## 1. Introduction

The Bender’s and Boettcher’s [

1] idea of replacement of Hermiticity

$H={H}^{\u2020}$ by parity-time symmetry (

$\mathcal{PT}$-symmetry)

$H\mathcal{PT}=\mathcal{PT}H$ of a Hamiltonian responsible for unitary evolution opened, after an appropriate mathematical completion [

2,

3,

4,

5] of the theory, a way towards the building of quantum models exhibiting non-Hermitian degeneracies [

6]

alias exceptional points (EPs, [

7]). For a fairly realistic illustrative example of possible applications opening multiple new horizons in phenomenology we could recall, e.g., the well-known phenomenon of Bose–Einstein condensation is a schematic simplification described by the non-Hermitian but

$\mathcal{PT}$-symmetric three-parametric Hamiltonian

This Hamiltonian represents an interesting analytic-continuation modification of the conventional Hermitian Bose–Hubbard Hamiltonian [

8,

9,

10]. In this form the model was recently paid detailed attention in Ref. [

11]. A consequent application of multiple, often fairly sophisticated forms of perturbation theory has been shown there to lead to surprising results. In particular, the behavior of the bound and resonant states of the system was found to lead to the new and unexpected phenomena in the dynamical regime characterized by the small coupling constant

c. The authors of Ref. [

11] emphasized that new physics may be expected to emerge precisely in the vicinity of the EP-related dynamical singularities.

These phenomena (simulating not necessarily just the Bose–Einstein condensation of course) were analyzed, in [

11], using several ad hoc, not entirely standard perturbation techniques. The role of an unperturbed Hamiltonian

${H}_{0}$ was assigned, typically, to the extreme EP limits of

H. Unfortunately, only too often the perturbed energies appeared to be complex as a consequence. In other words, the systems exhibiting

$\mathcal{PT}$-symmetry seemed to favor the spontaneous breakdown of this symmetry near EPs.

In the light of similar results one immediately must ask the question whether such a “wild behavior” of the EP-related quantum systems is generic. Indeed, an affirmative answer is often encountered in the studies by mathematicians (see, e.g., [

12]). A hidden reason is that they usually tacitly keep in mind just the “effective theory” and/or the so-called “open quantum system” dynamical scenario [

13].

In the more restrictive context of the unitary quantum mechanics the situation is different: the “wild behavior” of systems is usually not generic there (cf. also the recent explanatory commentary on the sources of possible misunderstandings in [

14]). Several non-numerical illustrative models may be found in [

15] where typically, the

$\mathcal{PT}$-symmetric Bose–Hubbard model of Equation (

1) (which behaves as unstable near its EP singularities [

11])) has been replaced by its “softly perturbed” alternative in which in an arbitrarily small vicinity of its EP singularity, the system remains stable and unitary under admissible perturbations.

The physics of stability covered by paper [

15] can be perceived as one of the main sources of inspiration of our present study. We intend to replace here the very specific model of Equation (

1) (in which the geometric multiplicity

L of all of its EP-related degeneracies was always equal to one) by a broader class of quantum systems. In a way motivated by the idea of a highly desirable extension of the currently available menu of the tractable and eligible dynamical scenarios beyond their

$L=1$ subclass, we will turn attention here to the EP-related degeneracies of the larger, nontrivial geometric multiplicities

$L\ge 2$. We will reveal that such a study opens new horizons not only in phenomenology (where the influence of perturbations becomes strongly dependent on the detailed structure of the non-Hermitian degeneracy) but also in mathematics (where a rich menu of physical consequences will be shown reflected by an unexpected adaptability of the geometry of the Hilbert space to the detailed structure of the perturbation).

The presentation of our results will be organized as follows. First, in

Section 2 we will recall a typical quantum system (viz., a version of the non-Hermitian Bose–Hubbard multi-bosonic model) in which the EP degeneracies play a decisive phenomenological role. We will explain that although the model itself only exhibits the maximal-order

$L=1$ EP degeneracies, such an option represents, from the purely formal point of view, just one of the eligible dynamical scenarios. In

Appendix A a full classification of the EPs is presented therefore, showing, i.a., that the number of the “anomalous” EPs of our present interest with

$L\ge 2$ exhibits an almost exponential growth at the larger matrix dimension

N.

The goals of our considerations are subsequently explained in

Section 3. For the sake of brevity, we just pick up the first nontrivial case with

$L=2$, and we emphasize that even in such a case the basic features of an appropriate adaptation of perturbation theory may be explained, exhibiting also, not quite expectedly, a survival of the fairly user-friendly mathematical structure.

In order to make our message self-contained, the known form of the EP-related perturbation formalism restricted to

$L=1$ is reviewed in

Appendix B. On this background, in a way based on a not too dissimilar constructive strategy, our present main

$L\ge 2$ results are then presented and described in

Section 4. We emphasize there the existence of the phenomenological as well as mathematical subtleties of the large-

L models. We show that in our generalized, degenerate-perturbation-theory formalism a key role is played by an interplay between its formal mathematical background (viz, the non-Hermiticity of the Hamiltonians) and its phenomenological aspects (typically, the knowledge of

L must be complemented by an explicit knowledge of the partitioning of Schrödinger equation).

In

Section 5, all these aspects of the

$L>1$ perturbation theory are summarized and illustrated by a detailed description of the characteristic, not always expected features of the leading-order approximations. Several related applicability aspects of our present degenerate-perturbation-theory formalism are finally discussed in

Section 6 and in two Appendices. We point out there that some of the features of the

$L>1$ theory (e.g., a qualitative, fairly counterintuitive clarification of the concept of the smallness of perturbations) may be treated as not too different from their

$L=1$ predecessors. At the same time, a wealth of new formal challenges is emphasized to emerge, in particular, in the analyses of the role of perturbations in the specific quantum systems which are required unitary.

## 5. Schrödinger Equation in Leading-Order Approximation

Two coupled equations (

29) determine the bound state. In the spirit of perturbation theory one may expect that the perturbations happen to be, in some sense, small. At the same time, even the analysis of the comparatively elementary

$L=1$ secular Equation (

A16) determining the single free variable (viz., the energy) led to the necessity of a strongly counterintuitive scaling (

A18) reflecting, near the extreme EP boundary of unitarity, the strong anisotropy of the geometry of the physical Hilbert space. Naturally, at least comparable complications must be expected to be encountered during the analysis of the more complicated set of two coupled equations (

29) representing the secular equation in its exact

$L=2$ version, constructed as particularly suitable for systematic approximations.

#### 5.1. Generic Case: Perturbations without Vanishing Elements

Even the most drastic truncation of the formal power series (

28) yields already a nontrivial ket vector

Needless to add that what must vanish are the auxiliary variables

${\mathsf{\Omega}}_{(a,b)}$ alias two functions which are available in closed form. Thus, we must solve the following two simplified equations

After the insertion of the respective matrix elements

${A}_{{N}_{(a,b)},k}^{(a,b)}\left(\epsilon \right)$ [cf. Equation (

A9)] we obtain the pair of relations

Obviously, this set can be read as a generalized eigenvalue problem which determines generalized eigenvectors $\overrightarrow{\omega}$ at a K-plet of eigenenergies $\u03f5$ which are all defined as roots of the corresponding generalized secular determinant.

#### 5.2. Hierarchy of Relevance and Reduced Approximations

In the generic case one must assume that the matrix elements of the perturbation do not vanish and that

$\lambda $ is small, i.e., that also the eigenvalues

$\epsilon $ remain small. This enables one to omit all the asymptotically subdominant corrections and to consider just the linear algebraic system

The solution of these two simplified coupled linear relations exists if and only if the determinant of the system vanishes,

Thus, an ordinary eigenvalue problem is encountered when $M=N$.

**Lemma** **1.** In the generic equipartitioned cases with $M=N\ge 3$ the spectrum ceases to be all real under bounded perturbations. The loss of unitarity is encountered.

**Proof.** Both roots

${\u03f5}^{N}=\lambda \phantom{\rule{0.166667em}{0ex}}x$ of the exactly solvable quadratic algebraic secular Equation (

36) may be guaranteed to be real in a certain domain of parameters. Still, some of the energies themselves are necessarily complex since

$\u03f5=\sqrt[N]{\lambda \phantom{\rule{0.166667em}{0ex}}x}$ is an

N-valued function with values lying on a complex circle. □

In the other, non-equipartitioned dynamical scenarios with, say, $M>N$, the behavior of the system in an immediate vicinity of its EP extreme is still determined by the asymptotically dominant part of the secular equation. After an appropriate modification, the above proof still applies.

**Lemma** **2.** In the generic case with $M>N\ge 3$ we get, from the dominant part of the generalized eigenvalue problem (36), a subset (N-plet) of asymptotically dominant eigenvalues $\epsilon =\mathcal{O}({\lambda}^{1/N})$ which cannot be all real. #### 5.3. Unitary Case: Re-Scaled Perturbations

The loss of unitarity occurring in the $L=1$ models was associated with the use of the too broad a class of norm-bounded perturbations. From our preliminary results described in preceding subsection one can conclude that a similar loss of unitarity may be also expected to occur, in the generic case, at $L=2$. Indeed, the anisotropy of the physical Hilbert space which reflects the influence of an EP-related singularity of the Hamiltonian may be expected to lead again to a selective enhancement of the weight of certain specific matrix elements of perturbations $\lambda \phantom{\rule{0.166667em}{0ex}}{V}^{\left(K\right)}$.

Once we recall the

$L=1$ scenario of

Appendix B.3 we immediately imagine that the main source of the apparent universality of the instability under norm-bounded perturbations should be sought, paradoxically, in the routine but, in our case, entirely inadequate norm-boundedness assumption itself. Indeed, in a way documented by Lemmas 1 and 2, the loss of the reality of spectra may directly be attributed to the conventional and comfortable but entirely random, unfounded and formal assumption of the uniform boundedness of the matrix elements of the perturbations.

Most easily the latter result may be illustrated using the drastically simplified version (

36) of the leading-order secular equation. Dominant role is played there by the quadruplet of matrix elements

${V}_{(P,1)}^{(P,Q)}$ with superscripts

P and

Q equal to

M or

N. Once they are assumed

$\lambda $-independent and non-vanishing, the leading-order energies read

$\u03f5=\sqrt[N]{\lambda \phantom{\rule{0.166667em}{0ex}}x}$. Thus, at any

$N\ge 3$ their

N-plet forms an equilateral

N-angle in the complex plane of

$\lambda $.

The latter observation inspires a remedy. In a way eliminating the

$N\ge 3$ complex-circle obstruction one simply has to re-scale the energies as well as all the relevant matrix elements of the perturbation. In this manner the ansatz

opens the possibility of the spectrum being real. Another multiplet of postulates

and

now contains a new partitioned matrix

$W={W}^{\left(K\right)}$ which is assumed uniformly bounded.

**Lemma** **3.** There always exists a non-empty $(2M+2N)$-dimensional domain $\mathcal{D}$ of the “physical” matrix elements of W for which the leading-order spectrum is all real and non-degenerate, i.e., in the language of physics, tractable as stable bound-state energies.

**Proof.** The main consequence of the amended, necessary-condition perturbation-smallness requirements (

38) and (

39) is that in our initial, unreduced leading-order generalized eigenvalue problem (

32) + (

33), all terms in the sums become of the same order of magnitude. By the scaling we managed to eliminate the explicit presence of the measure of smallness

$\lambda $. In other words the input information about dynamics is formed now by the re-scaled perturbation matrix

W which offers an

$(2M+2N)$-plet of free

$\mathcal{O}\left(1\right)$ parameters. At the same time, the spectral-definition output is given by the

$M+N$ roots of the corresponding secular equation, i.e., by the roots of an

$(M+N)$-th-degree polynomial in

E, with all its separate coefficients bounded and, in general, non-vanishing. Under these conditions the assertion of the lemma is obvious. □