Asymptotic Non-Hermitian Degeneracy Phenomenon and Its Exactly Solvable Simulation

Abstract

A conceptually consistent understanding is sought for the interactions sampled by the imaginary cubic oscillator with potential $V^{\left(ICO\right)}\left(x\right)=\mathrm{i}{x}^3$, which is by itself not acceptable as a meaningful quantum model due to a combination of its non-Hermiticity, unboundedness, and most of all the Riesz-basis non-diagonalizability of the Hamiltonian, known as its intrinsic exceptional point (IEP) feature. For the purposes of a perturbation-theory-based simulation of the emergence of such a singular system, a simplified (though not too strictly related) toy-model Hamiltonian is proposed. It combines an $N-$point discretization of the real line of coordinates with an ad hoc interaction in a two-parametric N-by-N-matrix Hamiltonian $H=H^{\left(N\right)}(A,B)$. After such a simplification, one can still encounter a somewhat weaker form of non-diagonalizability at the conventional Kato’s exceptional-point (EP) limit of parameters $(A,B)\to (A^{\left(EP\right)},B^{\left(EP\right)})$. The IEP-non-diagonalizability phenomenon itself appears mimicked by the less enigmatic EP degeneracy of the discrete toy model, especially at large $N\gg 1$. What we gain is that, in contrast to the IEP case, the regularization of the simplified toy model in vicinity to the black conventional EP becomes feasible.

1. Introduction

The concept of asymptotic non-Hermitian degeneracy, called the intrinsic exceptional point (IEP), was introduced by Siegl and Krejčiřík [1]. They detected the presence of this characteristic in the imaginary cubic oscillator (ICO) operator

$$H^{\left(ICO\right)}=-{\displaystyle \frac{d^2}{d{x}^2}}+\mathrm{i}{x}^3.$$

(1)

They proved that its eigenstates $|{\psi }_n^{\left(ICO\right)}\rangle$ do not form a Riesz basis, so they concluded that such an operator cannot be considered diagonalizable and that “there is no quantum-mechanical Hamiltonian associated with it” [1]. In this context we reconfirmed in [2] that the applicability of quantum theory at the non-Hermitian IEP dynamical extreme is questionable. We found that many sophisticated construction techniques which still do work for finite matrices cannot be transferred to the IEP dynamical regime.

A brief explanation is that near an N-by-N-matrix singularity with $N<\infty$, the “corridor of unitarity” (cf. [3]) becomes unacceptably (i.e., exponentially) narrow in the $N\to \infty$ limit. As a consequence, we felt forced to conclude that “the practical realizations of the standard quantum-mechanical ICO model remain … elusive” [2], and that “the currently unresolved status … of the IEP-related instabilities does not seem to have an easy resolution” [2].

In our present paper, we intend to propose another (partial) clarification of the puzzle. The not-quite-expected existence of such a new explanation of the problem has two roots. First, we imagined that regularization of the IEP singularity in the ICO model of Equation (1) and its multiple IEP-singular analogues cannot proceed directly via an immediate perturbative regularization of the IEP-singular differential operators themselves. In a “preparatory step” of our proposed treatment of the problem, we decided to mimic the relevant features of $H^{\left(ICO\right)}$ via certain limits of its discrete-coordinate N-by-N matrix alternatives. For this purpose, we are going to consider certain N-by-N-matrix Hamiltonians of the form $H^{\left(N\right)}=T^{\left(N\right)}+V^{\left(N\right)}$ in which the kinetic energy is represented by the discrete Laplacean,

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

(2)

Such an operator is Hermitian for any local and real potential $V^{\left(N\right)}=V^{\left(N\right)}\left(x\right)$. Moreover, its kinetic energy component (2) can be interpreted as an immediate discrete analogue of its continuous-coordinate partner $T=-d^2/d{x}^2$ (cf., e.g., the recent preprint [4] for details and/or further references; here we use units such that $\hslash =1$ and mass $m=1/2$).

The second root and step of our present IEP-simulation proposal was inspired by our other paper [5]. In a way proposed in this paper, Hamiltonians $H^{\left(N\right)}=T^{\left(N\right)}+V^{\left(N\right)}$ can be also perceived as the discrete-coordinate analogues of certain more general non-Hermitian differential-operator partners as sampled by the ICO model of Equation (1). The correspondence may be mediated, e.g., by the evaluations of a suitable given potential at the grid-point coordinates $x_1$, $x_2$, …, $x_N$ (for more details see Section 2 below). This may be expected to simplify the analysis. Thus, in [5] we studied these approximants, with the basic message being, from our present point of view, just a mixture of good and bad news (see a few further related comments below).

What remained encouraging was an observation that the construction of the necessary N-by-N matrix approximants might be a feasible task, especially when one admits ample use of computer-assisted symbolic manipulations. In parallel, a new discouragement emerged when we noticed that the memory-and-time costs of the construction appeared to increase so quickly with the growth of N that we were only able to test the performance up to comparatively small $N=6$. Thus, we had to conclude temporarily that the desirable “extrapolation to any hypothetical continuous-coordinate limit $N\to \infty$ does not seem to be feasible at present” [5].

Very recently, our skepticism faded away when we imagined that for a simulation of the IEP degeneracy it is in fact not necessary for us to insist on the entirely general form of the family of the discrete potentials $V^{\left(N\right)}\left(x_i\right)$, especially because the IEP degeneracy itself is merely an asymptotic higher-excitation phenomenon. Therefore, the consequences of the growth of our integer parameter N could be studied equally well using any suitable toy-model $V^{\left(N\right)}\left(x_i\right)$.

This idea immediately led to an innovated formulation of our “partial remedy” project (cf. Section 3). The essence of the innovation was that for the purposes of the IEP-degeneracy simulation during the continuous-coordinate limit $N\to \infty$, the structure of the degeneracy itself may and will be kept elementary and fixed. This led to the results which will be described in Section 4 and Section 5, with a compact summary added in Section 6.

2. Difference-Operator Hamiltonians

Although the IEP-singular ICO operator (1) cannot serve as a quantum Hamiltonian, its eigenstates form a complete set [1] so that one feels tempted to regularize the model using a suitable perturbation. Along such a path of considerations (see, e.g., [2]) one almost immediately discovers that besides the IEP-related asymptotic degeneracy (6), there emerges also another and possibly equally serious technical obstacle related to the unboundedness of the operator. This is a challenge which was already known to Dieudonné (cf. [6]). One of its resolutions was recommended and described in review [7] (cf. also [8]). As we already mentioned, a not too dissimilar trick based on the discretization of coordinates will also be used in what follows.

2.1. Discrete Large$-N$ Version of Square Well

At a fixed $N<\infty$ in Schrödinger equation

$$\left[T^{\left(N\right)}+V^{\left(N\right)}\right]|{\psi }_n\rangle =E\left(n\right)|{\psi }_n\rangle n=1,2,\dots ,$$

(3)

we may start our considerations by letting the potential, inside a finite interval of a fixed length L, vanish, $V^{\left(N\right)}=V_{squarewell}^{\left(N\right)}=0$. Then, the energy spectrum becomes strictly positive and can be written in closed form. Interested readers may find the details in the recent preprint [4]; for our present purposes, it is sufficient to recall just the ultimate energy-specifying formula Nr. 3.15 of loc. cit., viz.,

$$E^{\left(squarewell\right)}\left(n\right)={\left\{{\displaystyle \frac{sin[\pi n\lambda /L]}{\lambda }}\right\}}^2,n=1,2,\dots .$$

(4)

The real width of the infinitely deep square well is equal here to the product $L=N\lambda$ of N with the distance $\lambda$ between the grid points. This means that $\lambda \to 0$ is equivalent to $N\to \infty$.

Two aspects of the latter formula are relevant. First, the formula reproduces the continuous-coordinate square-well spectrum in the limit of $N\to \infty$ (see Figure 1). Second, the whole continuous-coordinate square-well spectrum can be perceived as a limit of the mere lower part of the respective $N<\infty$ spectrum (see Figure 2).

2.2. Inclusion of Non-Hermitian Potentials

One of the characteristic features of the square-well spectrum with $N=\infty$ is the steady growth of the energy-level differences $E_{n+1}^{\left(sqw\right)}-E_n^{\left(sqw\right)}$ with n. The inspection of the pictures reveals that at any finite $N<\infty$, such a feature is inadvertently lost beyond $n=n_{max}$ where $n_{max}=[N/2]$. Thus, once one decides to take a nontrivial $V^{\left(N\right)}\ne 0$ and study the spectra at the large $N\gg 1$, one can still safely ignore the excited states with $n>n_{max}$ as irrelevant.

For our present purposes it is important to know that all of the latter observations remain applicable especially after the inclusion of small perturbations $V^{\left(N\right)}\ne 0$. One should only keep in mind that even then, the spectrum need not be positive definite. In this sense the above-mentioned idea of the asymptotic irrelevance of the upper half of the spectrum of the discrete-coordinate toy models with finite $N<\infty$ must be properly modified.

The elusive IEP degeneracy can now be perceived as paralleled and mimicked by its finite$-N$ simulation mediated by the Kato’s degeneracy-causing exceptional points (EP, [9]) or more precisely, by the Kato’s exceptional points of order M (EPM). In this direction we already pointed out in [2] that the ICO-related singularity finds its analogue in the behavior of eigenstates $|{\psi }_n\rangle$ of a parameter-dependent non-Hermitian matrix $H^{\left(N\right)}=H^{\left(N\right)}\left(\kappa \right)$ in the vicinity of its exceptional-point singularity $H^{\left(N\right)}\left({\kappa }^{\left(EPM\right)}\right)$ at a suitable degree of degeneracy $M\ge 2$.

Some of these observations were already developed in the paper [5]. There, we revealed that a consequent and systematic analysis of the multiparametric and $\mathcal{PT}-$symmetric matrix models $H^{\left(N\right)}=T^{\left(N\right)}+V^{\left(N\right)}$ of the form depending on a real $[N/2]-$plet of parameters, viz.,

$$H^{\left(N\right)}=H^{\left(N\right)}(A,B,C,\dots )=\left[\begin{array}{ccccccc}-iA& -1& 0& \dots & & \dots & 0\\ -1& -iB& -1& 0& \dots & \dots & 0\\ 0& -1& -iC& -1& \ddots & & ⋮\\ 0& 0& \ddots & \ddots & \ddots & 0& 0\\ ⋮& & \ddots & -1& iC& -1& 0\\ 0& \dots & \dots & 0& -1& iB& -1\\ 0& \dots & & \dots & 0& -1& iA\end{array}\right]$$

(5)

ceases to be easy even in the first nontrivial three-parametric case with $N=6$ and $M=6$. On these grounds, we decided to circumvent the obstacles via a weakening of the strength of the assumptions. In place of the over-ambitious search for the EPMs “with a sufficiently large M”, we will consider a reduced task in which the number of the free parameters (determining also the maximal order M of the EPM degeneracies) will be kept fixed and restricted to the first few smallest integers.

2.3. Purely Imaginary Discrete Potentials

In any ordinary differential Schrödinger equation $H{\psi }_n=E_n{\psi }_n$ and for an arbitrary unbounded Hamiltonian $H=-d^2/d{x}^2+V\left(x\right)$, as we already pointed out, the real line of coordinates $x\in \mathbb{R}$ (or possibly its finite segment) may be replaced by a discrete and equidistant (and, say, finite) grid-point lattice $\{x_1,x_2,\dots ,x_N\}$. In such a setting, a return to the continuous-coordinate limit can be mediated by the growth of $N\to \infty$ in combination with a simultaneous decrease of the grid-point distance $\lambda \sim 1/N$.

For the special ICO-inspired class of models of our present interest, the continuous-coordinate potentials exhibiting the parity-time symmetry $H\mathcal{PT}=\mathcal{PT}H$ will be assumed purely imaginary. Then, the bounded-operator avatar of the initial Hamiltonian with such a symmetry acquires the matrix form of Equation (5) exhibiting the same antilinear symmetry.

Knowledge of the parameters might enable us to reconstruct (or better interpolate/approximate the continuous-coordinate potential $V\left(x\right)$ in the limit of large $N\to \infty$. For the sake of definiteness, we will assume that all of the constants A, B, … are real. At a fixed N, our non-Hermitian but $\mathcal{PT}$ symmetric quantum Hamiltonian will then have the N-by-N matrix form (5). Its spectrum will be real, discrete, and non-degenerate in an $N-$dependent domain $\mathcal{D}$ of parameters which may be called “physical”. This domain is not empty, since it contains a subset of the parameters which remain sufficiently small. Model (5) can be then perceived as a perturbation of the conventional square well with Hamiltonian $H_{\left(sqw\right)}^{\left(N\right)}=H^{\left(N\right)}(0,0,\dots )$ represented by a matrix which is Hermitian, $H_{\left(sqw\right)}^{\left(N\right)}={\left[H_{\left(sqw\right)}^{\left(N\right)}\right]}^{†}$.

2.4. Differential Difference Operator Correspondence

Incidentally, model (1) played the role of an important benchmark example for years in several branches of physics [10,11,12,13]. Therefore, the disproof of its probabilistic quantum-mechanical tractability was disturbing. Its unacceptability has also been reconfirmed by Günther with Stefani [14], who provided “clear complementary evidence” that models of such type “are not equivalent to Hermitian models”, mainly due to the IEP property. They called this property “non-Rieszian mode behavior”, and in a way which also inspired our present constructive considerations, they concluded that “what is still lacking is a simple physical explanation scheme for the non-Rieszian behavior of the eigenfunction sets” [14].

We found it important that the latter IEP-degeneracy property of the eigenstates $|{\psi }_n^{\left(ICO\right)}\rangle$ of $H^{\left(ICO\right)}$ can be visualized, roughly speaking, as a steady weakening of their mutual linear independence with the growth of excitation,

$$|{\psi }_n^{\left(ICO\right)}\rangle \approx |{\psi }_{n+1}^{\left(ICO\right)}\rangle ,n\gg 1.$$

(6)

A deeper insight is obtained when one recalls the notion of an exceptional point (EP) as introduced in Kato’s classical monograph on perturbation theory [9]. In spite of the fact that Kato’s attention was restricted to finite and parameter-dependent N-by-N matrices $H^{\left(N\right)}\left(\kappa \right)$, his concept of EP degeneracy can really be found analogous to its “intrinsic” asymptotic version in Equation (6).

The analogy is imperfect, of course; it only finds partial support in the fact that at a finite $N<\infty$, the EP can be also defined as an instant of parallelization of several (i.e., in general, of M) eigenvectors $|{\psi }_n^{\left(N\right)}\left(\kappa \right)\rangle$ of $H^{\left(N\right)}\left(\kappa \right)$ at $\kappa ={\kappa }^{\left(EPM\right)}$ with suitable $M\le N$. In comparison with IEP, the other formal difference is that in the EP limit (i.e., in the EPM limit) $\kappa \to {\kappa }^{\left(EPM\right)}$, the Kato’s finite$-N$ degeneracy involves not only an $M-$plet of certain properly normalized eigenvectors,

$$\underset{\kappa \to {\kappa }^{\left(EPM\right)}}{lim}|{\psi }_{m_j}^{\left(N\right)}\left(\kappa \right)\rangle =|{\psi }_{\left(EPM\right)}^{\left(N\right)}\rangle ,j=1,2,\dots ,M,$$

(7)

but also the related eigenvalues,

$$\underset{\kappa \to {\kappa }^{\left(EPM\right)}}{lim}{E}_{m_j}^{\left(N\right)}\left(\kappa \right)=E_{\left(EPM\right)}^{\left(N\right)},j=1,2,\dots ,M.$$

(8)

In [2] we found the “simplification” provided by the absence of the energy degeneracy (8) to be more than compensated in the IEP context by the technical complications arising from the “non-Rieszian mode behavior” (6). For this reason, Siegl with Krejčiřík emphasized that the IEP singularity is “much stronger than any EP associated with finite Jordan blocks” [1]. Thus, our present tentative replacement of the IEP-singular systems by their discrete EPM analogues could help us to understand at least some of the deeper mathematical roots of the rather subtle IEP-unacceptability enigma.

3. The Kato’s Exceptional Points at Finite N

3.1. The Domains of Unitarity

In our present paper we intend to study the regularization of the IEP models based on a simulation of their properties using matrices (5). Our motivation is threefold. First, we feel impressed by the empirical observation that all of these matrices share certain not-quite-expected “exact solvability” features. Second, we find it important that all of these models seem to admit a fall into an EPM singularity via smooth non-unitarity-violating passage through a strictly physical domain $\mathcal{D}$. This forces the system to evolve into a loss-of-observability collapse, i.e., towards the mathematically rather specific EPM-singularity extreme.

Third, besides the most elementary discrete-coordinate local-potential physics behind model (5), the study of the same finite-matrix form of a realistic Hamiltonian can also find motivation in a different phenomenological background. Pars pro toto, let us mention paper [15] in which Jin and Song introduced such a Hamiltonian (5) (in the special case with vanishing $B=C=\dots =0$) as a tight-binding chain model with possible applications in condensed matter physics, say, of Bloch’s electronic systems with impurities. These authors also mentioned that the same one-parametric matrices $H^{\left(N\right)}(A,0,0,\dots )$ find also another entirely different field of applicability in quantum information theory, where they describe the arrays of qubits. It is also worth adding that another immediate one-parametric generalization of the model has been proposed by Joglekar et al. [16].

In the virtually trivial one-parametric version of our model with $A\ne 0$ and with vanishing $B=C=\dots =0$, the localization of the central EP = EP2 singularity is feasible in closed form at an arbitrary finite matrix dimension N (cf. [17]). In the further, nontrivial versions of the two-parametric systems of our present interest, the localization of the physical domains $\mathcal{D}={\mathcal{D}}^{\left(N\right)}$ is no longer a trivial task. This is illustrated by Figure 3, where we see a perceivable difference between the shapes of boundaries $\partial {\mathcal{D}}^{\left(N\right)}$ at the two smallest N. The picture shows that in comparison with $N=4$, the domain of unitarity is more protruded and rotated to the left at $N=5$.

In light of the above-cited “no-go” observation that the choice of the triplet of variable parameters $A\ne 0$, $B\ne 0$, and $C\ne 0$ leading to the extreme EPM with $M=6$ already lies beyond the area of practical feasibility of the constructions at the larger matrix dimensions $N>6$ (see [5]), we are confronted with the remaining open question concerning the feasibility of the EPM constructions at arbitrary N in the two-parametric regime with $A\ne 0$ and $B\ne 0$ while $C=D=\dots =0$.

This is the question which is to be addressed and which will be affirmatively answered in what follows.

3.2. Secular Polynomials

In an overall introduction to our forthcoming localization of the non-Hermitian EPM degeneracies, we will separately consider the two-parametric N-by-N matrix Hamiltonians of Equation (5) with even and odd N. The differences are caused by the $\mathcal{PT}$-symmetry of the matrices, which enables us to evaluate the respective energy-dependent secular polynomials $P_{(A,B)}^{\left(N\right)}\left(E\right)$ in their slightly different respective forms.

3.2.1. Even $N=2K$

A brief inspection of Figure 3 makes the differences between even and odd N clearly visible. Thus, when we restrict our attention for the start to the even-dimensional models with $N=2K$, we may immediately reduce the solution of our basic eigenvalue problem (3) to searching the roots of the related secular polynomial

$$P_{(A,B)}^{\left(2K\right)}\left(E\right)=x^K+c_1(A,B)x^{K-1}+c_2(A,B)x^{K-2}+\dots +c_{K-1}(A,B)x+c_K(A,B).$$

(9)

By induction, we can then immediately prove the following result.

Lemma 1.

At even $N=2K=4,6,8,\dots$, the coefficients in (9) can be evaluated in closed form, yielding

$${(-1)}^K{c}_K(A,B)={(1+AB)}^2-A^2$$

(10)

and

$${(-1)}^K{c}_{K-1}(A,B)=B^2+K(K-1)A^2/2-(2K-1)-(K-1)(K-2){(1+AB)}^2/2,$$

(11)

etc.

Proof. 

The proof is simplified by the K-independence of Equation (10). This makes the induction step elementary, as it becomes restricted to just the single formula of Equation (11). □

Due to the $\mathcal{PT}$-symmetry, the EPM mergers of the M-plets of the energies $E_n\to E^{\left(EPM\right)}$ of interest to us here occur at the very center of the spectrum, $E^{\left(EPM\right)}=0$. This leads to the following easily-proved key conclusion.

Corollary 1.

The sufficient condition of the quadruple spectral degeneracy at even N (with $M=4$) has the form of coupled pair

$$c_{K-1}(A,B)=0,c_K(A,B)=0$$

(12)

of algebraic polynomial equations for $A=A^{\left(EPM\right)}$ and $B=B^{\left(EPM\right)}$, in which the respective polynomials are as specified in Lemma 1.

The latter corollary is slightly formal because at least some of the solutions $A=A^{\left(EPM\right)}$ and $B=B^{\left(EPM\right)}$ of Equation (12) may happen to be complex, and hence are not presently of interest. Second, in principle at least, some of the M-tuple spectral degeneracies may reflect just the presence of an exceptional point of the order smaller than M [9]. An explicit classification of these degeneracies as well as the analysis of the properties of the energies and/or wave functions near these boundaries of acceptability will have to be performed case-by-case. The process is sampled in what follows.

3.2.2. Odd $N=2K+1$

At odd $N=2K+1$, it obviously makes sense to set

$$P_{(A,B)}^{(2K+1)}\left(E\right)=E\varphi \left(E^2\right).$$

where we can still expand

$$\varphi \left(x\right)=x^K+c_1(A,B)x^{K-1}+c_2(A,B)x^{K-2}+\dots +c_{K-1}(A,B)x+c_K(A,B).$$

(13)

Then, the following result can be deduced.

Lemma 2.

At odd $N=2K+1=5,7,9,\dots$, the coefficients in (13) can be evaluated in closed form, yielding

$${(-1)}^K{c}_K=(K-1){(1+AB)}^2+2-K{A}^2$$

(14)

and

$${(-1)}^K{c}_{K-1}=(K-1)B^2+(K+1)K(K-1)A^2/6-K^2-K(K-1)(K-2){(1+AB)}^2/6,$$

(15)

etc.

Proof. 

By induction again, this becomes more complicated due to the explicit K-dependence of both the formulae. Still, no real complications emerge, since we may immediately use the tridiagonality of the matrix and inspection of the underlying secular determinant. □

Corollary 2.

The sufficient condition of the quintuple spectral degeneracy at odd N (with $M=5$) has the form of the coupled pair

$$c_{K-1}(A,B)=0,c_K(A,B)=0$$

(16)

of algebraic polynomial equations for $A=A^{\left(EPM\right)}$ and $B=B^{\left(EPM\right)}$, in which the respective polynomials are as specified in Lemma 2.

Now, we are finally prepared to move from the manifestly unphysical IEP-singular models with, schematically, $N=\infty$ to their EPM-singular analogues with a suitable (and not too small) $N<\infty$, and with just a fixed and not too large $M\le 4$ or $M\le 5$.

4. Two-Parametric Models with Arbitrary Even N = 2K

One of the beneficial consequences of the not overly complicated structure of our Hamiltonian matrices (5) is that irrespective of the parity of N, we always have to search for the squared-energy roots $x=E^2$ of a polynomial of degree K. At the same time, a deeper inspection of the problem reveals a not quite expected structural difference between the systems with the even $N=2K$ and those with the odd $N=2K+1$. In and only in the latter case, for example, there exists an odd central constant energy level $E_0=0$.

For this reason, we will study the respective two subsets of the Hamiltonians separately.

4.1. The Next-to-Elementary Hamiltonian with $N=6$

When we decide to keep our Hamiltonians just two-parametric, the most complicated three-parametric $N=6$ model of [5] becomes simplified:

$$H^{\left(6\right)}(A,B)=T^{\left(6\right)}+V^{\left(6\right)}(A,B)=\left[\begin{array}{cccccc}-iA& -1& 0& 0& 0& 0\\ -1& -iB& -1& 0& 0& 0\\ 0& -1& 0& -1& 0& 0\\ 0& 0& -1& 0& -1& 0\\ 0& 0& 0& -1& iB& -1\\ 0& 0& 0& 0& -1& iA\end{array}\right].$$

It is written here in partitioned form which emphasizes not only the absence of the third parameter, $C=0$, but also a certain enhancement of the symmetry of the matrix which encourages us to recall the $N=4$ analysis as a methodical guide.

At $N=6$, our first task is the evaluation of the secular polynomial

$$\begin{gathered} P_{(A,B)}^{\left(6\right)}\left(E\right)={\mathit{E}}^6+\left(-5+A^2+B^2\right){\mathit{E}}^4+\left(6-B^2+2BA-3A^2+B^2{A}^2\right){\mathit{E}}^2- \\ -1-2BA+A^2-B^2{A}^2. \end{gathered}$$

Its real and non-degenerate roots $E_{\pm m}\ge 0$ with $m=1,2,3$ (i.e., the sextuplet of bound-state energies) could be written in closed form. The formulae (easily generated using computer-assisted symbolic manipulations) become too long for a printed display. Still, whenever needed, their graphical presentation remains instructive and straightforward (cf., e.g., [18]).

In particular, it is important to notice and emphasize that the purely numerically determined star-shaped domain $\mathcal{D}$ of the reality of the energy spectrum at $N=6$ has been found to be similar and very close to its $N=4$ predecessor of Figure 3. Thus, although we are not going to provide a rigorous formal proof (which could be based on the increase-of-precision method of paper [19] and as such would be feasible), we believe that all of the similar numerical tests seem to confirm a hypothesis that the spikes of the boundaries of $\mathcal{D}$ really represent the non-degenerate EPM singularities of order four (at the even Ns) or five (at the odd Ns).

4.2. Arbitrary $N=2K$ and the EPMs with $M=4$

In what follows, the even matrix dimensions $N=2K$ will be allowed to be arbitrary, rendering our understanding of the behavior of the large-matrix models possible.

Theorem 1.

The degeneracy of the four central eigenvalues (i.e., of the four smallest eigenvalues) to $E^{\left(EP4\right)}(A^{\left(EP4\right)},B^{\left(EP4\right)})=0$ is encountered at any even $N=2K\ge 8$ when one considers the pair of polynomials

$$Z_{\left(2K\right)}\left(x_{+}\right)=(K-1)x_{+}^4-2(K-1)x_{+}^2+2x_{+}+1,$$

(17)

and

$$Z_{(-2K)}\left(x_{-}\right)=(K-1)x_{-}^4-2(K-1)x_{-}^2-2x_{-}+1,$$

(18)

and, making use of their exact solvability, when one determines the respective quadruplets of their roots $x_{+}=x_{+}^{\left(j\right)}$ and $x_{-}=x_{-}^{\left(j\right)}$ with $j=1,2,3,4$. Whenever one finds that these roots are real (which can be shown to be true for $K\ge 4$), the sets of the eligible EP4-supporting parameters in $H^{\left(2K\right)}$ become specified by the formulae

$$A^{\left(EP4\right)}=A_{\pm }^{\left(j\right)}=x_{\pm },B^{\left(EP4\right)}=B_{\pm }^{\left(j\right)}=(K-1)x_{\pm }^3-2(K-1)x_{\pm }\pm 1,j=1,2,3,4.$$

Proof. 

At any $N=2K$, the coupled pair of Equation (16) can immediately be solved using the elimination of the unknown quantity B from the second item of Equation (16). This can be done by taking the two alternative (viz., positive or negative) square roots of the latter constraint, which may be characterized by the subscripted sign to yield two options, viz.,

$$B=B_{\pm }=\pm 1-1/A.$$

After the elimination and insertion of $B_{\pm }=B_{\pm }\left(A\right)$ into the first item of Equation (16), we get the two respective polynomials (17), (18) and the two respective relations $Z_{(\pm 2K)}\left(x_{\pm }\right)=0$. The quadruple degeneracy of the eigenvalues at $E^{\left(EP4\right)}=0$ is guaranteed. □

A detailed classification based on the explicit proofs of the reality of the roots $x_{\pm }$ of the respective polynomials $Z_{(\pm 2K)}\left(x_{\pm }\right)$ remains K-dependent (i.e., N-dependent). Even though the polynomial is merely of the fourth order in x (i.e., exactly solvable), by far the most straightforward insight in the K- or N-dependence of the roots is provided graphically. In Figure 4, we display the six shapes of polynomial functions $Z\left(x\right)=Z_{(-2K)}\left(x\right)$ with $K=2,3,4,5$, and 6. As long as both of the local minima of these curves decrease with the growth of K, the picture clearly shows the convergence of the roots $x=A_{-}^{\left(EP4\right)}$ in the limit of $K\to \infty$. Numerically, this convergence is also confirmed by

Table 1. Eligible EP4-supporting real roots $A=A_{-}^{\left(EP4\right)}$ of polynomials $Z_{(-2K)}\left(x\right)$ and their $K\to \infty$ convergence (incomplete illustrative list; see also Equation (18) and Figure 4). For completion of the list, only the opposite-sign values $A_{+}^{\left(EP4\right)}=-A_{-}^{\left(EP4\right)}$ would have to be added.

K
2	−1.683771565	−0.3715069740	-	-
3	−1.560602400	−0.3133098559	-	-
4	−1.514868938	−0.2776482755	0.7925172140	1.000000000
5	−1.490937129	−0.2527091086	0.5611034016	1.182542836
6	−1.476207086	−0.2339114273	0.4652866201	1.244831894
7	−1.466224803	−0.2190384827	0.4055099224	1.279753364
∞	−$\sqrt{2}$	0	0	$\sqrt{2}$

.

The values of $B=B_{-}^{\left(EP4\right)}=-1-1/A_{-}^{\left(EP4\right)}$ are unique. For completeness, the table could also have been extended to contain the other sets of roots with opposite sign and subscript, viz., with $A_{+}^{\left(EP4\right)}=-A_{-}^{\left(EP4\right)}$ as well as with the related $B_{+}^{\left(EP4\right)}=1+1/A_{+}^{\left(EP4\right)}$.

4.3. Numerical Detour

Table 1 together with Figure 4 can be recalled as a sample of the use of a solvable model admitting an internally consistent extrapolation of an EPM-supporting quantum model to the large $N\gg 1$ or even towards the limit of $N=\infty$. This transition can be read as a tentative interpolation, approximation, or even just a mere simulation of a singular differential operator, with the idea of its possible (though not yet sufficiently well understood) regularization realized via a return to the simplified, non-EPM (i.e., diagonalizable and, as quantum Hamiltonians, acceptable) discrete perturbed-EPM $N<\infty$ models.

In light of Refs. [3] or [20], for instance, at least a few highly-excited eigenvectors of the $N\gg 1$ model tend to overlap at the EPM singularity. in this manner, they mimic the behavior of the deeply singular and manifestly unacceptable IEP-singular model as well as that of its eigenvectors, with their asymptotically-high-excitation degeneracy being characterized by Equation (6) above. Such an argument, even still just intuitive as it is at present, might explain the asymptotic (i.e., high-energy) parallelization of eigenvectors via its EPM-mimicked reinterpretation. Simultaneously, we also circumvent the very essence of the quantum-theoretical unacceptability of the IEP-singular operators as sampled by $H^{\left(ICO\right)}$, due to the finite-dimensional form of the EPM models. According to the dedicated perturbation theory as outlined in [3], the reason lies in the existence of a small-perturbation-induced and quasi-unitarity-compatible [7] regularization of the EPM singularity in any $N<\infty$ model. For example, in light of Figure 4 one can expect that at large $N=2K\gg 1$ the quickest asymptotic $K\to \infty$ convergence will be encountered in the case of the leftmost EP4 root $A=A_{-}^{\left(EP4\right)}\approx -\sqrt{2}$, while the slowest asymptotic $K\to \infty$ convergence can be expected to occur for the two central roots.

Therefore, for illustrative purposes we will pick up the value “in between” (i.e., the rightmost root $A_{+}^{\left(EP4\right)}\approx +\sqrt{2}$ of $Z_{(-N)}\left(x\right)$ alias), due to the left-right symmetry, the leftmost root $A_{-}^{\left(EP4\right)}\approx -\sqrt{2}$ of the other, plus the subscripted polynomial

$$Z=x^4-2x^2+\left(2x+1\right)g,$$

where we have changed the normalization factor and introduced a new asymptotically small parameter $g=1/\left(K-1\right)$.

We will always have $g\le 1$, so that for a purely numerical localization of any one of the four EP4 roots we may try to use an asymptotic-series ansatz

$$x=x\left(K\right)=-\sqrt{2}+c_{1}g+c_2{g}^2+c_3{g}^3+\dots .$$

(19)

As long as the very choice of our ansatz has been based on our knowledge of the solution of equation $Z=0$ in the limit of small $g\to 0$ (i.e., of large $K\to \infty$; cf. the last line of Table 1), the zero-order $\mathcal{O}\left(g^0\right)$ form of equation $Z=0$ is an identity. The next first-order $\mathcal{O}\left(g\right)$ component can be read as an explicit definition of coefficient

$$c_1=-\left(4-\sqrt{2}\right)/8\approx -0.3232233045.$$

Similarly, the second-order $\mathcal{O}\left(g^2\right)$ constraint leads to coefficient

$$c_2=\left(29\sqrt{2}-32\right)/128\approx 0.07040776030,$$

while on the third-order level of precision $\mathcal{O}\left(g^3\right)$ we get

$$c_3=-7\left(64-43\sqrt{2}\right)/1024\approx -0.02179855231,$$

etc.

In this context, we have to emphasize that although equation $Z=0$ is solvable in closed form, it still makes sense to construct and work with our “redundant” asymptotic expansion (19), becaus (a) all of its coefficients can be given a closed exact form and (b) even after its drastic truncation, the expansion yields a fairly reasonable numerical precision even when $N=4$ and $g=1$ is not too small (see

Table 2. Sample of convergence of the asymptotic-expansion approximants of the leftmost root of polynomial $Z_{(+N)}\left(x\right)$ at the smallest $N=2K$ (due to symmetry, we could just copy the last column from Table 1).

N	First Order	Second Order	Third Order	Exact
14	−1.466128	−1.46808	−1.4662292	−1.466224803
12	−1.47604	−1.4788	−1.476216	−1.476207086
10	−1.49062	−1.4950	−1.490959	−1.490937129
8	−1.51413	−1.5219	−1.51494	−1.514868938
6	−1.5582	−1.5758	−1.56095	−1.560602400
4	−1.6670	−1.737	−1.6888	−1.683771565

).

5. Two-Parametric Models with Odd N = 2K + 1

5.1. Algebraic Constructions at Small K

In [5], we thoroughly described the odd-N models with $K=1$ (i.e., the one-parametric case; see Section IV of loc. cit.) and with $K=2$ (which is the simplest two-parametric case yielding EPM with $M=5$; see subsection V. C in loc. cit.). This allows us to start directly from the odd-N system with $K=3$.

5.1.1. $K=3$ and $N=7$

The Hamiltonian

$$H^{\left(7\right)}(A,B)=\left[\begin{array}{ccccccc}-iA& -1& 0& 0& 0& 0& 0\\ -1& -iB& -1& 0& 0& 0& 0\\ 0& -1& 0& -1& 0& 0& 0\\ 0& 0& -1& 0& -1& 0& 0\\ 0& 0& 0& -1& 0& -1& 0\\ 0& 0& 0& 0& -1& iB& -1\\ 0& 0& 0& 0& 0& -1& iA\end{array}\right]$$

is an eligible candidate for an energy-representing observable in quasi-Hermitian quantum theory [7] for us only after we manage to specify the physical domain $\mathcal{D}$ of its real parameters A and B for which the spectrum remains real and non-degenerate.

This task requires evaluation of the secular polynomial

$$\begin{gathered} P_{(A,B)}^{\left(7\right)}\left(E\right)={\mathit{E}}^7-\left(6-A^2-B^2\right){\mathit{E}}^5-\left(-10+2B^2-2BA+4A^2-B^2{A}^2\right){\mathit{E}}^3- \\ -\left(4+4BA-3A^2+2B^2{A}^2\right)\mathit{E}. \end{gathered}$$

Thus, one obvious and constant central-energy root $E_0=0$ is accompanied by the sextuplet $E_{\pm m}$ with $m=1,2,3$. In closed form, these energies can be expressed using the well-known Cardano formulae. Although these formulae are already too long for display in print, they may be stored in the computer so that any form of the graphical or numerical representation of the energies still remains an entirely routine task. In the same sense, it is also entirely straightforward to find the shape of the physical parametric domain $\mathcal{D}$, most easily by use of the requirement of reality and non-negativity of the squares of the eigenvalues $E_{\pm m}^2$ at $m=1,2$, and 3. What one obtains is just a slightly modified analogue of Figure 3.

5.1.2. $K=4$ and $N=9$

The same techniques apply to the $N=9$ model yielding the secular polynomial

$$\begin{gathered} P_{(A,B)}^{\left(9\right)}\left(E\right)={\mathit{E}}^9-\left(8-A^2-B^2\right){\mathit{E}}^7-\left(-21+4B^2-2BA+6A^2-B^2{A}^2\right){\mathit{E}}^5- \\ -\left(20-3B^2+8BA-10A^2+4B^2{A}^2\right){\mathit{E}}^3-\left(-5+4A^2-6BA-3B^2{A}^2\right)\mathit{E}. \end{gathered}$$

Again, use of the (still-existing) closed formulae would be impractical. For the same reason, determination of the boundaries of the physical parametric domain $\mathcal{D}$ would be a purely numerical task.

For the purposes of the description of the energy-level mergers, fortunately one only has to know the boundaries of $\mathcal{D}$ in a small vicinity of the EPM singularity. This makes the use of approximate methods both sufficient and efficient.

5.2. Graphical and Numerical Constructions

We have noticed that an enormous growth in the complexity of the formulae already emerges at $N=2K+1$ with K as small as three. In contrast to the above-described elementary elimination of B from Equation (10) at an arbitrary even $N=2K$, the odd-N version of constraint $c_K(A,B)=0$ ceases to offer a sufficiently elementary elimination of one of the parameters. A full-fledged computer-assisted elimination technique must be used to solve the system of two coupled polynomial algebraic equations in (16).

Still, we can rely upon the computer-generated formulae. What is encouraging is that if one tries to formulate an odd-N analogue of Theorem 1, one still arrives at certain strict analogues of the respective auxiliary polynomials (17) and (18) which have to vanish at the EPM singularity. After an abbreviation $B^2=y$, we get the rule $Z_{(2K+1)}\left(y^{\left(EPM\right)}\right)=0$ with

$$Z_{\left(5\right)}\left(y\right)=y^4-12y^3+50y^2-76y+25,$$

(20)

$$Z_{\left(7\right)}\left(y\right)=4y^4-44y^3+169y^2-234y+49,$$

(21)

$$Z_{\left(9\right)}\left(y\right)=81y^4-936y^3+3748y^2-5360y+900,$$

(22)

etc. The shapes of the latter three curves are displayed in Figure 5, with the respective roots listed in 

Table 3. Numerical zeros of functions $Z=Z_{\left(N\right)}\left(B^2\right)$ of Figure 5.

N	$\mathit{B}={\mathit{B}}^{\left(\mathit{EP}5\right)}$
5	0.6683178062	1.607208567
7	0.5024794009	1.686679121
9	0.4388960232	1.818687904

.

In contrast to the left–right asymmetry of the curves at $N=2K$, their odd-N analogues are left–right symmetric (this is why only their right semi-axis halves are displayed in Figure 5). This implies that the roots of the new auxiliary polynomials $Z_{(2K+1)}\left(y\right)$ now define the eligible EPM parameters as the pairs of square roots $B=B_{\pm }^{\left(EP5\right)}=\pm \sqrt{y}$.

The good news is that the EPM construction is reduced to the search for roots of a polynomial of the fourth order in y. This means that these roots (cf. Table 3) as well as the further EP5 parameters are exact.

A few less pleasant complications emerge at larger K. In contrast to the even-N constructions, the higher-K polynomials $Z_{(2K+1)}\left(y\right)$ appear to be extrapolation-unfriendly, and they do not look sufficiently elementary anymore. Moreover, their optimal forms must be computer-generated at every separate value of K.

The task becomes more and more difficult with the growth of K. The computer-generated polynomials $Z_{\left(N\right)}\left(y\right)$ with $N=11$ and $N=13$ cease to look nice (so we do not display them here in print). Even the rather routine computer-assisted numerical localization of their EP5 roots becomes more costly in both of its graphical forms, as sampled in Figure 6, as well as in its purely numerical forms sampled in

Table 4. Numerical zeros of the two functions of Figure 6.

N	$\mathit{A}={\mathit{A}}^{\left(\mathit{EP}5\right)}$
11	0.8244776746	1.605982629
13	0.9057668676	1.574311228

.

Fortunately, all of these calculations are still routine. One can even decide to move beyond the scope of the present paper, trying to perform computer-assisted calculations at larger N and M. Interested readers can find a few technical comments on such a possible future project in Appendix A.

6. Summary

Detailed analysis of a family of comparatively elementary bound-state models enabled us to reveal and describe an intimate correspondence between the asymptotic behavior of certain non-Hermitian potentials $V\left(x\right)$ and a non-Hermiticity-related EP degeneracy of the spectra. As one byproduct, this led to a deeper understanding of the so-called intrinsic exceptional point (IEP) feature of certain maximally non-Hermitian but still $\mathcal{PT}$-symmetry-exhibiting operators.

IEP-singular behavior has recently been noticed to characterize a fairly broad family of ill-conceived candidates for quantum Hamiltonians. Even though such a property (i.e., basically the Riesz-basis loss of diagonalizability) makes every such operator unacceptable as a quantum observable, we proposed a partial remedy which lies in a perturbation-theory based weakening, if not removal, of its singular nature.

The regularization process has three parts. In the first, we follow theory and circumvent the unbounded-operator status of $H^{\left(IEP\right)}$ [7]. In our present paper, this goal has been achieved by discretization of the Schrödinger equation, thanks to which we can reinterpret operator $H^{\left(IEP\right)}$ as an $N\to \infty$ limit of a sequence of certain N-by-N matrices $H^{\left(N\right)}$.

In the second step we have made use of the elementary matrix nature of every $H^{\left(N\right)}$, which we reinterpret and generalize as parameter-dependent, $H^{\left(N\right)}\to H^{\left(N\right)}\left(g\right)$, admitting that the symbol g may represent an arbitrary multiplet of auxiliary variables, $g=\{A,B,\dots \}$. Even though our new matrices $H^{\left(N\right)}\left(g\right)$ forming a broader family are non-Hermitian by construction, their spectra should remain real and non-degenerate: This restricts the variability of g to a certain non-empty physical domain $\mathcal{D}$.

In the third step, we recall the fact that the boundary of the latter physical parametric domain is formed by the EP values at which one encounters the mergers of energy levels. Localization of these degeneracies appeared to be the most difficult part of our regularization recipe. Still, in our models we managed to localize the most relevant part of the EP boundary $\partial \mathcal{D}$ at which the number M of the merging energy levels was maximal.

The latter maxima appeared to be reached at the isolated EPs of order M (EPMs), and for the energies forming, at a given matrix-dimension $N<\infty$, a precise center of the spectrum. In other words, the key point was that at every separate N and in spite of the singular nature of every auxiliary operator $H^{\left(N\right)}\left(g\right)$ in the $N-$ and $M-$dependent EP limit of $g\to g^{\left(EPM\right(N\left)\right)}$, the physical unitary-evolution compatibility of every $H^{\left(N\right)}\left(g\right)$ becomes restored for $g\in \mathcal{D}$ in an arbitrarily small vicinity of $g^{\left(EPM\right)}\in \partial \mathcal{D}$.

On these grounds, becomes possible to conclude that in spite of the strictly unphysical nature of our preselected operator $H^{\left(IEP\right)}$, its phenomenologically acceptable vicinity could be understood as operators obtained as an $N\to \infty$ limit of their phenomenologically acceptable $N<\infty$ predecessors $H^{\left(N\right)}\left(g\right)$, provided only that one would be able to construct such a limit.

In our present paper, we have managed to perform such a constructive regularization, using a subset of matrices $H^{\left(N\right)}\left(g\right)$ which varied with two-parametric $g=\{A,B\}$. As a consequence, the finite-N EPM-related central-spectral parallelization of the middle-of-spectrum eigenstates of $H^{\left(N\right)}\left(g^{\left(EPM\right)}\right)$ appear successfully tractable as mimicking the IEP-related asymptotic parallelization in the continuous-coordinate limit of $N\to \infty$, i.e., as mimicking one of the most relevant characteristics of the IEP degeneracy singularity.