]. In conventional textbooks such a motion is often studied as restricted to a finite segment of the lattice, with the most common Dirichlet boundary conditions imposed at its ends. The energy levels can be then found as eigenvalues of a Hermitian quantum N-by-N-matrix Hamiltonian

$$H^{\left(N\right)}=\left[\begin{array}{ccccc}2& -1& 0& \dots & 0\\ -1& 2& -1& \ddots & ⋮\\ 0& -1& \ddots & \ddots & 0\\ ⋮& \ddots & \ddots & 2& -1\\ 0& \dots & 0& -1& 2\end{array}\right].$$

(1)

The model is exactly solvable because its eigenvectors can be sought in the form of superposition of classical Tschebyshev polynomials [14].

In our older paper [15] we revealed that the latter form of solvability survives a certain generalization. The essence of the generalization consists in a transition to the boundary-controlled parameter-dependent model and Hamiltonian

$$H^{\left(N\right)}\left(z\right)=\left[\begin{array}{ccccc}2-z& -1& 0& \dots & 0\\ -1& 2& -1& \ddots & ⋮\\ 0& -1& \ddots & \ddots & 0\\ ⋮& \ddots & \ddots & 2& -1\\ 0& \dots & 0& -1& 2-z^{\ast }\end{array}\right]$$

(2)

in which the parameter itself can be complex, $z\in \mathbb{C}$: a few other related technical details can be found summarized in Appendix B below.

Surprisingly enough, even the unconventional and manifestly non-Hermitian $N-$site lattice version of such a quantum square well model with $z\notin \mathbb{R}$ (i.e., in effect, with the complex Robin boundary conditions, cf. Appendix C) can be attributed a more or less conventional physical probabilistic interpretation. Indeed, in [15] we managed to show that there exists a non-empty complex domain $\mathcal{D}$ of parameters z at which the spectrum of the model remains strictly real and non-degenerate.

The operator can serve, therefore, as an exactly solvable stationary toy model Hamiltonian fitting the conventional quantum mechanics of unitary systems in its recent quasi-Hermitian reformulation (cf. [7] and also [6,8,9,16,17]). The manifest non-Hermiticity of matrix $H^{\left(N\right)}\left(z\right)$ with complex $z\in \mathcal{D}$, reflects merely the fact that our conventional tacit acceptance of the most common $N-$dimensional Hilbert space ${\mathcal{H}}_{mathematical}^{\left(N\right)}={\mathbb{C}}^N$ is unphysical. In a way recalled in Appendix A, its necessary conversion into another, acceptable physical Hilbert space ${\mathcal{H}}_{physical}^{\left(N\right)}$ is more or less straightforward.

The goal is to be achieved via an amended inner-product metric $\mathrm{\Theta }$. The details of the underlying theory can be found explained in [7] or in Appendix B below. On these grounds one can conclude that for the evolution that is generated by a preselected non-Hermitian but quasi-Hermitian Hamiltonian $H\ne H^{†}$ with real spectrum, the unitarity can be guaranteed by the condition

$$H^{†}\mathrm{\Theta }=\mathrm{\Theta }H,$$

(3)

i.e., by Dieudonné’s [17] quasi-Hermiticity property of H.

The assignment $H\to \mathrm{\Theta }$ is not unique. In applications, the construction of one or more operators $\mathrm{\Theta }$ may represent a decisive technical challenge (see, e.g., an extensive review of this item in [9]). For the stationary model (2), therefore, such an assignment has been performed, in [15], in an explicit manner. A brute-force solution of the finite set of the $N^2$ algebraic equations (3) for the unknown matrix elements of $\mathrm{\Theta }$ has been used for this purpose.

The demonstration of feasibility of the assignment $H\to \mathrm{\Theta }$ reconfirmed the appeal of quantum mechanics in its stationary quasi-Hermitian formulation in reviews [7,9,18]. Incidentally, the practical use of the formalism becomes much more technically complicated when one omits the condition of stationarity. Still, full conceptual consistency of quantum mechanics in its non-stationary quasi-Hermitian formulation can be achieved (cf. [19,20,21,22,23,24]).

Along the latter lines, the constructions based on the non-stationary and non-Hermitian observable Hamiltonians remained difficult but still feasible. Recently, fresh developments in the field were initiated by Fring et al. [25] and, independently, by Matrasulov et al. [26]. In both of these collaborations it has been clarified that it will make good sense to extend the applications of the quasi-Hermitian quantum mechanics to the unitary quantum systems in which the quasi-Hermitian quantum observables become manifestly time-dependent.

This new optimism has also been advocated in our recent study [27], where we decided to replace the stationary solvable toy model of Equation (2) with its non-stationary generalization containing a nontrivial, non-constant complex function of time $z=z\left(t\right)$. Still, a number of questions remained unanswered (cf. the presentation of [11] during a conference from last year). And precisely this survival of open questions also motivated our present return to the model and to its upgraded analysis, with the main attention shifted to the study of its genuine quantum singularities.

3. Specific Features of Boundary-Controlled Dynamics

One of the most visible and phenomenologically most deplorable gaps in our understanding of both the stationary and non-stationary versions of model (2) can be seen in the questions concerning the existence and, if they do exist, the localization of its singularities. These questions remained unanswered in [27]. Moreover, only a very few concise answers were provided later in [11].

In the latter study, indeed, the questions concerning the genuine quantum EPs caused by boundary conditions were only addressed via several numerical illustrative examples. The reason was not only the lack of non-numerical insight but also the lack of space provided by the proceedings. Both of these shortcomings appeared decisive.

More recently we returned to the problem, and we arrived at a much better and predominantly non-numerical understanding of the role and structure of singularities. Thus, we are now going to complement the key messages of [11] and to enrich and enhance this note to a full-paper format.

3.1. The Even$-N$ Models

Due to a certain favorable hidden symmetry of matrices (2) with a purely imaginary $z\left(t\right)$, the localization of EPs appeared comparatively easy at the even matrix dimensions $N=2, 4, \dots$. After a reparametrization of $z\left(t\right)=i\sqrt{1-r^2\left(t\right)}$ we also found, in [15], that it makes sense to treat the new variable $r\left(t\right)$ as a real and, say, non-decreasing function of time.

In loc. cit. we mentioned two main consequences of the reparametrization. First, the model only proved non-Hermitian (i.e., of our methodical interest) at $r^2\left(t\right)\le 1$. Second, the spectrum of $H^{\left(N\right)}\left(t\right)$ remains smoothly time-dependent and real at all of the real parameters $r\left(t\right)\in \mathbb{R}$. Thus, the non-Hermiticity–Hermiticity quantum phase transition (cf. [28]) is smooth. Here, the phenomenon can be found illustrated in Figure 1, where we choose $N=6$.

In a subsequent commentary [11] we emphasized that at any even N the spectrum remains discrete and non-degenerate, first of all, in the standard Hermitian regime (i.e., at $r^2>1$). In contrast, in a way well visible also in Figure 1, here the loss of the manifest Hermiticity at $r^2\le 1$ has been mentioned to open the possibility of a degeneracy of a pair of energy levels in the maximal non-Hermiticity limit of $r\to 0$.

Besides a numerical demonstration of these results, and besides their graphical representations, it would be also desirable to prove them analytically. Indeed, only then one can conclude that at any even dimension N we always encounter, at $r=0$, a genuine non-Hermitian degeneracy.

3.2. Odd$-N$ Problem

There were several reasons why we failed to describe the exceptional point singularities at the odd matrix dimensions $N=3, 5, \dots$ in [11]. In what follows, these cases will appear to be tractable, first of all, thanks to a simplification of the task. Trivial as it may look, it will consist of an elementary shift of the energy scale ($E\to E-2$) so that our toy model Hamiltonian will acquire its perceivably more transparent equivalent matrix form

$$H^{\left(N\right)}\left(t\right)=\left[\begin{array}{ccccc}-z\left(t\right)& -1& 0& \dots & 0\\ -1& 0& -1& \ddots & ⋮\\ 0& -1& \ddots & \ddots & 0\\ ⋮& \ddots & \ddots & 0& -1\\ 0& \dots & 0& -1& -z^{\ast }\left(t\right)\end{array}\right].$$

(4)

This will enable us to see that at odd N there exists an anomalous bound state energy root $E=0$ of the secular equation that is $r-$independent. For this reason one can immediately deduce that there is no level-crossing EP degeneracy at $r=0$.

In [11], the latter observation forced us to add a non-vanishing real part to $z\left(t\right)$. This, unfortunately, made the secular equation so complicated that we had to resort, in the major part of paper [11], to the mere purely numerical study of our toy model (2) (see also a concise summary of these efforts in Appendix C below).

This was a technical complication that certainly limited the appeal of our results. Due to these obstacles we only managed to describe and understand the mechanism of the emergence of the EP singularity, via two illustrative pictures, just at the first nontrivial choice of dimension $N=5$. Since then, remarkable progress has been achieved in the non-numerical forms to be reported in the present paper.

4. Solvability: Sturmians

Although the spectrum of Figure 1 as assigned to matrix (2) at $N=6$ does not seem to be too complicated, we did not manage to reveal in it any traces of the symmetries as observed in the picture. In [11] we still conjectured that the formal core of the feasibility of the localization of the EPs at $N=6$ has to be seen in the user-friendly structure of the related secular equation.

In loc. cit. there was no space to make the argument explicit and to support the claim by the formulae. This is to be carried out in what follows.

4.1. Non-Numerical Localizations of EPs

The formulae of paper [11] become almost miraculously simplified after the transition to the shifted-scale model (4). This can be found demonstrated in our present

Table 1. Sturmian solutions of secular equations for the present simplified model (4).

N	${\mathit{r}}^2\left({\mathit{E}}^2\right)$
2	$E^2$
3	$E^2-1$
4	$E^2(E^2-2)/(E^2-1)$
5	$(E^4-3E^2+1)/(E^2-2)$
6	$E^2(E^2-1)(E^2-3)/(E^4-3E^2+1)$
7	$(E^6-5E^4+6E^2-1)/\left[(E^2-1)(E^2-3)\right]$
8	$E^2(E^6-6E^4+10E^2-4)/(E^6-5E^4+6E^2-1)$
9	$(E^8-7E^6+15E^4-10E^2+1)/(E^6-6E^4+10E^2-4)$

and

Table 2. Sample of further auxiliary factorizations.

N	Denominator of ${\mathit{r}}^2\left({\mathit{E}}^2\right)$
6	$E^4-3E^2+1=(E^2-1+E)(E^2-1-E)$
8	$E^6-5E^4+6E^2-1=(E^3-2E+E^2-1)(E^3-2E-E^2+1)$
9	$E^6-6E^4+10E^2-4=(E^2-2)(E^4-4E^2+2)$

. We see there, in particular, that the upgraded $N=6$ item is much more compact and transparent than its unshifted-scale predecessor of paper [11]. Also the existence of certain additional parity-symmetry-breaking sub-factorizations of Sturmian [29] couplings $r^2\left(E^2\right)$ as sampled in Table 2 appeared not only equally unexpected but also fairly useful, especially for our present purposes of the localization of the EPs (see the details below).

The manifest $E\to -E$ symmetry of the Sturmians $r^2=r^2\left(E^2\right)$ is visible also in Figure 1. This is a benefit of the model that remains visible even after the factorization of the formulae, as displayed in Table 1. Serendipitously, one also reveals another “hidden” symmetry of these results, by which, up to a certain $E^2-$factor anomaly, all of the factors found in the numerators at some N are found relocated into denominators at $N+1$.

In [11], one of our main goals was to prove that the EP degeneracy can be localized even when the matrix dimension N is odd. Unfortunately, the task remained unfulfilled. Indeed, we only managed to explain that one has to use a shifted complex form of parameter $z\left(t\right)=y\left(t\right)+\mathrm{i}\sqrt{1-r^2\left(t\right)}$ containing a non-vanishing constant or time-dependent real shift $y\left(t\right)\ne 0$.

We only found that the central level crossing as presented in Figure 1 at $N=6$ disappears at odd N. For illustration we chose $N=5$ and used a purely numerical approach—see a compact outline of the argumentation in Appendix C below. Now, these results will be complemented by their study using rigorous analytic techniques.

4.2. Example

For the purposes of our present search of the EPs, the role of the shift u in

$$z=-u+\mathrm{i}\sqrt{1-r^2}$$

(5)

is trivial at $N=2$. Its change just moves the origin of the energy scale. This means that it is sufficient to confirm the existence of the EP singularity at $u=E=0$ (cf. the first line in Table 1).

This is an elementary but explicit confirmation of the existence of a singularity tractable as Kato’s exceptional point. This is a mathematically rigorous result that is an immediate consequence of the following elementary observation and construction.

Lemma 1. 

Matrix

$$H_{\left(EP\right)}^{\left(2\right)}\left(u\right)=\left[\begin{array}{cc}u-i& -1\\ -1& u+i\end{array}\right]$$

(6)

is not diagonalizable. It can only be given the canonical Jordan form via a matrix-intertwining relation

$$H_{\left(EP\right)}^{\left(2\right)}{Q}_{\left(EP\right)}^{\left(2\right)}=Q_{\left(EP\right)}^{\left(2\right)}\left[\begin{array}{cc}u& 1\\ 0& u\end{array}\right]$$

(7)

where the invertible intertwiner

$$Q_{\left(EP\right)}^{\left(2\right)}=\left[\begin{array}{cc}-i& 1\\ -1& 0\end{array}\right]$$

(8)

is usually called a transition matrix.

5. Analytically Solvable Benchmark Models

5.1. The First Nontrivial Model: $N=3$

Even the choice of a matrix dimension as small as $N=3$ makes our insight in the spectrum perceivably worsened. The reason is that the necessary Cardano formulae yielding the energies are far from nice.

The $r\to -r$ symmetry of the spectrum (usable, after all, at any N) enables us to deduce that the deformation of the spectral curves as caused by the changes in the shift u can only lead to a degeneracy of some levels at $r=0$. Hence, it is sufficient to study the spectrum of matrix

$$H^{\left(3\right)}\left(u\right)=\left[\begin{array}{ccc}u-i& -1& 0\\ -1& 0& -1\\ 0& -1& u+i\end{array}\right]$$

(9)

i.e., the roots of its characteristic polynomial

$$P(u,E)={\mathit{E}}^3-2u{\mathit{E}}^2-\left(1-u^2\right)\mathit{E}+2u.$$

(10)

An intuitive insight into the form of the spectrum is provided, in Sturmian representation, by Figure 2. In this picture we see that a pairwise confluence of the levels can only be achieved at the minimum or maximum of the bounded part of the curve given by one of the Sturmian roots of equation $P(u,E)=0$, viz., by formula

$$u\left(E\right)={\displaystyle \frac{{\mathit{E}}^2-1+\sqrt{-{\mathit{E}}^2+1}}{\mathit{E}}}.$$

(11)

For the rigorous proof of the fact that the confluence of the energies is of Kato’s type, i.e., that it is also accompanied by the confluence of the respective eigenvectors, it is again sufficient to construct the transition matrix and/or to prove the non-diagonalizability of the Hamiltonian.

Lemma 2. 

Matrix

$$\left[\begin{array}{ccc}-1/4{\left(\sqrt{-2+2\sqrt{5}}\right)}^3& 0& 0\\ 0& 1/2\sqrt{-2+2\sqrt{5}}& 1\\ 0& 0& 1/2\sqrt{-2+2\sqrt{5}}\end{array}\right]$$

(12)

is the Jordan block representation of the toy model (9) at its right EP singularity.

Proof. 

This proof is straightforward and its short version could proceed just by insertion in the $N=3$ analog of Equation (7). What led to the result was the exact and unique specification (11) of the root of the characteristic polynomial. The (say, positive) maximum of function (11) then appeared determined by the standard rule $u^{\prime }\left(E\right)=0$, i.e., by the cubic equation for $x=E^2$,

$$(1-x){(1+x)}^2=1$$

(13)

possessing the unique positive closed-form solution

$$x=1/2\sqrt{5}-1/2\approx 0.6180339887.$$

(14)

The EP energy $E\approx 0.7861513775$ is related to the reconstructed EP-supporting shift

$$u^{\left(EP\right)}=1/2\sqrt{-2+2\sqrt{5}}-2{\displaystyle \frac{1}{\sqrt{-2+2\sqrt{5}}}}+\sqrt{4{\left(-2+2\sqrt{5}\right)}^{-1}-1}\approx 0.3002831061.$$

□

5.2. Benchmark Model with $N=4$

The same procedure can be applied to the $N=4$ toy model Hamiltonian matrix

$$H^{\left(4\right)}\left(u\right)=\left[\begin{array}{cccc}u-i& -1& 0& 0\\ -1& 0& -1& 0\\ 0& -1& 0& -1\\ 0& 0& -1& u+i\end{array}\right]$$

(15)

yielding the secular polynomial of the fourth order in the energy

$$E^4-2u{E}^3+(-2+u^2)E^2+4uE-u^2.$$

(16)

Its form is compatible with the existence of the trivial EP singularity at $E=0$ and $u=0$.

A nontrivial task can be now formulated as the question and proof of existence of the other “off-central” EP singularity or singularities at some nontrivial shift or shifts $u\ne 0$.

A non-rigorous answer is provided by Figure 3, in which we see that in close parallel with the preceding case of $N=3$, the $N=4$ Sturmian curve

$$u\left(E\right)=E+{\displaystyle \frac{\sqrt{2-{\mathit{E}}^2}-1}{{\mathit{E}}^2-1}}E$$

(17)

also has two off-central $u\ne 0$ extremes, indicating the emergence of the EPs.

For any practical purposes it is sufficient to localize the off-central EP coordinates $u^{\left(EP\right)}$ and $E^{\left(EP\right)}$ approximatively, using a suitable magnification of the graph of Figure 3 (see Figure 4 as a sample of such a magnification and graphical localization). Nevertheless, the rigorous answer is also accessible. Along the same lines as above, it can be obtained in a comparatively compact form of expression

$$E^{\left(EP\right)}=1/3\sqrt{3\sqrt[3]{26+6\sqrt{33}}-24{\displaystyle \frac{1}{\sqrt[3]{26+6\sqrt{33}}}}+6}\approx 1.138243270.$$

(18)

We can conclude that this result is fully compatible with the graphical solution as shown in Figure 4.

6. Beyond $N=4$

6.1. Odd Versus Even Matrix Dimensions N

Comparison of Figure 2 and Figure 3 reveals a certain intimate qualitative correspondence between the positions of the EP singularities at $N=3$ and $N=4$. Indeed, the information about the EPs that is carried by the function $u\left(E\right)$ at $N=3$ only differs from the information about the EPs at $N=4$ by the emergence of an additional third trivial EP at $E=0$.

It is, naturally, tempting to conjecture that such a correspondence might be extensible to any larger pair of neighboring matrix dimensions $N=2k-1$ and $N=2k$. This, really, opens the possibility of the generalization of our results to the models with $k=3, 4, \dots$ and, in principle, even with very large pairs of matrix dimensions with $k\gg 1$.

For a verification of the validity and of the possible explicit forms of such a type of conjecture, one can feel encouraged by the survival of simplicity of the corresponding general Hamiltonians at $r=0$,

$$H^{\left(N\right)}\left(u\right)=\left[\begin{array}{cccccc}u-i& -1& 0& \dots & 0& 0\\ -1& 0& -1& 0& \dots & 0\\ 0& -1& 0& \ddots & \ddots & ⋮\\ 0& 0& \ddots & \ddots & -1& 0\\ ⋮& \ddots & \ddots & -1& 0& -1\\ 0& \dots & 0& 0& -1& u+i\end{array}\right]$$

(19)

yielding the related explicit forms of the secular polynomials in a more or less routine manner (the task is left to the readers).

6.2. The $N=5$ Model Revisited

A verification of the latter conjecture has to start in the first truly nontrivial model with $k=3$ and odd $N=5$. The purely numerical analysis of the spectrum of such a “generic odd$-N$” example of our toy model (2) was performed in [11]. In light of Table 1, as well as in light of our preceding, purely analytic description of the “generic even$-N$” model with $N=4$, it is possible to expect that a certain increase in the complexity of Sturmians $r^2\left(E^2\right)$ would already enter the scene at $N=6$.

This expectation can be further supported by

Table 3. Visualization-friendly re-arrangements of some formulae of Table 1.

N	${\mathit{r}}^2\left({\mathit{E}}^2\right)$
4	$E^2-1-1/(E^2-1)$
5	$E^2-1-1/(E^2-2)$
6	$E^2-1-(E^2-1)/(E^4-3E^2+1)$
	$=E^2-1-1/(E^2-2-1/(E^2-1))$

, in which we display certain partial simplifications of the Sturmians $r^2\left(E^2\right)$ at $N=4$, $N=5$, and $N=6$. Thus, along the same methodical lines as used above, the basic orientation in the structure and parameter dependence of the $N=5$ spectrum can be obtained in full analogy with its $N=4$ predecessor.

What is to be expected is the emergence of two off-central non-Hermitian EP degeneracies at $r=0$ and at two critical shifts $u_{(\pm )}^{\left(EP\right)}=\pm \left|u_{(\pm )}^{\left(EP\right)}\right|$. This expectation is fully confirmed by the numerical experiments of [11] as well as by our new numerically generated Figure 5. Its inspection reveals that the interval of u inside which the whole $N=5$ spectrum remains real is rather small.

Outside of this interval (with the endpoints representing the two EP singularities) the $r=0$ spectrum becomes composed of three real and two complex eigenvalues. After we return to the analytic approach, we obtain the following formula for the Sturmian of relevance:

$$u\left(E\right)={\displaystyle \frac{1-3{\mathit{E}}^2+{\mathit{E}}^4-\sqrt{1-4{\mathit{E}}^2+4{\mathit{E}}^4-{\mathit{E}}^6}}{{\mathit{E}}^3-2\mathit{E}}}$$

(20)

Again, the approximate graphical search for the positions of the EPs can be based on Figure 6. The validity of the approximation published in [11] is confirmed by Figure 7, which is just the magnified version of the relevant part of Figure 6, which is, by itself, just a magnified version of the relevant part of Figure 5.

7. Beyond N = 5

After one compares, once more, Figure 2 (where $N=3$ is odd) and Figure 3 (where $N=4$ is even), one easily accepts the assumption that having now, at our disposal, the analytic as well as numerical characteristics of the $k=3$ model with $N=5$, one can hardly expect the emergence of any surprise at $N=6$. Obviously, much more exciting becomes the project of the study of the “next$-k$” model with $N=7$.

In some sense, the result of the $N=7$ calculations is truly surprising. Even though such a result could have been given here, again, in a closed and explicit analytic form (after all, this task may again be left to the readers), a much more concise and persuasive message is mediated and provided by Figure 8, in which we clearly see a decisive qualitative difference from its $N=5$ predecessor of Figure 5.

First of all, we notice that the number of EP degeneracies grew from two at $N=5$ to six at $N=7$. Secondly, from a complementary point of view, the picture clearly demonstrates that the whole spectrum remains real (i.e., that the evolution of the underlying quantum system remains unitary) not only near $u=0$ (when the real part of parameter z or of function $z\left(t\right)$ in the Hamiltonian of Equation (4) remains small) but also inside the two small intervals where the values of $u\approx \pm 0.46$ are safely non-vanishing.

What can be also considered remarkable is that our three “intervals of unitarity” are separated by the “gaps of non-unitarity” in which the spectrum ceases to be all real. An apparent paradox is clarified easily because the phenomenon just reflects the fact that the energy levels merging at the EP boundaries of the separate intervals of u are different.

In our last comment on the phenomenon, we have to add that virtually all of the later features of the $N=7$ model seem to be generic. Indeed, we draw several $k>4$ descendants of Figure 8 (where $k=4$), and we found that all that is added at $k>4$ are just the decoupled “outer observer” energy levels: at $k=6$ (i.e., at $N=11$) this is illustrated in our last Figure 9.

8. Discussion

The basic idea of our present project of the search for certain specific EP singularities was twofold. The first one was theoretical. Its essence can be seen in the admissibility of quantum models using, formally, non-Hermitian operators. This, indeed, extended the scope of the theory while opening the possibility of control of the fate of classical singularities after quantization.

On the experimental physics side, various experimental simulations have been performed recently, ranging from rather elementary coupled LRC circuits [30] and systems of ultracold atoms [31] to truly sophisticated coupled optical waveguides [32], etc. Still, in our present paper, our initial idea was purely pragmatic. Reflecting the conventional wisdom that the essence of many puzzling technical questions (emerging only during the practical implementations of abstract considerations) becomes fully clarified only when one tests the theory on a sufficiently simplified schematic toy model.

In the past, similar combinations of ambitious theoretical considerations with equally ambitious experiments and observations were accompanied by scepticism, as expressed in our brief note [11]. We worked there with several elementary illustrative examples but we only managed to describe the properties of the models using just some brute-force numerical methods.

Our insight in the problem only proved amended when we managed to unify the ideas. We realized that one of the decisive shortcomings of the current quantum theory (predicting the absence of singularities after quantization) has to be seen in the comparatively less developed techniques of working with non-Hermitian operators. We imagined that the use of non-numerical descriptions of non-Hermitian solvable models (cf., e.g., [33]) could really open the way towards a synthesis of the theory with its sufficiently transparent interpretations.

In the related literature we noticed that too many singularities emerging in classical physical systems (with their most prominent sample being Einstein’s theory of gravity and cosmology) are widely believed to disappear and get smeared out after quantization. In this sense, our main aim was a search of the models in which the solvability is combined with the existence of genuine quantum EP degeneracies.

For a long time, a key obstruction excluding the models (2) and (4) with a purely imaginary parameter z from our consideration was the absence of EPs at the odd matrix dimensions N. We found the difference between models with respective even and odd N puzzling. Fortunately, what we had in mind was just a more or less inessential difference between the respective presence and absence of the EP singularity at a central part of the energy spectrum with $E=0$. Thus, a broadening of the perspective was the key to the ultimate decisive progress and success.

A correct insight into the mechanism of the emergence of the EP singularity has been achieved via a return to the numerical tests as presented in our note [11]. This inspired us to add a non-vanishing real part to z. Thus, in our present final resolution of the puzzles as formulated in [11], we finally found a unified approach to the model at both odd and even N. We were able to conclude that irrespectively of the parity of N, the quantum singularities supported by the model have an entirely analogous structure realized via genuine quantum Kato’s EP singularities.

With the prominent example of the quantized Big Bang singularity being, presumably, too complicated for qualitative analysis at present, we restricted our attention to a much narrower problem of the emergence and construction of the non-Hermitian EP degeneracies to the most elementary boundary-controlled toy model in which it was possible to simulate the emergence and unfolding of the EP singularity by purely analytic non-numerical means.

This enabled us to conclude that the intuitive perception of the existence of a singularity can also be given a fully consistent probabilistic quantum theoretical background and interpretation. Naturally, with such a possibility being clarified on a toy model level, one has to expect that in the near future, the study of some more realistic models might open a Pandora’s box of multiple new and difficult mathematical challenges.

Among them, it is already possible to mention the currently well-known enormous sensitivity of the systems near EPs to perturbations (cf. [34,35] or a few remarks in Appendix D) as well as all of the related deeper conceptual, physical, and phenomenological questions as formulated and discussed in the related older as well as newer literature (cf., e.g., [10,32,36,37,38]).