The Intrinsic Exceptional Point: A Challenge in Quantum Theory

Abstract

In spite of its unbroken PT symmetry, the popular imaginary cubic oscillator Hamiltonian $H^{\left(IC\right)}=p^2+\mathrm{i}{x}^3$ does not satisfy all of the necessary postulates of quantum mechanics. This failure is due to the “intrinsic exceptional point” (IEP) features of $H^{\left(IC\right)}$ and, in particular, to the phenomenon of a high-energy asymptotic parallelization of its bound-state-mimicking eigenvectors. In this paper, it is argued that the operator $H^{\left(IC\right)}$ (and the like) can only be interpreted as a manifestly unphysical, singular IEP limit of a hypothetical one-parametric family of certain standard quantum Hamiltonians. For explanation, ample use is made of perturbation theory and of multiple analogies between IEPs and conventional Kato’s exceptional points.

1. Introduction

The concept of the so-called “intrinsic exceptional point” (IEP) was introduced by Siegl and Krejčiřík who, in their paper [1], studied the “prominent” imaginary cubic (IC) Schrödinger equation:

$$H^{\left(IC\right)}|{\psi }_n^{\left(IC\right)}\rangle =E_n^{\left(IC\right)}|{\psi }_n^{\left(IC\right)}\rangle ,n=0,1,\dots ,H^{\left(IC\right)}=-{\displaystyle \frac{d^2}{d{x}^2}}+\mathrm{i}{x}^3.$$

(1)

They felt motivated by the instability of the IC spectrum under perturbations [2]. They were able to complement such a numerically supported observation by several rigorous mathematical proofs (cf. also [3]). They found that “the eigenvectors of the imaginary cubic oscillator do not form a Riesz basis” [1]. In spite of having a spectrum that is real, discrete and bounded below [4,5,6], the manifestly non-Hermitian IC Hamiltonian appeared not to be, in the Riesz-basis sense, diagonalizable.

Siegl, with Krejčiřík, concluded that “there is no quantum-mechanical Hamiltonian associated with it” [1]. The same authors also recalled the standard mathematical terminology, and they reformulated their conclusion: “In the language of exceptional points, the imaginary cubic oscillator possesses an ‘intrinsic exceptional point’” which is, as a singularity, “much stronger than any exceptional point associated with finite Jordan blocks” [1].

These words are truly challenging, having also motivated our study of the role of IEPs in the deepest conceptual foundations of contemporary quantum physics. It makes sense to add that Siegl, with Krejčiřík, only introduced the concept via the above-cited remark, i.e., without giving a formal definition. They specified IEP as an $N=\infty$ descendant of the conventional exceptional point of order N (EPN, [7]). In this sense, the linear IEP differential operator $H^{\left(IC\right)}$ is really “essentially different with respect to self-adjoint Hamiltonians” [1].

The problem of the interpretation of all of the non-standard, IEP-related quantum bound-state problems remains, at present, open. In our contribution to the currently running discussion of this topic (cf. also [8]), we will study and describe, more deeply, the parallels, as well as the differences, between the two (viz., the IEP and EPN) concepts.

We will start, in Section 2, with a brief account of what is known about the linear-algebraic EPN analogs of the ordinary differential IEP Equation (1). We will consider a class of Hamiltonians (depending on a real or complex parameter g) that admit a singular EPN limit when $g\to g^{\left(EPN\right)}$. We will recall a few recent results of the studies of this problem, in which a suitable parameter-dependent N-by-N matrix quantum Hamiltonian $H^{\left(N\right)}\left(g\right)$ is considered at a finite $N<\infty$. We will emphasize the possibility and relevance of its canonical representation by an N-by-N matrix Jordan block when $g\to g^{\left(EPN\right)}$.

In the latter limit, operator $H^{\left(N\right)}\left(g\right)$ ceases to be diagonalizable and, hence, it ceases to be acceptable as an eligible quantum Hamiltonian. In Section 3, we will emphasize that many of its properties become particularly reminiscent of the IEP features of the differential operator model (1), where the corresponding Hilbert space of states is infinite-dimensional, $N=\infty$. We will remind the readers of the existing results concerning physical meaning and impact of the EPN-related finite-dimensional models. We will explain that, in many (often called “quasi-Hermitian” [9,10]) quantum models of such a type, the limiting transition $g\to g^{\left(EPN\right)}$ can be interpreted as one of the most natural realizations of a genuine quantum phase transition (cf., e.g., the description of a class of exactly solvable models of such a process in [11]).

In Section 4, the emphasis will be shifted to the $N\to \infty$ scenarios and to the existence of several very useful analogies between both the IEP and EPN singular extremes. We will point out that, in such a comparison, the key role of a methodical guide may be expected to be played by (and a possibly amended) perturbation theory. Interested readers will be recommended to find a phenomenologically oriented inspiration, as well as many related technical details, in an older paper [12]. The authors in that paper studied a fairly realistic non-Hermitian Hamiltonian, describing an N-particle Bose–Einstein condensate that is generated by a sink and a source in interaction. Using a combination of several complementary numerical, as well as analytic and perturbation methods, they managed to detect the presence of the EPN singularities in their model. They also revealed and explained that, under small perturbations, these singularities did unfold in a very specific manner.

These results appeared encouraging because, as the authors mentioned, the “further investigations” of the EPN-related problems “remain tasks for future research”. In our present paper, we decided to follow just such a recommendation. In Section 5 and Section 6, we will, in particular, address the main technical challenge, and we will also propose an IEP-related generalization of the well-known perturbation-theory-based description of a generic unitary quantum system near its IEP singularity. We will succeed in showing that many of the known technical tricks that are used and tested near EPN at $N<\infty$ can immediately be transferred to the quantum-dynamical scenarios in which the generic Hamiltonian $H\left(g\right)$ lies very close to its IEP limiting extreme.

In our last two Section 7 and Section 8, and also in the series of six brief Appendices, we will finally complement our considerations by several quantum-physics-oriented contextual remarks.

2. Conventional Exceptional Points Associated with Finite Jordan Blocks

In [1], we read that the existence of the IEP singularity “does not restrict to the particular Hamiltonian” of Equation (1) so that some “new directions in physical interpretation” of all of the analogous non-Hermitian quantum models have to be sought “since their properties are essentially different with respect to self-adjoint Hamiltonians” [1].

This makes the IC model important as a genuine methodical, as well as conceptual, challenge. Here, we intend to propose and advocate the idea that the resolution of the problem could be guided by another EPN-related “good basis” problem and by the existence of the parallels between quantum systems near their respective IEP and EPN singularities.

The study of these parallels could proceed in several independent directions (cf. the samples of some of them in [8] or in [13]). In what follows, we will explain that and how one of these directions could make use of perturbation expansion techniques.

2.1. The Phenomenon of EPN Degeneracy

In a review paper [14], the very first word of the abstract emphasized that every operator H that is eligible as an observable Hamiltonian of a unitary quantum system in a Schrödinger picture [15] must be diagonalizable. For any specific one-parametric family of Hamiltonians $H\left(g\right)$, such a requirement is not satisfied in the EPN limit $g\to g^{\left(EPN\right)}$. Then, the operator can consistently be treated as Hamiltonian only when $g\ne g^{\left(EPN\right)}$.

In an opposite direction of argumentation, one could recall the existence of exactly solvable quasi-Hermitian N-by-N matrix models $H^{\left(N\right)}\left(g\right)$, as shown in paper [11], for which there exists a vicinity of $g^{\left(EPN\right)}$ (i.e., a suitable compact and simply connected real or complex open domain $\mathcal{D}$ that does not, of course, contain $g^{\left(EPN\right)}$), inside which the respective quantum system is found to admit the standard physical probabilistic interpretation. For $g\in \mathcal{D}$, the diagonalizability of Hamiltonians $H^{\left(N\right)}\left(g\right)$, then, implies that we may construct all of the bound-state solutions of the so-called time-independent Schrödinger equation:

$$H^{\left(N\right)}\left(g\right)|{\psi }_n\left(g\right)\rangle =|{\psi }_n\left(g\right)\rangle E_n^{\left(N\right)}\left(g\right),n=0,1,\dots ,N-1.$$

(2)

Now, whenever the dimension of the Hilbert space of states is finite, $N<\infty$, we may immediately notice that, even in the generic non-degenerate case, all of the eigenvalues $E_n^{\left(N\right)}\left(g\right)$ and eigenvectors $|{\psi }_n\left(g\right)\rangle$ with $g\in \mathcal{D}$ degenerate in the ultimate (albeit manifestly unphysical) EPN limit:

$$\underset{g\to g^{\left(EPN\right)}}{lim}{E}_n^{\left(N\right)}\left(g\right)=E^{\left(EPN\right)},\underset{g\to g^{\left(EPN\right)}}{lim}|{\psi }_n\left(g\right)\rangle =|{\Psi }^{\left(EPN\right)}\rangle ,n=0,1,\dots ,N-1.$$

(3)

This leads to the following observations:

• [1] For all of the “acceptable” $g\ne g^{\left(EPN\right)}$ lying in the “physical”, unitarity-compatible vicinity of the EPN value, $g\in \mathcal{D}$, the normalized eigenvectors $|{\psi }_n\left(g\right)\rangle$ of $H^{\left(N\right)}\left(g\right)$ are almost parallel to each other.
• [2] At the “unacceptable” value of $g=g^{\left(EPN\right)}\notin \mathcal{D}$, their set ceases to serve, as a basis suitable, say, for the purposes of perturbation theory.
• [3] At $g=g^{\left(EPN\right)}$, one can still construct a “good basis” composed of the single remaining (degenerate) eigenvector $|{\psi }_0\left(g^{\left(EPN\right)}\right)\rangle =|{\Psi }_0\rangle$ and of an $(N-1)$-plet of linearly independent associated vectors $|{\Psi }_j\rangle$ with $j=1,2,\dots ,N-1$.

2.2. EPN and Modified Schrödinger Equation

The latter “good basis” can be perceived as an N-plet of column vectors. They may be arranged into the following formal N-by-N matrix:

$$\{|{\Psi }_0\rangle ,|{\Psi }_1\rangle ,\dots ,|{\Psi }_{N-1}\rangle \}:=R^{\left(EPN\right)},$$

(4)

which is called, usually, a transition matrix. Thus, we may introduce the two-diagonal Jordan block:

$$J^{\left(N\right)}\left(\eta \right)=\left(\begin{array}{ccccc}\eta & 1& 0& \dots & 0\\ 0& \eta & 1& \ddots & ⋮\\ 0& 0& \eta & \ddots & 0\\ ⋮& \ddots & \ddots & \ddots & 1\\ 0& \dots & 0& 0& \eta \end{array}\right),$$

(5)

and define the transition matrix as a solution of the following Schrödinger-like equation:

$$H^{\left(N\right)}\left(g^{\left(EPN\right)}\right)R^{\left(EPN\right)}=R^{\left(EPN\right)}{J}^{\left(N\right)}\left(E^{\left(EPN\right)}\right).$$

(6)

Interested readers are recommended to find a constructive illustration of the reconstruction of the transition matrix $R^{\left(EPN\right)}$ from the Hamiltonian in [16], where the illustrative solvable Hamiltonians are real matrices that are tridiagonal and multiparametric. At $N=2J$, one has

$$H^{\left(2J\right)}(a,b,\dots ,z)=\left[\begin{array}{cccccccc}2J-1& z& 0& \dots & & & & \\ -z& \ddots & \ddots & \ddots & ⋮& & & \\ 0& \ddots & 3& b& 0& \dots & & \\ ⋮& \ddots & -b& 1& a& 0& \dots & \\ & \dots & 0& -a& -1& b& 0& \dots \\ & & \dots & 0& -b& -3& \ddots & \\ & & & ⋮& \ddots & \ddots & \ddots & z\\ & & & & \dots & 0& -z& 1-2J\end{array}\right],$$

(7)

etc., (see also Appendix A.1 and Appendix A.2 below for a few further related comments).

3. The Mechanism of the Unfolding of the EPN Degeneracy

3.1. The Hypothesis of Admissibility of at Least Some $g\approx g^{\left(EPN\right)}$

The purpose of the above-outlined choice of the basis is twofold. Firstly, it enables us to re-read our Schrödinger-like Equation (6) as an equivalent linear-algebraic relation:

$${\left[R^{\left(EPN\right)}\right]}^{-1}H\left(g^{\left(EPN\right)}\right)R^{\left(EPN\right)}=J^{\left(N\right)}\left(E^{\left(EPN\right)}\right),$$

(8)

i.e., as a definition of a canonical Jordan-block representation of any non-Hermitian Hamiltonian of interest at its EPN singularity. Secondly, the amended basis will find application in a reformulation of standard perturbation theory. In such a reformulation, the role of the unperturbed Hamiltonian will be played by its unphysical, singular EPN limit. The trick is that we use the columns of $R^{\left(EPN\right)}$ as elements of an unperturbed basis. In the overall perturbation theory spirit, the perturbed system acquires, then, a standard phenomenological interpretation for the parameters g lying inside a suitable “physical” vicinity $\mathcal{D}$ of the EPN singularity.

The latter philosophy is to be advocated and used in what follows. We will only assume the knowledge of the transition matrix $R^{\left(EPN\right)}$ at an exceptional point of finite order, and we will then extend the use of this basis to a vicinity of the singularity. This will enable us to invert the limiting process $g\to g^{\left(EPN\right)}$ and to consider the original Hamiltonians at some $g\ne g^{\left(EPN\right)}$. Our knowledge of the transition matrix will yield the model described as a perturbation of the Jordan block matrix:

$${\left[R^{\left(EPN\right)}\right]}^{-1}{H}^{\left(N\right)}\left(g\right)R^{\left(EPN\right)}=J^{\left(N\right)}\left(E^{\left(EPN\right)}\right)+\lambda V^{\left(N\right)}\left(g\right).$$

(9)

A priori, we will only have to demand that the auxiliary variable $\lambda =\mathcal{O}(g-g^{\left(EPN\right)})$ remains small.

3.2. The Possibility of Keeping the Perturbed Spectrum Real

In papers [17,18], we considered the above-mentioned quantum dynamics scenarios. We studied there the criteria of smallness of perturbations $V^{\left(N\right)}$. The conditions of the stability and unitarity of the system were given there a mathematically as well as a phenomenologically consistent form.

For illustration, let us set $E^{\left(EPN\right)}=0$, and let us consider the bound-state problem as a perturbation of its EPN limit:

$$\left[J^{\left(N\right)}\left(0\right)+\lambda V^{\left(N\right)}\right]|\Psi \left(\lambda \right)\rangle =ϵ\left(\lambda \right)|\Psi \left(\lambda \right)\rangle .$$

(10)

With the energy levels counted by a subscript or superscript, we will never use this index, keeping it as just a dummy. We will rather introduce another subscript, which will run, say, from 1 to N, and which will number the components ${\Psi }_j$ of the ket vector $|\Psi \rangle$ (we are also dropping argument $\lambda$ as redundant). This convention enables us to fix the norm of $|\Psi \rangle$ by the choice of ${\Psi }_1=1$ and to define another “shifted” column vector:

$$\left(\begin{array}{c}{\Psi }_2\\ {\Psi }_3\\ ⋮\\ {\Psi }_N\\ {\Omega }_N\end{array}\right):=\left(\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_{N-1}\\ y_N\end{array}\right)=\overrightarrow{y},$$

(11)

where ${\Omega }_N$ is a new auxiliary variable. Next, we notice that the N-by-N matrix

$$A=A(N,ϵ)=\left(\begin{array}{ccccc}1& 0& 0& \dots & 0\\ ϵ& 1& 0& \ddots & ⋮\\ {ϵ}^2& ϵ& \ddots & \ddots & 0\\ ⋮& \ddots & \ddots & 1& 0\\ {ϵ}^{N-1}& \dots & {ϵ}^2& ϵ& 1\end{array}\right),$$

(12)

is just an inverse of the two-diagonal matrix

$$A^{-1}=\left(\begin{array}{ccccc}1& 0& 0& \dots & 0\\ -ϵ& 1& 0& \ddots & ⋮\\ 0& -ϵ& \ddots & \ddots & 0\\ ⋮& \ddots & \ddots & 1& 0\\ 0& \dots & 0& -ϵ& 1\end{array}\right).$$

(13)

Finally, we selected the first column of the matrix in Equation (10), and we denoted it by another dedicated symbol:

$$\left(\begin{array}{c}ϵ-\lambda V_{1,1}\\ -\lambda V_{2,1}\\ ⋮\\ -\lambda V_{N,1}\end{array}\right):=\overrightarrow{r}=\overrightarrow{r}\left(\lambda \right).$$

(14)

All of these abbreviations convert our initial homogeneous Schrödinger Equation (10) into its equivalent matrix form:

$$(A^{-1}+\lambda Z)\overrightarrow{y}=\overrightarrow{r},$$

(15)

or, better,

$$(I+\lambda AZ)\overrightarrow{y}=A\overrightarrow{r},$$

(16)

where the symbol Z stands for a modified form of the matrix of perturbation:

$$V^{\left(N\right)}\to Z=\left(\begin{array}{ccccc}{V}_{1,2}& V_{1,3}& \dots & V_{1,N}& 0\\ V_{2,2}& V_{2,3}& \dots & V_{2,N}& 0\\ \dots & \dots & \dots & ⋮& ⋮\\ V_{N,2}& V_{N,3}& \dots & V_{N,N}& 0\end{array}\right).$$

(17)

In paper [17], we proved that the construction of the perturbation corrections is reduced to self-consistency condition

$${\Omega }_N=0.$$

(18)

In the same reference, interested readers may also find an explicit form of the construction in the leading-order approximation.

Its basic aspects are worth recalling because they immediately help us to clarify the meaning of the rather vague assumption of the smallness of perturbation. It is sufficient to employ the Taylor-series expansion of the resolvent that yields the following formula:

$${\overrightarrow{y}}^{\left(solution\right)}\left(ϵ\right)=A\overrightarrow{r}-\lambda AZA\overrightarrow{r}+{\lambda }^{2}AZAZA\overrightarrow{r}-\dots .$$

(19)

Such a wave-function-representing ket vector depends on the variable parameter $ϵ$. But, ultimately, all of the eligible values of $ϵ$ become fixed by Equation (18).

The latter constraint plays the role of secular equation, which has the single vector-component form

$${y_N}^{\left(solution\right)}\left(ϵ\right)=0.$$

(20)

In the last step of the construction, we had to solve such an explicit transcendental equation in order to obtain all of the alternative perturbation-generated energy corrections $ϵ$. In a direct dependence on the model in question, the study of the roots of this equation also precisely offers the criterion of the reality of the whole spectrum in the leading-order approximation.

4. Large N and Anomalous Hamiltonians

The message to be extracted from the above-outlined EPN-based construction is that, for the purposes of transition to its IEP analog, we may try to make use of the EPN–IEP similarities. The main one will consist in the unperturbed Hamiltonian interpretation of the singular IEP operator tractable as, in some sense, a large-N descendant of its finite-N EPN analogs.

In the analysis of both of the EPN and IEP singularities, a central role is certainly played by the phenomenon of the asymptotic confluence of finitely or infinitely many eigenvectors, which are not accompanied by the confluence of the eigenvalues in the IEP case. This can be found confirmed in [8], where we read that “for matrices approaching an exceptional point, it is known [19] that the corresponding eigenvectors are tending to coalesce. For the infinite-dimensional Hilbert space (and Krein space) setup of the IC model, the eigenfunctions of the Hamiltonian having diverging projector norms and asymptotically approaching a PT phase transition region at spectral infinity signal a possible tendency toward collinearity and isotropy of an infinite number of these eigenfunctions” [8].

4.1. The Phenomenon of the Asymptotic Degeneracy of Eigenvectors

The EPN–IEP parallels are certainly incomplete. Still, in both cases, an amendment of the notation might prove useful. Here, we will follow the notation convention that was proposed in our comprehensive review paper [20]. In this spirit, the first mathematical subtlety that we will have to keep in mind is that, for a generic IEP model, the spectrum itself remains non-degenerate. Still, in a way sampled by the IC example, the generic IEP Schrödinger equation,

$$H^{\left(IEP\right)}|{\psi }_n^{\left(IEP\right)}\rangle =E_n^{\left(IEP\right)}|{\psi }_n^{\left(IEP\right)}\rangle ,n=0,1,\dots ,$$

(21)

can be considered analogous to its finite-dimensional, EPN-supporting partners.

Once the spectrum is found to be real and discrete (which is precisely the case for our illustrative IC Schrödinger Equation (1)), the same property also characterizes the formally independent Hermitian conjugate Schrödinger equation problem:

$${\left[H^{\left(IEP\right)}\right]}^{†}|{\psi }_n^{\left(IEP\right)}\rangle \rangle =E_n^{\left(IEP\right)}|{\psi }_n^{\left(IEP\right)}\rangle \rangle ,n=0,1,\dots .$$

(22)

Here, our use of the “ketket” symbol $\rangle \rangle$ deserves an immediate comment and explanation. Mainly, because it is closely connected with its role played in the three-Hilbert-space reformulation of the conventional quantum mechanics of unitary systems, as described, e.g., in review paper [20]. For the reasons explained in our three dedicated Appendix A.4, Appendix A.5 and Appendix A.6 below, the latter formalism is also—implicitly—recalled and used in our present paper. In these Appendices, interested readers may find a more extensive commentary on the entirely equivalent three-Hilbert-space version of the standard textbook quantum theory, with more emphasis put upon some of the questions of the physical probabilistic interpretation of the illustrative physical models of our present methodical interest.

The IEP-characterizing phenomenon of asymptotic degeneracy enables us to re-establish the above-mentioned analogy with the EPN form of the confluence of eigenfunctions. This phenomenon involves, first of all, the right eigenvectors $|{\psi }_n^{\left(IEP\right)}\rangle$ of $H^{\left(IEP\right)}$. For them, we have

$$|{\psi }_{M+k}^{\left(IEP\right)}\rangle \approx |{\psi }_{M+k+1}^{\left(IEP\right)}\rangle ,k=1,2,\dots$$

(23)

at $M\gg 1$. Similarly, the degeneracy also concerns the left eigenvectors $\langle \langle {\psi }_n^{\left(IEP\right)}|$ of the same non-Hermitian operator $H^{\left(IEP\right)}$. Often, we refer to their conjugate form $|{\psi }_n^{\left(IEP\right)}\rangle \rangle$ of “ketket” eigenvectors of conjugate ${\left[H^{\left(IEP\right)}\right]}^{†}$. In this representation, we encountered an entirely analogous IEP-related confluence of the eigenvectors

$$|{\psi }_{M+k}^{\left(IEP\right)}\rangle \rangle \approx |{\psi }_{M+k+1}^{\left(IEP\right)}\rangle \rangle ,k=1,2,\dots .$$

(24)

In both Equations (23) and (24), the degree of confluence depends on the Hamiltonian, and it may be expected to grow with the growth of M.

The phenomenon of confluences (23) and (24) finds its formal predecessor in the finite-dimensional case in which, during the transition to singularity $g\to g^{\left(EPN\right)}$, the elements of the N-plet of eigenvectors of any preselected N-by-N Hamiltonian matrix $H=H^{\left(N\right)}\left(g\right)$ do, in fact, lose their mutual linear independence. Still, the analogy of a genuine IEP system with the IEP-mimicking $N=\infty$ EPN extreme is incomplete since, in the former case, the spectrum remains non-degenerate. A more explicit analysis, thus, is necessary.

4.2. Canonical Representation of $H^{\left(IEP\right)}$

The IEP-characterizing non-degeneracy of eigenvalues can be perceived as a serendipitous simplification of their study. Still, a decisive IEP-related difficulty results from the effective asymptotic parallelization of the unlimited number of eigenvectors.

This forces us to recall, as our main source of inspiration, Equations (5) and (6) of Section 2 above. In the IEP case, our key task can be now identified as an appropriate generalization of the transition matrices $R^{\left(EPN\right)}$, which play a key role in the perturbation theory considerations of Section 3. In other words, we have to replace Equation (6) with a modified eigenvalue-like problem:

$$H^{\left(IEP\right)}{\mathcal{R}}^{\left(IEP\right)}={\mathcal{R}}^{\left(IEP\right)}{\mathcal{J}}^{\left(IEP\right)},$$

(25)

in which the low-lying eigenstates do not require any specific attention. Thus, the conventional Jordan-block-like bidiagonal (i.e., minimally non-diagonal) canonical matrix structure of Equation (5) will only reemerge here in a infinite-dimensional submatrix of an upgraded

$${\mathcal{J}}^{\left(IEP\right)}=\left(\begin{array}{cccccccc}{E}_0& 0& \dots & 0& 0& \dots & & \\ 0& E_1& \ddots & ⋮& ⋮& & & \\ ⋮& \ddots & \ddots & 0& 0& \dots & & \\ 0& \dots & 0& E_{K-1}& 0& 0& \dots & \\ 0& \dots & 0& 0& E_K& 1& 0& \dots \\ ⋮& & ⋮& 0& 0& E_{K+1}& 1& \ddots \\ & & & ⋮& 0& 0& E_{K+2}& \ddots \\ & & & & ⋮& ⋮& \ddots & \ddots \end{array}\right).$$

(26)

In this arrangement, the partitioning of the basis may be characterized by projectors P (on the first K lowest eigenstates of $H^{\left(IEP\right)}$) and Q (such that the unit operator I in Hilbert space can be decomposed as $I=Q+P$).

In a certain parallel with EPN, a key technical step will be a suitable perturbation-mediated weakening or removal of the asymptotic degeneracies (23) and (24) of the asymptotic eigenstates of $H^{\left(IEP\right)}$.

5. Toward a Regularization of ${\mathit{H}}^{\left(\mathit{IEP}\right)}$s by Perturbation

From a purely historical point of view, the idea of “prominence” of the IEP-sampling Schrödinger Equation (1) dates back to its methodical role in field theory [21] and to the Bessis’ and Zinn-Justin’s empirically revealed conjecture (cf. [4], which is also cited in [5]), where the spectrum $\left\{E_n^{\left(IC\right)}\right\}$ of $H^{\left(IC\right)}$ is real, discrete and bounded from below, i.e., tractable, in principle at least, as a set of observable energy levels. In spite of the manifest non-Hermiticity of Hamiltonian $H^{\left(IC\right)}$, the model was temporarily accepted as potentially compatible with all of the principles and postulates of quantum mechanics. The corresponding technical details may be found in the review paper [22].

Unfortunately, the excitement came to an end after Siegl’s and Krejčiřík’s rigorous proof that the IC model cannot, in fact, be assigned any form of conventional probabilistic interpretation in a mathematically consistent manner [1]. More or less the same conclusion has been also made by Günther and Stefani in a not yet published preprint [8]. At present, in the context of the unitary evolution part of non-Hermitian quantum mechanics, the problem of a correct physical interpretation of the IC model itself remains unresolved.

Concerning future developments, we remain a bit skeptical because the IEP nature of the IC model looks, in many respects, only too similar to its much better understood (and manifestly singular and unphysical) EPN-related finite-dimensional analogs.

5.1. The EPN–IEP Differences and Parallels

The essence of the anomalous nature of any IEP-related Hamiltonian $H^{\left(IEP\right)}$ lies in the asymptotic degeneracies (23) and (24) of its, respective, right and left eigenvectors. At the same time, a weak point of the amendment of the basis, as mediated by the choice of the non-diagonal matrix (26), may be seen in the necessity of the specification of a “sufficiently large” onset $K\gg 1$ of the de-parallelization. Such a specification is just numerically and computer-precision motivated. In contrast to the above-outlined treatment of the EPN scenarios, where dimension N was fixed, the IEP-implied choice of any finite K is purely pragmatic, immanently approximative and virtually arbitrary.

We now intend to show that, surprisingly enough, the apparently more or less accidental flexibility of our choice of K can, in fact, become an important mathematical tool facilitating an EPN-resembling regularization and consistent interpretation of quantum systems near their IEP singular extreme.

First of all, there is no doubt about the necessity of transition from the less suitable unperturbed basis (composed of eigenvectors) to an “anomalous” basis resembling Equation (4). The reason for this is provided by Equations (25) and (26): only a rectification of the underlying biorthogonal or biorthonormal basis [23] can re-establish the EPN–IEP parallels—even when achieved, as well as in the latter case—at the expense of non-diagonality and non-Hermiticity of the matrix of Equation (26).

Although the EPN singularity encountered at finite matrix dimensions $N<\infty$ is, according to paper [1], perceivably weaker than its IEP analog, the essence of our present message will be complementary. Essentially, we will emphasize that one can also reveal and make a productive use of certain partial similarities between the two concepts. In particular, we propose that the above-outlined possibility and feasibility of treating the manifestly unphysical finite-dimensional singular matrices $H^{\left(N\right)}\left(g^{\left(EPN\right)}\right)$ as formally acceptable unperturbed Hamiltonians is to also be transferred to the IEP context. An anomalous “good” basis composed of the columns of transition matrix should also be, mutatis mutandis, reconstructed from any given Hamiltonian $H^{\left(IEP\right)}$.

5.2. IEP-Unfolding Bases

The IEP (i.e., $N=\infty$) and EPN (i.e., $N<\infty$) singularities share the phenomenon of the parallelization of eigenvectors. In a small vicinity of the singularity, the analysis has to rely upon a properly adapted form of perturbation theory. Our present proposal of transfer of this idea from EPN to IEP will be inspired, therefore, by Section 3.

The parallels are, naturally, incomplete, such that, in the IEP setting, certain truly specific features have to be expected to emerge. For the purposes of clarification, let us mention that, even if we fix a finite $K\gg 1$, it remains far from obvious how to follow the analogy with Equations (2) and (4) and how to also treat $H^{\left(IEP\right)}$ as an unperturbed Hamiltonian. The reason for this is that we do not have any immediate analog of Equation (9). In the models sampled by the IC oscillator, we also miss a parameter g or $\lambda$, which would control the form and size of perturbations needed for a phenomenologically motivated unfolding of the manifestly unphysical IEP singularity.

This being admitted, we may still be guided by the EPN dynamical scenario, as outlined in the preceding Section 2 and Section 3. Therefore, in the study of the IEP systems, first of all, we should construct a good unperturbed basis in Hilbert space. The most natural IEP analog of the EPN-related Jordan-block matrix (5) is to be seen in its IEP-related amendment (26), rendering the EPN-related unperturbed Schrödinger-like Equation (6), which is replaced by its IEP-related alternative (25).

In connection with the standard and unmodified conjugate eigenvalue problems (21) and (22), the difficulty is that, in Siegl’s and Krejčiřík’s words, “the eigenvectors, despite possibly being complete, do not form a ‘good’ basis, i.e., an unconditional (Riesz) basis” [1]. Thus, the left and right eigenvectors of $H^{\left(IEP\right)}$ can only be used as a basis in the P-projected subspace of the Hilbert space. Otherwise, the EPN–IEP parallelism has to be fully taken into account, i.e., one has to recall the EPN-related definition (4) in Equation (25), as well as define, in Equation (25), its present IEP-related calligraphic symbol partner ${\mathcal{R}}^{\left(IEP\right)}$ as the following concatenated infinite set of column vectors:

$${\mathcal{R}}^{\left(IEP\right)}=\{|{\psi }_0\rangle ,|{\psi }_1\rangle ,\dots ,|{\psi }_{K-1}\rangle ,|f_K\rangle ,|f_{K+1}\rangle ,|f_{K+2}\rangle ,\dots \}.$$

(27)

This array is composed of the mere first K eigenkets $|{\psi }_j\rangle$, which are complemented by the modified, associated-like ket vectors $|f_{K+k}\rangle$ with $k=0,1,2,\dots$.

5.3. Recurrences

The possibility (as well as, in some sense, the necessity) of the explicit construction of the latter subfamily of the new ket vectors is, in fact, the very core of our present innovation of the foundations of the formalism of quantum mechanics. Briefly, our basic message is that, in a way which parallels the EPN-related considerations of Section 3 above, our present introduction of the nontrivial IEP-motivated transition matrix of Equation (27) may be expected to play a key role in the regularization of any singular $H^{\left(IEP\right)}$ via its perturbation.

The replacement of eigenvectors $|{\psi }_{K+k}\rangle$ by non-eigenvectors $|f_{K+k}\rangle$ in (27) has to weaken the asymptotically increasing parallelism between the subsequent columns of the transition matrix ${\mathcal{R}}^{\left(IEP\right)}$. Its infinite-dimensional matrix form makes this task different from its EPN predecessor. In technical terms, the insertion of (27) may be used to reduce the nontrivial part of Equation (25) to the sequence of recurrences

$$\left(H^{\left(IEP\right)}-E_{K+m}^{\left(IEP\right)}\right)|f_{K+m}\rangle =|f_{K+m-1}\rangle ,m=1,2,\dots ,$$

(28)

with the initial choice of $|f_K\rangle =c_{0,0}|{\psi }_K\rangle$ using any $c_{0,0}\ne 0$.

The solution of these relations can be then given the form of a finite sum:

$$|f_{K+p}\rangle =\sum _{n=0}^p{c}_{p,n}|{\psi }_{K+n}\rangle ,p=0,1,\dots ,$$

(29)

where the leading coefficient $c_{k,k}$ is arbitrary. Now, we assume and recall the biorthonormality of the eigenbasis, yielding $\langle \langle {\psi }_m|{\psi }_n\rangle ={\delta }_{m,n}$ and enabling us to convert Equation (28), i.e., the recurrences for kets into the recurrences for the following coefficients:

$$c_{k,m}={(E_{K+m}-E_{K+k})}^{-1}{c}_{k-1,m},m=0,1,\dots ,k-1,k=1,2,\dots .$$

(30)

Our freedom of choice of the highest-component coefficients $c_{k,k}$ enable us to suppress the IEP-accompanying asymptotic parallelization of the vectors of basis in Hilbert space. The goal is achieved. For every particular IEP model, we may recall the recurrences in (30) and replace the Q-projected part of the basis composed of eigenvectors by the Q-projected part of the basis composed, up to the first item $|f_K\rangle =c_{0,0}|{\psi }_K\rangle$, of non-eigenvectors. And this is precisely what was done in Equation (27).

6. Constructive IEP-Perturbation Considerations

6.1. Formulation of the Problem

Günther with Stefani [8] stated that “what is still lacking” in the IEP setup “is a simple physical explanation scheme for the non-Rieszian behavior of the eigenfunction sets”. We agree. We are even more skeptical because we would instead say that the expected ‘simple physical explanation’ making, in particular, the popular IC oscillators (1) physical need not exist at all. Indeed, we believe that a consistent physical, closed quantum system interpretation could much more easily be assigned to suitable perturbations of the “seed” IEP oscillators with uncertain interpretation (cf. [24] in this respect also).

Our belief is supported by the existence of parallels between the IEP and EPN scenarios. On this ground, one could, in fact, become able to assign a sound phenomenological meaning to many hypothetical parameter-dependent Hamiltonians ${\mathcal{H}}^{\left(new\right)}\left(\lambda \right)$, which are defined as certain “admissible” perturbations of the extreme IEP reference operators:

$$H^{\left(IEP\right)}\equiv {\mathcal{H}}^{\left(new\right)}\left(0\right).$$

In this respect, the methodical guidance, as provided by the illustrative EPN-related Equation (42), is also present in Section 7 below.

Open questions emerge when we fix a sufficiently large K, separate the Hilbert space of states into its two more or less decoupled subspaces and when we, finally, introduce a hypothetical, perturbed Hamiltonian ${\mathcal{H}}^{\left(new\right)}\left(\lambda \right)$ and the following IEP analog of Equation (9):

$${\left[{\mathcal{R}}^{\left(IEP\right)}\right]}^{-1}{\mathcal{H}}^{\left(new\right)}\left(\lambda \right){\mathcal{R}}^{\left(IEP\right)}={\mathcal{J}}^{\left(IEP\right)}+\lambda \mathcal{V}.$$

(31)

The analogy with EPNs is incomplete because, here, the spectrum of the unperturbed zero-order Hamiltonian remains non-degenerate. This is a simplification that can be perceived as partially compensating the increase in the overall complexity of the IEP problem.

Incidentally, a similar simplification was also detected in the realistic EPN-supporting Bose–Hubbard model of paper [12], where, in dependence on the parameters, the authors had to use both the degenerate and non-degenerate versions of perturbation theory. Thus, no abstract conceptual problems have to be expected to emerge after one returns to generic IEP-related dynamics. Still, as long as the IEP-related problems are infinite-dimensional, the perturbed IEP spectrum cannot be deduced from any analog of the EPN-based implicit-definition constraint (18). The study of properties of the vicinity of the IEP singularity cannot be based on a direct reference to its EPN analog. The methods of construction have to be amended.

6.2. Structure of Solutions

In light of the $P+Q$ partitioning of matrix ${\mathcal{J}}^{\left(IEP\right)}$ in Equation (26), the construction of the perturbed forms of the low-lying bound states remains standard. The P-projected states may be ignored just as certain decoupled observers. Only the treatment of the “asymptotic”, Q-projected components of the quantum system in question becomes difficult and singular, such as, “for instance, due to spectral instabilities” [1].

This leads to the necessity in solving the perturbed Schrödinger equation:

$$\left[{\mathcal{J}}^{\left(IEP\right)}+\lambda \mathcal{V}\right]|\psi \left(\lambda \right)\rangle =E\left(\lambda \right)|\psi \left(\lambda \right)\rangle ,$$

(32)

where $\lambda \ne 0$ (such that we avoid the IEP singularity) and where we have to insert the following:

$$|\psi \left(\lambda \right)\rangle =|\psi \left(0\right)\rangle +\lambda |{\psi }^{\left[1\right]}\rangle +{\lambda }^2|{\psi }^{\left[2\right]}\rangle +\dots$$

(33)

and

$$E\left(\lambda \right)=E\left(0\right)+\lambda E^{\left[1\right]}+{\lambda }^2{E}^{\left[2\right]}\rangle +\dots .$$

(34)

In the light of definition (26) the P-projected part of our unperturbed Hamiltonian ${\mathcal{J}}^{\left(IEP\right)}$ is a diagonal matrix containing the unperturbed bound-state energy eigenvalues $E_0$, $E_1$, …$E_K$. All of the related perturbed low-lying bound states can be then constructed using the conventional Rayleigh-Schrödinger perturbation theory of textbooks [15]. Therefore, for any practical purposes, it is fully acceptable just to make the choice of a sufficiently large dimension K of the P-projected subspace.

At present, in conducting conceptually more ambitious analyses of the problem, it makes sense to turn attention to the states with high-lying zero-order unperturbed energies. The related unperturbed ket vectors $|\psi (0)\rangle$ will lie in the complementary (and infinite-dimensional) Q-projected subspace of Hilbert space. The more-or-less conventional construction of its perturbed descendant, as given by Equation (33), will then, of course, possess several anomalous features.

The first anomaly is that the Q-projected part $Q{\mathcal{J}}^{\left(IEP\right)}Q$ of our unperturbed Hamiltonian is manifestly non-Hermitian. Even after a tentative finite-matrix truncation of the perturbed eigenvalue problem using a sufficiently large cut-off $N\gg K\gg 1$ of the Hilbert space bases, the implementation of the conventional Rayleigh-Schrödinger recipe would require a spectral representation of the unperturbed Hamiltonian operator.

Due to the specific upper-triangular structure of matrix $Q{\mathcal{J}}^{\left(IEP\right)}Q$, the construction of a biorthonormalized basis would be needed. Thus, the left eigenvectors of $Q{\mathcal{J}}^{\left(IEP\right)}Q$ (i.e., “brabras” $\langle \langle {\chi }_j|$ [20]) will be different from their right-eigenvector biorthogonal partners $|{\chi }_j\rangle$. This means that the conventional Rayleigh-Schrödinger elementary unperturbed projectors $|{\chi }_j\rangle \langle \langle {\chi }_j|$ (which are needed during the construction) will also have a practically prohibitively complicated explicit matrix structure.

One could also find another, more immediate, indication of the possible emergence of irregularities in Section 3 where Equation (20) plays the role of an ultimate transcendental equation determining all of the perturbed EPN eigenvalues as just a constraint imposed upon the very last, N-th component of a relevant ket vector. Needless to add that, in the IEP setting, one should consider $N\to \infty$ so that the direct analogy with EPNs is broken.

Another independent word of warning might originate from the fact that, for all of the truly high energy levels $E_{K+m}\left(\lambda \right)$, with $m\gg 1$, the use of the explicit Rayleigh-Schrödinger recipe would require the derivation of formulae, which would be m-dependent and different for the different, i.e., for the $(K+m)$-th, m-numbered excitations. Fortunately, the latter technical obstacle and difficulty has a comparatively elementary resolution because what is fully at our disposal is our choice of the value of K. We may feel free to work, exclusively, with the properly innovated Rayleigh-Schrödinger formulae that may be deduced at a single value of m, i.e., say, at $m=0$.

This certainly simplifies our task. In a methodical setting, it will be sufficient to work with the Hamiltonian of Equation (26) at $K=0$. Thus, one simply has to solve the following Schrödinger equation:

$$\left[\left(\begin{array}{cccc}{E}_0-E\left(\lambda \right)& 1& 0& \dots \\ 0& E_1-E\left(\lambda \right)& 1& \ddots \\ 0& 0& E_2-E\left(\lambda \right)& \ddots \\ ⋮& \ddots & \ddots & \ddots \end{array}\right)+\lambda \left(\begin{array}{cccc}{\mathcal{V}}_{00}& {\mathcal{V}}_{01}& {\mathcal{V}}_{02}& \dots \\ {\mathcal{V}}_{10}& {\mathcal{V}}_{11}& {\mathcal{V}}_{12}& \ddots \\ {\mathcal{V}}_{201}& {\mathcal{V}}_{21}& {\mathcal{V}}_{22}& \ddots \\ ⋮& \ddots & \ddots & \ddots \end{array}\right)\right]|\psi \left(\lambda \right)\rangle =0,$$

(35)

where

$$|\psi \left(\lambda \right)\rangle =\left(\begin{array}{c}{\psi }_0\left(0\right)\\ {\psi }_1\left(0\right)\\ ⋮\end{array}\right)+\sum _{k=1}^{\infty }{\lambda }^k\left(\begin{array}{c}{\psi }_0^{\left[k\right]}\\ {\psi }_1^{\left[k\right]}\\ ⋮\end{array}\right).$$

(36)

As long as $E_0=E\left(0\right)$ and $|{\psi }_j\left(0\right)\rangle =0$ at all $j\ne 0$, it makes sense to abbreviate $E_k-E\left(\lambda \right):=F_k\left(\lambda \right)$ and remember that $F_0\left(\lambda \right)=\lambda E^{\left[1\right]}+\mathcal{O}\left({\lambda }^2\right)$.

In the first-order approximation we have, therefore, the following Equation:

$$\left(\begin{array}{cccc}-\lambda E^{\left[1\right]}& 1& 0& \dots \\ 0& F_1\left(0\right)& 1& \ddots \\ 0& 0& F_2\left(0\right)& \ddots \\ ⋮& \ddots & \ddots & \ddots \end{array}\right)\left[\left(\begin{array}{c}{\psi }_0\left(0\right)\\ 0\\ 0\\ ⋮\end{array}\right)+\lambda \left(\begin{array}{c}{\psi }_0^{\left[1\right]}\\ {\psi }_1^{\left[1\right]}\\ {\psi }_2^{\left[1\right]}\\ ⋮\end{array}\right)\right]=$$

(37)

$$=\lambda \left(\begin{array}{c}-E^{\left[1\right]}{\psi }_0\left(0\right)\\ 0\\ 0\\ ⋮\end{array}\right)+\lambda \left(\begin{array}{cccc}1& 0& o& \dots \\ F_1\left(0\right)& 1& 0& \ddots \\ 0& F_2\left(0\right)& 1& \ddots \\ ⋮& \ddots & \ddots & \ddots \end{array}\right)\left(\begin{array}{c}{\psi }_1^{\left[1\right]}\\ {\psi }_2^{\left[1\right]}\\ {\psi }_3^{\left[1\right]}\\ ⋮\end{array}\right)=$$

(38)

$$=-\lambda \left(\begin{array}{cccc}{\mathcal{V}}_{00}& {\mathcal{V}}_{01}& {\mathcal{V}}_{02}& \dots \\ {\mathcal{V}}_{10}& {\mathcal{V}}_{11}& {\mathcal{V}}_{12}& \ddots \\ {\mathcal{V}}_{201}& {\mathcal{V}}_{21}& {\mathcal{V}}_{22}& \ddots \\ ⋮& \ddots & \ddots & \ddots \end{array}\right)\left(\begin{array}{c}{\psi }_0\left(0\right)\\ 0\\ 0\\ ⋮\end{array}\right).$$

(39)

In the context of the conventional Rayleigh-Schrödinger perturbation-expansion recipe, this is precisely the equation that would yield the explicit formula for the Coefficient $E^{\left[1\right]}$, which is defined in terms of the matrix elements of Perturbation $\mathcal{V}$. Nevertheless, as long as our present unconventional unperturbed Hamiltonian is a non-diagonal (and also an infinite-dimensional) matrix, we have to pay the price: The left eigenvector $\langle \langle \chi \left(0\right)|$ of our unperturbed Hamiltonian is not at our disposal. We cannot use it for the standard pre-multiplication of Equation (39) from the left. This means that, without the knowledge of $\langle \langle \chi \left(0\right)|$, the first line of Equation (39) is, viz., the following relation:

$$E^{\left[1\right]}={\mathcal{V}}_{00}+{\psi }_1^{\left[1\right]}/{\psi }_0\left(0\right),$$

(40)

which only enables us to extract the value of $E^{\left[1\right]}$ in the form of the function of an unknown Parameter ${\psi }_1^{\left[1\right]}$. This is the ambiguity that can be perceived as mimicking the unaccounted influence of the rest of the matrix elements of Perturbation $\mathcal{V}$.

The latter formal disadvantage is partially compensated by the presence of an easily invertible triangular matrix in Equation (39). This suggests that the role of a variable parameter could be played, in a partial resemblance of the EPN recipe, by the energy correction $E^{\left[1\right]}$ itself. We would, then, have

$${\psi }_1^{\left[1\right]}={\psi }_1^{\left[1\right]}\left(E^{\left[1\right]}\right)=(E^{\left[1\right]}-{\mathcal{V}}_{00}){\psi }_0\left(0\right).$$

Similarly, from the second row of Equation (39), we would obtain the value of the second wave-function component

$${\psi }_2^{\left[1\right]}=(E\left(0\right)-E_1){\psi }_1^{\left[1\right]}\left(E^{\left[1\right]}\right)-{\mathcal{V}}_{10}{\psi }_0\left(0\right),$$

(41)

etc.

7. Discussion

We can conclude that, in both the EPN- and IEP-related unitary-evolution scenarios, the properly amended form of perturbation theory seems to be able to provide, even in its leading-order form, some explicit and useful criteria of the acceptability or unacceptability of various preselected perturbations of phenomenological interest.

7.1. Benign Perturbations

Between the EPN and IEP alternatives, there still exists a crucial difference. Indeed, in a typical EPN-related analysis, our considerations usually start from our knowledge of the “physical” family of models $H\left(g\right)$. Then, the only truly difficult problem is to localize the EPN singularity, especially when the values of N are not too small. In the case of the IEP singularities, in contrast, we only know, typically, the unperturbed Hamiltonian, as sampled here by the IC operator. It is possible to conclude that this precisely makes the IEP-related models perceivably more difficult to study.

From a purely pragmatic point of view, a source of certain optimism could be drawn from the leading-order perturbation-approximation criteria. Their key strength lies in the possibility of identification of the “malign” IEP perturbations that would destroy the reality of the spectrum and would make the evolution non-unitary.

The complementary reliable identification of the “benign” perturbations is a mathematically much more difficult open problem. Incidentally, qualitatively, the same conclusions have already been obtained in a simpler EPN context. For example, in the above-mentioned study [12] of a specific Bose–Hubbard model near its EPN dynamical extreme, the authors did not insist on the reality of the spectrum. They decided to treat their mathematical results as applicable and valid in a broader, not necessarily unitary, open-system context.

In a narrower, closed-system setting, a deeper analysis was performed, and a resolution of the apparent EPN-related instability paradox was described in paper [18]. We studied, there, the exact, as well as approximate, secular equations in more detail. Our ultimate conclusion was that the necessary smallness condition specifying the class of the admissible, unitarity non-violating perturbations does not involve their upper-triangular matrix part at all. In contrast, for the perturbed EPN model in question, the lower triangular matrix part of all of the “benign” (i.e., unitarity-compatible) perturbations was shown to have the following element-dependent matrix form of condition of the sufficient smallness of $\lambda$:

$$\lambda V_{\left(admissible\right)}^{\left(N\right)}=\left[\begin{array}{cccccc}{\lambda }^{1/2}{\mu }_{11}& 0& \dots & 0& 0& 0\\ \lambda {\mu }_{21}& {\lambda }^{1/2}{\mu }_{22}& \dots & 0& 0& 0\\ {\lambda }^{3/2}{\mu }_{31}& \lambda {\mu }_{32}& \ddots & ⋮& ⋮& 0\\ {\lambda }^2{\mu }_{41}& {\lambda }^{3/2}{\mu }_{42}& \ddots & \ddots & 0& 0\\ ⋮& ⋮& \ddots & \lambda {\mu }_{N-1N-2}& {\lambda }^{1/2}{\mu }_{N-1N-1}& 0\\ {\lambda }^{N/2}{\mu }_{N1}& {\lambda }^{(N-1)/2}{\mu }_{N2}& \dots & {\lambda }^{3/2}{\mu }_{NN-2}& \lambda {\mu }_{NN-1}& {\lambda }^{1/2}{\mu }_{NN}\end{array}\right].$$

(42)

The matrix structure (42) may be interpreted as manifesting a characteristic anisotropy and the hierarchically ordered weights of influence of the separate matrix elements because, during the decrease in $\lambda \to 0$, all of the “benign” matrix-element parameters have to have bounded components ${\mu }_{j,k}=\mathcal{O}\left(1\right)$. For a more explicit explanation, we may rescale

$$\lambda V_{\left(admissible\right)}^{\left(N\right)}={\lambda }^{1/2}B\left(\lambda \right)V^{\left(reduced\right)}{B}^{-1}\left(\lambda \right),$$

(43)

where $B\left(\lambda \right)$ would be a diagonal matrix with elements $B_{jj}\left(\lambda \right)={\lambda }^{j/2}$ and where the whole reduced “benign” matrix of perturbation would be bounded, $V_{jk}^{\left(reduced\right)}=\mathcal{O}\left(1\right)$.

On this necessary-condition background, which is valid at all dimensions N, the samples of sufficient conditions retain a purely numerical trial-and-error character with the small-N non-numerical exceptions that were discussed in [18] for the matrix dimensions up to $N=5$.

7.2. The IC Oscillator as a Popular Toy Model

In order to elucidate the benchmark model role of the IC IEP oscillator, let us recall paper [5], in which Bender, with Boettcher, examined a rather broad family of time-independent non-Hermitian toy-model Hamiltonians (cf. Equation (A1) in Appendix A.2 below). They felt guided by the postulate of the (antilinear) symmetry of their models:

$$\mathcal{PT}{H}^{\left(BB\right)}=H^{\left(BB\right)}\mathcal{PT}.$$

(44)

The linear operator $\mathcal{P}$ was treated as parity (causing the space reflection $x\to -x$), while $\mathcal{T}$ had to mimic the anti-linear time reversal.

The authors proposed to treat their operators $H^{\left(BB\right)}$ as “Hamiltonians whose spectra are real and positive” so that “these $\mathcal{PT}$-symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space” [5]. During the subsequent wave of development of the related mathematics, it has been revealed that, in the language of functional analysis, the $\mathcal{PT}$ symmetry of Equation (44) can be re-read as pseudo-Hermiticity [14], as well as a self-adjointness, in the Krein space that is endowed with indefinite pseudo-metric $\mathcal{P}$ [25,26,27].

A deeper mathematical insight in the class of $\mathcal{PT}$-symmetric models was obtained. In the narrower context of quantum mechanics of closed systems, in contrast, the IEP-possessing IC model, itself, was not assigned up to now, nor are there any sufficiently consistent phenomenological interpretations yet [8]. Still, in retrospective, its temporary popularity was enormous. Its roots may be dated back to the Bessis and Zinn-Justin unpublished [4] but widely communicated [5] discovery that, in spite of the manifest non-Hermiticity of the IC Hamiltonian, its spectrum appeared to be real and bound-state-like, i.e., discrete and bounded from below.

In the extensive existing literature devoted to the study of systems with $\mathcal{PT}$ symmetry (cf., for example, the reviews of [28,29]), a great deal of attention has been paid to the non-Hermitian but still sufficiently realistic ordinary differential Hamiltonians of the form $H=T+V$, which are reminiscent of the IC oscillator in which the entirely conventional kinetic-energy term $T=-d^2/d{x}^2$ is combined with a suitable local complex one-dimensional potential $V=V\left(x\right)$. By many authors, the latter models were sampled by the field-theory-mimicking oscillator Hamiltonian (1), in which the purely imaginary form of the asymptotically growing potential is a truly puzzling mathematical curiosity.

This was also the feature that attracted a great deal of attention among physicists [5,22,26,28], precisely because they happened to generate the purely real, discrete and non-negative (i.e., hypothetically, observable and bound-state-like) energy spectra. Still, the ultimate verdict by mathematicians [1,8] was discouraging because they proved that the IC Hamiltonian cannot be assigned any isospectral self-adjoint avatar $\mathfrak{h}\left(t\right)$ or acceptable physical inner-product metric [1]. Thus, the rigorous mathematical analysis finally led to the loss of some of the most optimistic phenomenological expectations.

8. Summary

Not too surprisingly, the highly desirable proofs of the so-called unbroken form of $\mathcal{PT}$ symmetry (in which, by definition [22], the spectrum remains real) appeared to be, in numerous applications, strongly model-dependent. There seemed to be no universal criteria guaranteeing the existence of the unbroken $\mathcal{PT}$ symmetry in dependence on a suitable measure of degree of the non-Hermiticity of the Hamiltonian.

During the preparation of our present study, we came to the conclusion that the lack of a deeper understanding of correspondence between the (apparent) non-Hermiticity and (hidden) unitarity might have been caused by an overambitious generality of the choice of the models in the literature. For this reason, we decided to narrow the scope of our analysis, and we restricted our attention to the Hamiltonians living in a vicinity of their singular EPN or IEP extreme.

In our present paper, we have endeavored to explain how such a decision enabled us, not only to pick up a rather natural measure of the non-Hermiticity (which is characterized simply by the inverse distance of the variable physical parameter $g\in \mathcal{D}$ from its unphysical exceptional value), but also to formulate a well-defined project in which we developed and applied, consequently, some innovative and suitable perturbation-approximation techniques. In this framework, we managed to show that the unbroken $\mathcal{PT}$ symmetry of our models can, in fact, survive inside an open parametric domain on a point of boundary of which our measure of non-Hermiticity reaches its maximum. The latter point (which remains manifestly unphysical) has been shown to coincide either with the Kato’s [7] exceptional point of a finite-order N or with its hypothetical IEP analog.

Such an approach has been found productive. Using certain slightly modified techniques of perturbation theory of linear operators, with finite N, we found that, paradoxically, the restriction of attention to the smallest vicinity of the singularity (in which the Hamiltonians become maximally non-Hermitian) leads to a remarkable simplification of perturbation-approximation constructions. In spite of being singular and unacceptable as observables at $g=g^{EPN}$, the special, “exceptional” non-diagonalizable operators appeared to be eligible as unperturbed Hamiltonians. In their vicinity, such that $g\in \mathcal{D}$, their diagonalizability, as well as the observability status (i.e., the standard physical status), were re-established.

In this sense, the core of our present message is that the same perturbation-regularization physical interpretation should be also attributed to the IEP models where $N=\infty$. For the purposes of an illustrative example, we chose the popular imaginary oscillator Hamiltonian. Such a choice was motivated, first of all, by its long-lasting theoretical significance. It ranges from the more-or-less purely formal role of the model in mathematics and functional analysis [30] up to its deeper phenomenological significance in quantum statistics where the imaginary ${\varphi }^3$ interaction mimics the Yang–Lee edge singularity [5,21]. The same Hamiltonian also plays an important theoretical role of a benchmark model in conformal quantum field theory [31] and in Reggeon field theory [32].

In all of these contexts, our present results imply that the IEP property of the IC-like models means unphysicality. Only a suitable perturbation can reinstall the (possibly “hidden” alias “quasi-”) unitarity and physicality. Thus, the practical realizations of the standard quantum-mechanical IC model remain, in a way and for the reasons outlined in [1], elusive, at least in the unitary-theory context [33]. At the same time, one might still expect that some of its realizations could emerge out of the realm of quantum mechanics, e.g., in optics [29].

Constructively, we also managed to clarify some of the consequences of our present perturbative regularization recipe. The emergence of qualitative, as well as quantitative, EPN–IEP parallels helped us to complement and better understand the twelve-year-old disproof of the internal mathematical consistency of the IC IEP quantum oscillator [1]. Such a clarification can be perceived as being of a fundamental importance in quantum theory. Indeed, potentially, most of our observations might also immediately be extended to many other currently popular, but still IEP-singular, non-Hermitian quantum models.