Confluences of exceptional points and a systematic classification of quantum
catastrophes

Miloslav Znojil

The Czech Academy of Sciences, Nuclear Physics Institute,

Hlavní 130, 250 68 Řež, Czech Republic

and

Department of Physics, Faculty of Science, University of Hradec Králové,

Rokitanského 62, 50003 Hradec Králové, Czech Republic

and

Institute of System Science, Durban University of Technology, P. O. Box 1334,
Durban, 4000, South Africa

e-mail<EMAIL_ADDRESS>
## Abstract

In the problem of classification of the parameter-controlled quantum phase
transitions, attention is turned from the conventional manipulations with the
energy-level mergers at exceptional points to the control of mergers of the
exceptional points themselves. What is obtained is an exhaustive
classification which characterizes every phase transition by the algebraic and
geometric multiplicity of the underlying confluent exceptional point. Typical
qualitative characteristics of non-equivalent phase transitions are
illustrated via a few elementary toy models.

## Keywords

.

quasi-Hermitian quantum Hamiltonians;

loss of observability processes;

exceptional point multiplicities;

phase transition classification.

##  1 Introduction

In the conventional descriptions of unitary evolution of quantum systems in
Schrödinger picture (SP, [1]) the information about dynamics is all carried by
the Hamiltonian. The predictions of experiments are all based on the solution
of Schrödinger equation

${\rm i}\frac{d}{dt}|\psi(t)\rangle=H\,|\psi(t)\rangle\,,\ \ \ \ \ \ \
|\psi(t)\rangle\in{\cal V}\,.$ (1)

A broader perception of the concept of quantum dynamics will be advocated
here, with emphasis upon qualitative  aspects of the role of parameters.

Among several sources of inspiration of our project the most obvious one is
the Thom’s theory of catastrophes [2, 3]. As long as such a theory is of a
purely geometric nature [4], it is applicable, predominantly, to the classical
dynamical systems. In this domain it offers, first of all, a systematic
classification of structures and of the possible changes of structures of the
long-term classical equilibria [5]. The applicability and/or an immediate
transfer of these ideas to quantum systems are limited [6, 7, 8, 9, 10].

According to papers [11, 12, 13, 14, 15, 16, 17], new eligible directions of
research emerged after a turn of attention to the Kato’s concept of
exceptional point (EP, [18]). After a small change $g\to g^{(EP)}$ of a real
or complex parameter in Hamiltonian $H=H(g)$, such an operator ceases to be
diagonalizable. Hence, the bifurcation of the Thom’s classical equilibria can
find its genuine quantum analogue in the passage of the parameter through its
real or complex value $g^{(EP)}$. The latter possibility is, in essence, also
the key point of our present paper. In our text we will try to develop the
idea in a certain more systematic and constructive manner.

A decisive key to the realizability of the project can be seen in the Bender’s
and Boettcher’s change of the traditional paradigms [19]. Indeed, it was them
who conjectured that the unitary evolution could be, under certain conditions,
realized and described even when the generator $H$ of evolution of the wave
function in Schrödinger Eq. (1) becomes, in an apparent contradiction to the
well known Stone theorem [20], manifestly non-Hermitian. And precisely this
change of paradigm (cf. also the detailed outline of the resulting consistent
quantum theory of closed systems as reviewed, more than ten years ago, in
papers [21, 22]) opened also the way towards the change of the status of the
notion of EPs from a strictly mathematical tool as developed in the Kato’s
book to one of the most important concepts in experimental physics – see,
e.g., paper [23] outlining the related “roadmap for future studies and
potential applications”.

The core of our present message will lie in a combination of the purposeful
theoretical use of parameter-dependent non-Hermitian Hamiltonians with a
detailed analysis of some of the consequences in the ambitious
phenomenological context of description and classification of a broad class of
phenomena called quantum phase transitions [24]. Naturally, the feasibility of
our project will require a certain methodically motivated narrowing of its
scope. Thus, in contrast to the more conventional perception of the quantum
phase transition phenomena as described in textbook [24] (and, typically,
associated with the spontaneous symmetry breaking), our present approach to
the problem of phases will be slightly different, more closely associated with
the potential passage of the quantum system in question strictly through its
EP singularity.

In the context of such a theory (rendered consistent by the non-Hermiticity of
$H$) we will only consider a number of elementary toy models, with emphasis
upon the quick, non-numerical solvability of the related Schrödinger Eq. (1).
It is worth adding that in such a case (to be related here, for the sake of
simplicity, just to the spectral phase transitions) one of the phases may
(though need not) be ill-defined in a way depending on the respective presence
or absence of the complexification of the spectrum near the EP (see, e.g.,
[25] for a few illustrative examples of the latter, slightly less well known
possibility of having no complexification).

In the currently highly popular pragmatic context of the phenomenological
applicabilty and of the proposals and predictions of the results of
experiments, the unavoidable methodical limitations of our present
considerations using oversimplified toy models will be even more visible and
restrictive. In this respect the readers may be recommended to fill the
experiment-related gaps in our text by following the currently existing and
rich specialized literature (see, e.g., the freshmost reviews of non-Hermitian
physics in [26, 27]).

In the latter frame we will only emphasize the central role played, in the
underlying mathematics and physics, by the ubiquitous [12] Kato’s notion of
EPs. In the language of mathematics we only intend to complement the
contemporary popular but rather formal reference to EPs in various realistic
models by a slightly more ambitious theoretical interpretation of the EP
concept referring to its non-equivalent realizations.

##  2 Unitarity-of-evolution constraint

Among the existing applications of qualitative considerations to quantum
dynamics we felt particularly addressed by the mathematical studies in which
the EP limits were of order two (EP2). In this scenario, just some two
neighboring eigenvalues $E_{n}(g)$ and $E_{n+1}(g)$ of $H(g)$ are assumed to
merge and complexify at $g=g^{(EP)}=g^{(EP2)}$.

The latter studies were often motivated by the physics of systems exhibiting a
genuine quantum phase transition [28, 29]. According to our most recent
commentary [30], most of these systems have been considered “open”,
interacting with a certain not too well specified “environment”. As a
consequence, the bound states remained unstable, with the energies which need
not be kept real [31]. In such an open-system setup a typical Hamiltonian
$H(g)$ is non-Hermitian so that its EP singularities of the $N-$th-order may
be complex, $g=g^{(EPN)}\in\mathbb{C}$. One can, nevertheless, hardly speak
about fundamental theory because the “input” information about the open-system
dynamics (and, in particular, about the environment) remains incomplete.

Our present attention will be restricted to the closed systems characterized
by the unitarity of their evolution. One of the key technical consequences is
that the postulate of unitarity must be, due to the Stone theorem [20],
necessarily connected with the postulate of Hermiticity of the Hamiltonian.

The way out of the apparent impasse has only been discovered very recently. It
appeared sufficient to replace the conventional textbook SP paradigm by its
straightforward upgrade which works with the two  non-equivalent Hermitian
conjugations and which may be called pseudo-Hermitian quantum mechanics (PHQM,
see its review [22]).

###  2.1 Quantum observables in pseudo-Hermitian representation.

In the PHQM SP approach the EP singularity may mark a natural boundary of the
formal acceptability of any candidate for quantum Hamiltonian. The theory
emphasizes that the mere specification of the linear space ${\cal V}$ and the
related knowledge of the ket-vector solutions $|\psi(t)\rangle\in{\cal V}$ of
Schrödinger Eq. (1) are insufficient. What is considered equally important is
the freedom of the choice of the physical inner product between states. This
is equivalent to the specification of a correct dual space ${\cal V}^{\prime}$
of the linear functionals in ${\cal V}$. Such a choice is known to be
ambiguous (see, e.g., p. 246 in [1]). Still, in contrast to the widespread
beliefs, this ambiguity may be useful, bringing several immediate theoretical
challenges as well as practical benefits [32].

Some of the subtler aspects of the problem did not find their ultimate
clarification yet [33]. Nevertheless, under certain additional technical
assumptions the apparent paradox has already been resolved, almost thirty
years ago, in review paper [34]. The ambiguity of the abstract theory, i.e.,
the ambiguity of the choice of the correct physical Hilbert space ${\cal
H}_{\rm(physical)}=[{\cal V},{\cal V}^{\prime}_{\rm(physical)}]$ has been
shown removable. It has been explained that there exists a very natural method
of the necessary unique specification of the correct antilinear duality map
${\cal T}_{\rm(physical)}:\,{\cal V}\ \to\ {\cal V}^{\prime}_{\rm(physical)}$.

In the literature one still encounters a few obstinate terminological
misunderstandings. One of their sources lies in the fact that the operator
${\cal T}_{\rm(physical)}$ of the correct Hermitian conjugation need not
necessarily have an easily obtainable realization (cf., e.g., [35, 36, 37,
38]). The reconstruction of the physical inner-product space ${\cal
H}_{\rm(physical)}$ is, therefore, most often postponed till the very end of
the calculations. Temporarily, the correct physical space is being replaced by
its simplified, user-friendlier alternative ${\cal H}_{\rm(auxiliary)}$. In
spite of being manifestly unphysical, the key advantage of the latter choice
lies in the simplification of the conjugation. Its most straightforward form
${\cal T}_{\rm(auxiliary)}:\,{\cal V}\ \to\ {\cal
V}^{\prime}_{\rm(auxiliary)}$ is realized as the action which transforms the
column-vector alias  ket-vector $|\psi\rangle\in{\cal V}$ into its
conventional “Dirac’s” conjugate of textbooks, i.e., into its bra-vector
partner $\langle\psi|\in{\cal V}^{\prime}_{\rm(auxiliary)}$ which is
constructed as a row-vector composed of the complex-conjugate elements of its
partner $|\psi\rangle$.

For the users of the PHQM SP theory it is sufficient to know that the decisive
simplification of it applications is achieved via a consequent representation
of all of the states in ${\cal H}_{(auxiliary)}$ rather than in ${\cal
H}_{(physical)}$. The only  space in which one performs calculations is ${\cal
H}_{(auxiliary)}$. Hence, the use of the Dirac’s bra-ket notation conventions
cannot lead to any contradictions. Under this convention it is easy to
evaluate any correct inner product $(\psi_{1},\psi_{2})_{(physical)}$ in
${\cal H}_{(physical)}$ in terms of its unphysical partner in ${\cal
H}_{(auxiliary)}$ because we are allowed to abbreviate
$(\psi_{1},\psi_{2})_{(auxiliary)}=\langle\psi_{1}|\psi_{2}\rangle$. The
representation of the amended inner product $(\psi_{1},\psi_{2})_{(physical)}$
remains based on the definition
$(\psi_{1},\psi_{2})_{(physical)}=\langle\psi_{1}|\Theta|\psi_{2}\rangle$ in
which the new symbol $\Theta$ (called Hilbert space metric) can in fact carry
a nontrivial part of the information about dynamics.

The picture of reality remains internally consistent. Whenever one considers a
parameter-dependent (and, say, analytic) family of SP Hamiltonians $H(g)$
admitting an EP singularity at a (real or complex) EP value $g=g^{(EP)}$, one
has to localize the domain ${\cal D}_{(physical)}$ of admissible parameter(s)
$g$ at which the spectrum remains real and discrete, i.e., unitarity- and
closed-system-compatible. The theory is completed when one specifies also the
Hilbert-space metric which is, in general, $g-$dependent, $\Theta=\Theta(g)$.

The necessary mathematical  properties of the metric operator can be found
thoroughly discussed in [22, 33, 34] and in [39]. Among these properties a key
role is played by the ambiguity of the assignment of the metric to a
preselected SP Hamiltonian symbolized, whenever needed, by the introduction of
another formal parameter $c\,$ in $\Theta=\Theta(g,c)$.

Irrespectively of the latter ambiguity, any  Hamiltonian-compatible metric
will necessarily cease to exist  in the EP limit [40]. In parallel, our
operator $H(g)$ will cease to be diagonalizable and it will lose its status of
an acceptable Hamiltonian in the same  limit of $g\to g^{(EP)}$.

###  2.2 Quantum phase transitions at exceptional points.

Several well known quantum effects can be connected with some EP
singularities. The limit of $g\to g^{(EP)}$ implies the end (or at least
interruption) of the observability of the quantum system. In such a limit,
typically (cf., e.g., the schematic model in [41]), at least one pair of
energy levels merges and complexifies, i.e., the system ceases to be unitary.
Besides the schematic models there also exist multiple entirely realistic
samples of such a phenomenon. The best known ones are encountered in
relativistic quantum mechanics. The emergence of the singularity requires
there an abrupt redefinition of the Hamiltonian in which one has to
incorporate the new, “unfrozen” dynamical degrees of freedom.

The necessary matching of the old (= “before EP”) and new (= “after EP”)
dynamics (i.e., between the respective ad hoc  Hamiltonians) is usually
performed on a pragmatic, effective-Hamiltonian basis. The realization of the
transition becomes less counterintuitive when the EP-caused loss of the
observability happens to involve more than two levels. One of the most
characteristic illustrative examples is the well known Landau’s [42] strongly
singular harmonic oscillator with potential

$V^{(HO)}(x)=x^{2}-g/x^{2}\,.$ (2)

The system collapses, in suitable units, at $g=1/4$. One of the ways towards
the resolution of the puzzle has been described in Ref. [43]. At $x=0$ we
regularized the potential in the spirit of pseudo-Hermitian quantum theory.
The collapse of the spectrum then acquired an immediate EP-related form. With
the growth of attraction $g$ the levels were found to merge and to form,
subsequently, the complex conjugate pairs (cf. Fig. 1).

Figure 1: The low-lying part of the bound-state spectrum of the exactly
solvable quasi-Hermitian harmonic oscillator of Eq. (2). The model possesses
the confluent exceptional point tractable as the loss-of-unitarity quantum
phase transition at $g=1/4$.

In our recent follow-up paper [44] a closer connection has been established
between the harmonic-oscillator physics of collapse and the mathematics of its
exceptional points. Near $g=1/4$, in particular, explicit form has been found
of all of the admissible duality maps ${\cal T}$ defining all of the available
physical Hilbert spaces and metrics $\Theta(g,c)$. With an auxiliary
regularization shift $\varepsilon>0$ of coordinates $x\to x-{\rm
i}\varepsilon$ in (2) (which does not influence the results and can be
arbitrary) the HO model has been finally shown to support the desirable
degeneracies of our present interest,

$\lim_{{{g}}\to{{g}}^{(EP2)}_{m}}E_{2m}({{g}})=\lim_{{{g}}\to{{g}}^{(EP2)}_{m}}E_{2m+1}({{g}})=E^{(EP2)}_{m}<E^{(EP2)}_{m+1}\,,\
\ \ \ m=0,1,\ldots\,.$ (3)

From the point of view of mathematics it is utterly nontrivial that all of
these exceptional-point couplings coincide,

${g}^{(EP2)}_{0}={g}^{(EP2)}_{1}={g}^{(EP2)}_{2}=\ldots={g}^{(EP)}_{(confluent)}\,$
(4)

(see, once more, Fig. 1). By far the most interesting physics behind the
latter “degeneracy of degeneracies” occurs in a small vicinity of the
confluent EP value. At the slightly weaker couplings ${g}<1/4$ the whole
spectrum is non-degenerate and real, i.e., the system is unitary. Whenever we
choose just a slightly stronger attraction ${g}>1/4$, the reality of all of
the individual energy levels gets simultaneously lost.

For a complementary, qualitatively different introductory illustration let us
recall the exactly solvable square-well model $H^{(SQW)}({g})$ of Ref. [41].
The process of the loss of the observability starts there at the two lowermost
bound states. With the growth of $g$ (i.e., of the non-Hermiticity), the model
produces an infinite sequence of the energy mergers of order two (EP2). They
are real, well separated and ordered as follows,

$0<{g}^{(EP2-SQW)}_{0}<{g}^{(EP2-SQW)}_{1}<{g}^{(EP2-SQW)}_{2}<\ldots\,.$ (5)

This is a typical, generic scenario. In many other quantum models with a
variable parameter (cf., e.g., [45, 46, 47]), the EP-caused quantum-phase-
transition phenomenon just exclusively involves the pairs of the merging
levels. Obviously, the gradually emerging EP2s are characterized by their
nonzero distance from the parametric domain ${\cal D}_{(physical)}$ so that
they remain phenomenologically irrelevant. At the same time, their confluence
as sampled by Fig. 1 has always been considered improbable and next to
impossible to achieve in the laboratory [12].

A decisive return to optimism as sampled by the theoretical results of Ref.
[25] is only of a very recent date. One has to emphasize that also in the
parallel context of the possible experimental simulations the recent progress
is quick. In [48], for example, the authors argued that the models as sampled,
up to some similarity transformations, in Ref. [25] could really find an
immediate experimental realization. In detail these authors have shown that
the matrices controlling the evolution of the higher-order field moments of
certain two-mode systems could be realizable in the zero-dimensional bosonic
anti-${\cal PT}$-symmetric dimers.

###  2.3 The mergers of the mergers.

In our present paper we will ignore the isolated EP2 mergers of a single pair
of energies

$\lim_{{{g}}\to{{g}}^{(EP2)}}[E_{j}({{g}})-E_{j+1}({{g}})]=0$

occurring at a single excitation $j$ as not too interesting. Our search will
be redirected to the models exhibiting certain “mergers of the mergers”. More
explicitly, we will introduce at least one other variable parameter (say, $p$)
and we will search for the confluence of the EP2s themselves, i.e., of
${g}^{(EP2)}_{a}(p)$ with ${g}^{(EP2)}_{b}(p)$, etc. Thus, in an “upgraded”
dynamical scenario we will search, say, for four-level merger
EP4=EP2$\oplus$EP2 such that

$\lim_{{{p}}\to{{p}}^{(EP4)}}[{g}^{(EP2)}_{a}(p)-{g}^{(EP2)}_{b}(p)]=0$

etc.

In the framework of such a project, both of our previous illustrative models
proved unsatisfactory. In the former, harmonic-oscillator case, the “merger of
all mergers” did occur but it remained rigid, parameter-independent, i.e., not
usable for any active control of dynamics. In the other, SQW model, what
remained rigid was the separation of the exceptional points. The absence of
any auxiliary parameter $p$ did not allow us to convert at least some of the
sharp inequalities into equal signs in Eq. (5).

An encouraging partial resolution of the puzzle only came with paper [25]. We
managed to match there the evolution “before EP” with the evolution “after
EP”. The goal has been realized via an extreme and brutal maximal fine-tuning
procedure. The graduality formula (5) has been made parameter-dependent, i.e.,
in our present notation, $p-$dependent. Next, the $p-$supported limiting-
confluence conversion of the sharp inequality signs “$<$” into equal signs
“$=$” in (5) has been imposed upon all  of the separate EP2s. The EPN
degeneracy involved all of the states (the number $N$ of which was chosen
even). The “gradual” pattern of Eq. (5) has been replaced by its “confluent”
predecessor (4).

In the models of Ref. [25] where $N={\rm dim\,}\,H^{(before\,EP)}(g)={\rm
dim\,}\,H^{(after\,EP)}(g)$, the construction implied the complete degeneracy
of the energies,

$\lim_{g\to g^{(EPN)}}\,E_{n}(g)=\eta\,,\ \ \ \ n=0,1,\ldots,N-1\,.$ (6)

The phase-transition-mediating Hamiltonians acquired, at the matching EP
instant, the same, strongly fine-tuned canonical form of a single $N$ by $N$
non-diagonal Jordan-block matrix,

$\lim_{g\to g^{(EP)}}\,H^{(before/after\,EP)}(g)\sim
J^{({N})}(\eta)=\left[\begin{array}[]{ccccc}\eta&1&0&\ldots&0\\\
{}0&\eta&1&\ddots&\vdots\\\ {}0&0&\eta&\ddots&0\\\
{}\vdots&\ddots&\ddots&\ddots&1\\\ {}0&\ldots&0&0&\eta\end{array}\right]\,.$
(7)

The explicit construction of a genuine quantum energy-level-degeneracy
catastrophe proved successful and involved all of the levels in the spectrum.

In technical terms, the feasibility of the construction reflected the finite-
dimensional nature of Hamiltonians $H^{(before\,EP)}(g)$ and
$H^{(after\,EP)}(g)$. Although the mechanisms causing the collapse remained
unchanged, the specific simultaneous EPN-based phase-transition effect (6)
itself has been rendered possible. Hamiltonians $H^{(before\,EP)}$ and
$H^{(after\,EP)}$ were connected and matched in a strictly continuous and
both-sided “fundamental-Hamiltonian” manner.

In the early applications of the non-Hermitian degeneracies (7), many of them
retained their pragmatic effective-operator open-system physical background
admitting a virtually arbitrary complex $\eta$. Only in a small minority of
the closed-system models with strictly real spectra the authors emphasized
their dynamically complete description as well as their fundamental-theory
character.

In the latter context of our present exclusive interest, multiple further new
questions emerged. Some of them will be re-opened and answered in what
follows.

##  3 Results

Our present project is aimed at the search for new forms of manipulation and
control of important qualitative features of quantum dynamics. Our main result
can be characterized as a proposal of an EP-based quantum alternative to the
classical Thom’s catastrophe theory. The essence of such a classification
concerning the quantum phase transitions will lie, in its present form, in the
control of the EP2-related singularities and, in particular, in the control of
their confluences and/or restructuralizations.

###  3.1 Purpose: Unitary access to EPs in closed quantum systems.

In the literature, many authors (cf., e.g., Trefethen and Embree [49] or
Krejčiřík et al [50]) studied the PHQM-related quantum systems far from their
EP singularities. For this reason they did not need to distinguish too
carefully between the open (i.e., intrinsically non-unitary) and closed (i.e.,
intrinsically unitary) quantum systems. As a consequence, several
interpretations of their results happened to be unclear or even,
involuntarily, misleading. Typically, whenever they correctly identified
“unexpectedly wild” reaction to “small” perturbations [50], they did not
emphasize that such a scenario is only encountered in the non-unitary open-
quantum-system setup.

The clarification of the apparent puzzle was published in Refs. [51, 52]. For
the sake of clarity we picked up there just the “extreme” matrix (7) as an
unperturbed operator. Then, for any perturbed Hamiltonian

$H(g)=J^{({N})}(\eta)+V(g)$ (8)

we showed that the class of perturbations $V(g)={\cal O}(g)$ characterized as
“sufficiently small” in the conventional open-system norm of the unphysical
Hilbert space ${\cal H}_{(auxiliary)}$ has to be re-classified as
unacceptable, always containing perturbations which prove unbounded when
measured in the correct closed-system norm of space ${\cal H}_{(physical)}$.

In Ref. [51] these observations were complemented by the consistent closed-
system interpretation of the perturbed models (8) in ${\cal H}_{(physical)}$.
We demonstrated that the standard requirement of the smallness of the norm of
$V(g)$ in ${\cal H}_{(physical)}$ offers a natural picture of reality in the
vicinity of EP. We argued that in connection with the evolution of models (8)
in ${\cal H}_{(physical)}$ one can localize certain non-empty corridors of
unitary access to the quantum phase transition extremes at EPs.

A clear separation between the open- and closed-system theories must always be
kept sufficiently well verbalized. Partially, what is to be blamed for the
existing misunderstandings is the currently widely accepted terminology. Even
our present conventional usage of the term “non-Hermitian” should be taken cum
grano salis, i.e., with understanding of its true meaning. The point is that
in the closed-system context our considerations will never contradict the
conventional formulations of quantum mechanics. The operators of observables
will always be self-adjoint. The only necessary clarification is that in the
upgraded PHQM SP framework, all of the computations are realized in a
manifestly unphysical Hilbert space ${\cal H}_{(auxiliary)}$ [22, 34]. The
conventional and correct physical Hilbert space (say, ${\cal H}_{(physical)}$)
remains only available indirectly, via its representation in ${\cal
H}_{(auxiliary)}$.

One of the key technical merits of the PHQM amendment of the theory is that
the latter representation of ${\cal H}_{(physical)}$ is mediated by the mere
amendment of the inner product. The resulting re-arrangements of the usual SP
model-building recipes then really work with the operators which are non-
Hermitian (in ${\cal H}_{(auxiliary)}$).

###  3.2 Tool: Schrödinger equations on discrete lattices.

The main purpose of our present message is to show that the PHQM enhancement
of the flexibility of the SP formalism leads, near the EP singularities, to
some particularly important consequences. This will be illustrated by a few
not too complicated benchmark gain + loss Hamiltonians in which we will
postulate the existence of two parameters controlling the strength of the two
separate, independent gain + loss subcomponents.

Via these toy models we will demonstrate that one can achieve several
desirable transmutations of the EPs (i.e., of the dynamics in their vicinity)
via the mere fine-tuned interference between the remote and central gain-plus-
loss interactions.

For introduction let us recall the ordinary-differential-operator non-
Hermitian square-well model of Ref. [41]. The unitarity (i.e., the reality of
the spectrum of bound states) has only been guaranteed there in a finite
interval of the strength (say, $g$) of the non-Hermiticity. The reality (i.e.,
observability) has been lost due to the EP-related mechanism of the merger of
the ground state with the first excited state, $\lim_{g\to
g^{(EP)}_{0}}\,[E_{0}(g)-E_{1}(g)]=0$.

With the further growth of the non-Hermiticity of $H(g)$ beyond its EP value
$g^{(EP)}=g^{(EP)}_{0}$, further mergers occurred, and all of them were
followed by the complexifications of the energies of the higher and higher
excited states. The process involved, gradually, the whole spectrum, resulting
in the formation of an infinite sequence of exceptional points $g^{(EP)}$ such
that $\lim_{g\to g^{(EP)}}\,[E_{n}(g)-E_{n+1}(g)]=0$.

Such a behavior of the EPs appeared to be generic. Typically, the
phenomenologically relevant boundary of ${\cal D}_{(physical)}$ only
contained, in the vast majority of the elementary closed-system models, a
single isolated EP singularity. A richer, multi-parametric structure of the
Hamiltonian appeared necessary for the realization of any more interesting
scenario.

In order to avoid the loss of the easy mathematical tractability of the
desirable toy models we decided to redirect our attention from the
differential Hamiltonians $H=-\triangle+V(x)$ to their difference-operator
analogues. The most straightforward implementation of such an idea is easy:
One simply replaces the continuous real line of coordinates $x\in\mathbb{R}$
by an equidistant lattice of grid points

$x_{k+1}=x_{k}+h\,,\ \ \ \ \ \ \ \ k=0,1,\ldots,N\,.$ (9)

This opens the possibility of replacement of the conventional differential
Schrödinger equation by its difference-equation analogue

$-\frac{\psi(x_{k+1})-2\,\psi(x_{k})+\psi(x_{k-1})}{h^{2}}+V(x_{k})\,\psi(x_{k})=E\,\psi(x_{k})\,,\
\ \ \ k=1,2,\ldots,N\,.$ (10)

With the equally conventional Dirichlet asymptotic boundary conditions
$\psi(x_{0})=\psi(x_{N+1})=0$ the construction of bound states is then reduced
to the mere linear algebraic problem

$\left(\begin{array}[]{ccccc}v_{1}&-1&&&\\\ -1&v_{2}&-1&&\\\
&-1&\ddots&\ddots&\\\ &&\ddots&v_{N-1}&-1\\\
&&&-1&v_{N}\end{array}\right)\,\left(\begin{array}[]{c}\psi_{1}\\\ \psi_{2}\\\
\vdots\\\ \psi_{N}\end{array}\right)=F\,\left(\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\\\ \vdots\\\ \psi_{N}\end{array}\right)\,.$ (11)

In this local-interaction model the Hamiltonian contains just an $N-$plet of
the dynamics-determining diagonal matrix elements $v_{k}=h^{2}V(x_{k})$
yielding the spectrum of the re-scaled and shifted bound-state energies
$F_{n}=h^{2}E_{n}-2\ $ with $n=0,1,\ldots,N-1$.

Our interest in Eq. (11) was initially inspired by the popularity of the non-
Hermitian Hamiltonians with real spectra [21, 27]. Various non-analytic
square-well realizations of the potentials have been studied in this direction
of research [53, 54, 55, 56]. In these analyses an important role was played
by the discrete models as sampled by Eq. (11) [57, 58, 59, 60].

In an introductory methodical remark let us pick up $N=6$ and let us consider
Eq. (11) with one of the most elementary constant-interaction Hamiltonians

$H^{(6)}(w)=\left[\begin{array}[]{cccccc}-iw&-1&0&0&0&0\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr-1&-iw&-1&0&0&0\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&-1&-iw&-1&0&0\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr 0&0&-1&iw&-1&0\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr 0&0&0&-1&iw&-1\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr 0&0&0&0&-1&iw\end{array}\right]\,.$

The brute-force numerical analysis reveals that in spite of the non-
Hermiticity of the matrix, its spectrum is real (i.e., in principle,
observable) inside a unique unitarity-compatible interval of

$w\in{\cal D}_{(physical)}\approx(-0.322,0.322).$

Along the whole real line of parameters the model supports the existence of as
many as four separate exceptional points, viz.,

$\\{-0.54006,\,-0.32215,\,0.32215,\,0.54006\\}\,.$

The distance of the outer pair of these EPs from ${\cal D}_{(physical)}$ is
not zero so that they cannot play any immediate physical role. Their existence
is only considered interesting in mathematics (or in the open-system physical
setup) where people are trying to describe, irrespectively of the condition of
unitarity, the whole spectrum.

##  4 Realization: Models with two free parameters.

Most of the above-mentioned studies confirmed the expectations that at the
sufficiently small non-Hermiticities the spectra of the energy eigenvalues
should be real [21, 22, 61]. In our present paper we intend to complement
these results by the study of models in which, via the manipulation of the
EPs, one could control the qualitative dynamics directly. In a search for such
models one intends to control the positions of EPs using several independent
variable parameters. In such a project the main obstacles would be technical
because besides a few most elementary matrix structures the brute-force
numerical localization of the EPs is a badly ill-conditioned problem [62, 63,
64].

This being said, the comparatively transparent and feasible study of the EPs
can still be based on Schrödinger Eq. (11) in which almost all of the matrix
elements $v_{k}=h^{2}V(x_{k})$ of the local interaction term would be assumed
to vanish. In our present paper we will study, first of all, the two-
parametric family of Hamiltonians

$H^{(N)}(\varrho,w)=\left[\begin{array}[]{ccccc|ccccc}-{\rm
i}{{}\varrho}&-1&0&\ldots&0&0&\ldots&&\ldots&0\\\
-1&{{}{0}}&-1&\ddots&\vdots&\vdots&\ddots&&&\vdots\\\
0&\ddots&\ddots&\ddots&0&\vdots&&&&\\\
\vdots&\ddots&-1&{{}{0}}&-1&0&&&\ddots&\vdots\\\ 0&\ldots&0&-1&-{\rm
i}\,w&-1&0&\ldots&\ldots&0\\\ \hline\cr 0&\ldots&\ldots&0&-1&{\rm
i}\,w&-1&0&\ldots&0\\\ \vdots&\ddots&&&0&-1&{{}{0}}&-1&\ddots&\vdots\\\
&&&&\vdots&0&\ddots&\ddots&\ddots&0\\\
\vdots&&&\ddots&\vdots&\vdots&\ddots&-1&{{}{0}}&-1\\\
{}0&\ldots&&\ldots&0&0&\ldots&0&-1&{\rm i}{{}\varrho}\end{array}\right]\,.$
(12)

in which $N=2K$ is even and in which the central and remote parts of the
interaction (with the respective strengths $w$ and $\varrho$) are well
separated.

###  4.1 The confluence of EPs controlled by the fine-tuning of the remote
gain-and-loss interaction.

Technically, the separation of the influence of $w$ and $\varrho$ can simply
be strengthened, whenever needed, by the choice of a sufficiently large matrix
dimension $N$. At the same time, the potentially adverse aspect of the growth
of $N$ (making the secular equation less easily tractable) can very easily be
suppressed using the dedicated $N-$independent matching method of Ref. [60].
Using this method one can always try to test whether the bound-state spectrum
of the closed-system toy-model Hamiltonian (12) is real.

Usually, the answer becomes affirmative for the parameters lying inside a two-
dimensional unitarity-compatible (and, say, real) domain ${\cal
D}_{(physical)}$. Within the framework of our present project we will only be
interested in the situations in which one of the parameters is fixed while the
other one approaches the boundary $\partial{\cal D}_{(physical)}$ of the
energy-reality domain. What one then a priori  expects is that for the
different choices of the fixed parameter the mergers of the energies
encountered at the EP boundary might be of different types.

Figure 2: The $N-$plets of the real energy eigenvalues $E_{n}({\varrho},w)$ of
Hamiltonian (12) with $N=10$ and $n=0,1,\ldots,N-1$ at constant
${\varrho}=-0.28$ (the rightmost curves), at constant ${\varrho}=0$ (the
curves in the middle) and at constant ${\varrho}=0.56$ (the leftmost curves).

In the first test of the hypothesis let us choose $N=10$. Once we fix the
remote-gain-and-loss parameter ${\varrho}$ we may study the spectra of
energies $E_{n}({\varrho},w)$, $n=0,1,\ldots,N-1$ as functions of the
remaining variable parameter $w$. Numerically we evaluated several
characteristic samples of such a type. In Fig. 2 we displayed three of them.
The energies are shown there as functions of $w$, calculated at the three
different values of the remote-non-Hermiticity parameter ${\varrho}$, viz., at
${\varrho}=-0.28$ (the rightmost curves), of ${\varrho}=0$ (the middle-
positioned curves), and of ${\varrho}=0.56$ (the leftmost curves). After
inspection of this picture it is possible to formulate the following
observation.

###### Conjecture 1

At the sufficiently small real values of the remote-interaction parameter
${\varrho}$ the separate EP2 coordinates $w=w^{(EP2)}_{j}({\varrho})$ of the
pairwise mergers of the neighboring energies $E_{2j}^{(10)}({\varrho},w)$ and
$E_{2j+1}^{(10)}({\varrho},w)$ are all strictly decreasing functions of
${\varrho}$ such that

$w^{(EP2)}_{2}({\varrho})\leq
w^{(EP2)}_{1}({\varrho})=w^{(EP2)}_{3}({\varrho})\leq
w^{(EP2)}_{0}({\varrho})=w^{(EP2)}_{4}({\varrho})\,.$ (13)

The inequalities become sharp at ${\varrho}\neq 0$. At larger
$w>w^{(EP2)}_{j}({\varrho})$ the respective pairs of energies become complex
so that the local boundary of ${\cal D}_{(physical)}$ becomes determined by
function $w^{(EP2)}_{2}({\varrho})$.

Beyond such a purely numerically supported hypothesis (which could probably be
generalized to hold at any matrix dimension $N=2K$), the inspection of Fig. 2
also inspired the formulation of the following exact result valid at all $K$s.

###### Proposition 2

During the passage of the remote coupling $\,\varrho\,$ through the origin at
$\varrho=0$, Hamiltonian (12) encounters the instantaneous confluence of all
of the separate exceptional points of order two,

$w^{(EP2)}_{j}(0)=1\,,\ \ \ j=0,1,\ldots,N-1\,.$ (14)

The canonical form of the $w\to 1$ limit of matrix (12) then acquires the $N$
by $N$ matrix form

$H^{(2K)}(0,1)\ \sim\ \bigoplus_{j=1}^{K}\,J^{(2)}(\eta_{j})$ (15)

of a direct sum of $K$ two-dimensional Jordan blocks as defined in Eq. (7).

In the limit of ${\varrho}\to 0$, we witness the complete degeneracy alias
confluence of all of the separate EP2s. The rigorous proof will be given in
section 5 below. As a byproduct of this proof, also the values of the limiting
energies $\eta_{j}$ will be shown obtainable in closed form.

###  4.2 The confluence of EPs caused by the fine-tuning of the central gain-
and-loss interaction.

Our above-outlined projection of the motion of the EP boundaries of the two-
dimensional domain ${\cal D}_{(physical)}$ can be complemented by the
perpendicular sections in which the value of $w$ is fixed. We may expect that
at the boundary $\partial{\cal D}_{(physical)}$ the energies will merge in a
way reflecting the characteristics of the underlying EPs.

Figure 3: The $N-$plets of the real energy eigenvalues $E_{n}({\varrho})$ of
Hamiltonian (12) with $N=10$ and $n=0,1,\ldots,N-1$ at constant ${w}=2/3$ (the
rightmost curves), at constant ${w}=1$ (the curves in the middle) and at
constant ${w}=1.4$ (the leftmost curves).

The resulting scenario sampled in Fig. 3 is qualitatively not too different
from its predecessor of Fig. 2. At every fixed value of $w$, the single
central EP2 energy-merger ${\varrho}^{(EP)}_{2}$ is smaller than its first
off-central doubly-degenerate partner
${\varrho}^{(EP)}_{1}={\varrho}^{(EP)}_{3}$ which is, in its turn, smaller
than the second, most off-central partner doublet
${\varrho}^{(EP)}_{0}={\varrho}^{(EP)}_{4}$. With the growth of $w$ sampled,
in the picture, by the choice of $w=2/3$, $w=1$ and $w=1.4$, all of the energy
mergers move leftwards.

###### Conjecture 3

In a vicinity of $w=1$ the separate EP2 coordinates
$\varrho=\varrho^{(EP2)}_{j}({w})$ of the pairwise mergers of the neighboring
energies $E_{2j}^{(10)}({\varrho},w)$ and $E_{2j+1}^{(10)}({\varrho},w)$ are
all strictly decreasing functions of ${w}$ such that

$\varrho^{(EP2)}_{2}({w})\leq\varrho^{(EP2)}_{1}({w})=\varrho^{(EP2)}_{3}({w})\leq\varrho^{(EP2)}_{0}({w})=\varrho^{(EP2)}_{4}({w})\,.$
(16)

The inequalities are certainly sharp at ${w}\neq 1$. At the small and positive
difference $\varrho-\varrho^{(EP2)}_{j}({w})>0$ the respective pairs of
energies cease to be real so that the local boundary of ${\cal
D}_{(physical)}$ is prescribed by function $\varrho^{(EP2)}_{2}({w})$.

In the limit of $w\to 1$ all of the EP2 singularities may be guessed to
coincide forming a degenerate quintuplet EP10. Such a possibility advised by
the inspection of Fig. 3 inspired the following proposition in which the value
of the even matrix dimension $N=2K$ can be arbitrary.

###### Proposition 4

During the passage of the central coupling $\,w\,$ through the value of
$w=w^{(EP)}=1$, Hamiltonian (12) encounters the instantaneous confluence of
all of the separate exceptional points of order two,

$\varrho^{(EP2)}_{j}(1)=0\,,\ \ \ j=0,1,\ldots,N-1\,.$ (17)

The canonical form of the $\varrho\to 0$ limit of matrix (12) acquires the
same $N$ by $N$ matrix form (15) as in Proposition 2

Proof. In the light of Proposition 2 the proof is elementary because our
Hamiltonian matrix $H^{(2K)}(\varrho,w)$ can be transformed into matrix
$H^{(2K)}(w,\varrho)$ using an elementary block-diagonal unitary-
transformation matrix ${\cal U}^{(2K)}$ defined as a direct sum of two $K$ by
$K$ antidiagonal unit matrices ${\cal I}^{(K)}$ with Kronecker-delta elements
${\cal I}^{(K)}_{i,j}={\delta}_{i,K+1-i}$, $i=1,2,\ldots,K$. $\square$

##  5 Exact solutions at $\varrho=0$.

Our illustrative Hamiltonian (12) is the special case of a broader class of
models of Eq. (11). Some of them might be analytically solvable by the
matching method of Ref. [60]. What would be required is a special choice of
the matrix elements $v_{j}$ of the interaction. For the sake of simplicity we
decided to consider just the very special model (12), with the study of its
possible generalizations left to the readers.

###  5.1 Constructive proof of proposition 2.

The independent variability of the two real parameters $\varrho$ and $w$ in
(12) proved sufficient for our present illustration purposes. Now we only have
to prove Proposition 2 in which our $\varrho=0$ toy-model Hamiltonian has even
$N=2J+2$ and mere two nonzero values of $v_{j}$,

$H^{(2J+2)}(w)=\left(\begin{array}[]{cccc|cc|cccc}0&-1&0&\ldots&0&0&0&\ldots&\ldots&0\\\
{}-1&0&-1&\ddots&\vdots&\vdots&\vdots&&&\vdots\\\
{}0&-1&\ddots&\ddots&0&\vdots&&&&\\\
{}\vdots&\ddots&\ddots&0&-1&0&\vdots&&&\vdots\\\
\hline\cr{}0&\ldots&0&-1&-{\rm i}w&-1&0&\ldots&\ldots&0\\\
{}0&\ldots&\ldots&0&-1&{\rm i}w&-1&0&\ldots&0\\\
\hline\cr{}\vdots&&&\vdots&0&-1&0&-1&\ddots&\vdots\\\
{}\hfil&&&&\vdots&0&-1&\ddots&\ddots&0\\\
{}\vdots&&&\vdots&\vdots&\vdots&\ddots&\ddots&0&-1\\\
{}0&\ldots&\ldots&0&0&0&\ldots&0&-1&0\end{array}\right)$ (18)

The related Schrödinger equation is tractable by the standard numerical
diagonalization techniques. The task becomes simplified when one introduces,
in the spirit of Refs. [19, 65], the requirement ${\cal
PT}\,H^{(2J+2)}(w)=H^{(2J+2)}(w)\,{\cal PT}$ of ${\cal PT}-$symmetry defined
in terms of the antidiagonal-unit matrix ${\cal P}$ [changing the parity and
causing the left-right inversion of the spatial lattice (9)] and of the
antilinear complex-conjugation operator ${\cal T}$ (which simulates the time
reversal in Schrödinger equation). Whenever our real parameter $w$ is such
that the spectrum remains real and non-degenerate, Schrödinger equation (11)
will then yield, exclusively, just the ${\cal PT}-$symmetric eigenstates,

${\cal PT}\,\left(\begin{array}[]{c}\psi_{1}\\\ \psi_{2}\\\ \vdots\\\
\psi_{N}\end{array}\right)\sim\left(\begin{array}[]{c}\psi_{1}\\\ \psi_{2}\\\
\vdots\\\ \psi_{N}\end{array}\right)\,.$ (19)

We may set $\psi_{N}=\psi_{1}^{*}$, $\psi_{N-1}=\psi_{2}^{*}$ (etc), we may
abbreviate $h^{2}E=-2x=-2\,\cos\theta$, and we may recall the definition of
the Tshebyshev plynomials of the second kind,

$U_{k}(\cos\theta)=\frac{\sin(k+1)\theta}{\sin\theta}\,,\ \ \ \
k=0,1,\ldots\,.$ (20)

The recurrences satisfied by these polynomials [66, 67] enable us to guess the
ansatz

$\psi_{k+1}=(\alpha+{\rm i}\,\beta)\,U_{k}(x)\,,\ \ \ \ k=0,1,\ldots,J\,$

containing just a pair of unspecified real parameters $\alpha$ and $\beta$.
Its use converts the first $J$ lines of relations (11) into identities.
Recalling the ${\cal PT}-$symmetry of the model we may also write down the
rest of the components of the eigenvector in closed form reflecting the
validity of the last $J$ lines of relations (11),

$\psi_{N-k}=(\alpha-{\rm i}\,\beta)\,U_{k}(x)\,,\ \ \ \ k=0,1,\ldots,J\,.$

What remains to be satisfied are the two middle lines of Schrödinger equation
(11),

$-(\alpha+{\rm i}\,\beta)\,U_{J-1}(x)+[(2x-{\rm i}w)\,(\alpha+{\rm
i}\,\beta)-(\alpha-{\rm i}\,\beta)]\,U_{J}(x)=0\,,$ $-(\alpha-{\rm
i}\,\beta)\,U_{J-1}(x)+[(2x+{\rm i}w)\,(\alpha-{\rm i}\,\beta)-(\alpha+{\rm
i}\,\beta)]\,U_{J}(x)=0\,.$

The separation of the real and imaginary components yields

$-\alpha\,U_{J-1}(x)+(2x\alpha+w\,\beta)-\alpha)\,U_{J}(x)=0\,$

and

$-\beta\,U_{J-1}(x)+(2x\beta-w\,\alpha+\beta)\,U_{J}(x)=0\,.$

After a premultiplication by suitable constants, the sum of the latter two
relations yields

$-2\,\alpha\,\beta\,U_{J-1}(x)+[4\,\alpha\,\beta\,x+(\beta^{2}-\alpha^{2})\,w]\,U_{J}(x)=0$
(21)

while their difference only leads to elementary relation

$(\alpha^{2}+\beta^{2})\,w=2\,\alpha\,\beta\,.$ (22)

This enables us to reparametrize $\alpha=\alpha(\tau)=\cos\tau$ and
$\beta=\beta(\tau)=\sin\tau$ and to deduce that $w=w(\tau)=\sin 2\tau$.

One can treat the auxiliary angle $\tau$ as an alternative dynamical-input
information about the strength of the non-Hermiticity. We are now only left
with the secular equation (21), i.e.,

$2\,\alpha\,\beta\,U_{J+1}(x)+(\beta^{2}-\alpha^{2})\,w\,\,U_{J}(x)=0\,.$ (23)

The insertion of $w$ reduces it to the relation

$U_{J+1}(x)=[\alpha^{2}(\tau)-\beta^{2}(\tau)]\,U_{J}(x)\,.$ (24)

This is our ultimate implicit definition of the spectrum of the energies
$h^{2}E=-2x$ at arbitrary matrix dimension $N=2J+2$.

At the ${\cal PT}-$symmetry-breakdown boundaries with $w=w^{(EP)}=\pm 1$ or
$\tau=\tau^{(EP)}=\pm\pi/4$, we have
$\alpha^{2}(\tau^{(EP)})=\beta^{2}(\tau^{(EP)})$ so that the EPN-related
energy values coincide with the roots of a single polynomial,

$U_{J+1}\left(x^{(EP)}\right)=0\,.$ (25)

These roots can be given an elementary form given by formula (20).

The availability of such an explicit parameter-dependence of the spectrum in
the EPN limit can be extended to cover also, in an approximative form, a small
vicinity of the singularity. In this vicinity the difference
$\alpha^{2}(\tau)-\beta^{2}(\tau)$ entering Eq. (24) will be a small number.
The well known intertwining property of the roots of the polynomials
$U_{J+1}(x)$ and $U_{J}(x)$ will then immediately imply the correct
qualitative understanding of the branching of the levels at $|w|\lessapprox 1$
as sampled, at $N=10$, in Fig. 4.

Figure 4: The left-right symmetry of the low-lying spectrum of model (18) at
$N=10$. The spectral locus is also symmetric with respect to the $F=0$ axis,
so we did not need to display the upper, high-excitation half of the spectrum.

###  5.2 Spectral curves at $N=10$.

The interval $w\in(-1,1)$ of the unitarity-compatible physical parameters is
the same for all  of the bound states at an arbitrary  even matrix dimension
$N=2J+2$. Whenever the value of $w$ leaves this interval, all  of the energies
cease to be real so that abruptly, the whole  spectrum becomes unobservable.
At all of the lattice-size-determining integers $J$, even the parameter-
dependence of the numerically evaluated spectra remains qualitatively the
same, characterized by the specific pairwise  degeneracies of all  of the
energy levels at $w=1$ and $w=-1$.

At $N=10$ the model still remains non-numerical. Its secular polynomial
$P^{(10)}({F},{w})$ is a polynomial of the fifth degree in the energy-
representing variable $F^{2}$,

${{\it{F}}}^{10}+\left(-9+{{\it{w}}}^{2}\right){{\it{F}}}^{8}+\left(28-6\,{{\it{w}}}^{2}\right){{\it{F}}}^{6}+\left(-35+11\,{{\it{w}}}^{2}\right){{\it{F}}}^{4}+\left(15-6\,{{\it{w}}}^{2}\right){{\it{F}}}^{2}-1+{{\it{w}}}^{2}\,$
(26)

but this does not imply that the search for its roots is complicated. Their
brute-force numerical localization is not necessary. It is sufficient to
notice that the secular polynomial is just a linear function of the square of
the coupling constant $w^{2}$. This implies that the spectrum can be
constructed, in the implicit-function form, non-numerically.

The shape and symmetry of the spectrum are sampled in Fig. 4. The picture just
reconfirms the existence of the strictly two Kato’s exceptional points
$w=w^{(EP)}_{\pm}=\pm 1$. One can visualize the coupling $w$ as an elementary
function

$w=w_{\pm}(F)=\pm\,[\left({{F}}^{2}-1\right)^{2}-{{F}}^{2}]^{-1}\,{\sqrt{1-{{F}}^{10}+9\,{{F}}^{8}-28\,{{F}}^{6}+35\,{{F}}^{4}-15\,{{F}}^{2}}}\,$

of the energy. At both of the EPN extremes with $|w|=1$ the $N=10$ secular
polynomial can be factorized,

$P^{(10)}({F},\pm 1)={F}^{2}\,({F}^{2}-1)^{2}\,({F}^{2}-3)^{2}\,.$ (27)

The function $w=w(F)$ can be also Taylor-expanded. Near $F=0$ this yields the
symmetric and “deeper-than-quadratic” well,

$w({F})\approx-1+{\frac{9}{2}}{{F}}^{2}+{\frac{201}{8}}{{F}}^{4}+O\left({{F}}^{5}\right)\,.$

Similarly, off the origin we get, in agreement with the picture, the two
narrower and asymmetric wells which are steeper than the one near the origin.
Thus, we get

$w({F})\approx-1+8\,\left({F}-1\right)^{2}-24\,\left({F}-1\right)^{3}+122\,\left({F}-1\right)^{4}+O\left(\left({F}-1\right)^{5}\right)\,,$

etc. Finally, the outer wells have just the more pronounced shapes of the same
form, with

$w({F})\approx-1+72\,\left({F}-\sqrt{3}\right)^{2}-648\,\sqrt{3}\,\left({F}-\sqrt{3}\right)^{3}+16782\,\left({F}-\sqrt{3}\right)^{4}+O\left(\left({F}-\sqrt{3}\right)^{5}\right)\,$

etc. All of these observations gave birth to their generalizations valid in
any analogous EP-supporting $N-$level quantum system with arbitrary finite
$N<\infty$.

##  6 Discussion

From the purely methodical point of view the choice of the discrete local-
interaction model of Eq. (11) has its weaknesses. Firstly, its variable
parameters only lie on the main diagonal. This lowers the flexibility of
dynamics leading, typically, just to the EP2 energy mergers. Secondly,
additional antilinear symmetries [sampled here by ${\cal PT}-$symmetry of Eq.
(19)] had to be imposed in order to guarantee the reality of the spectrum.
Thirdly, the well known numerically ill-conditioned nature of the study of the
limiting transition $g\to g^{(EP)}$ often forces us to use certain truly
sophisticated construction methods in a way sampled, say, in Ref. [68].

For all of these reasons it will make sense to turn attention to the more
general matrix models in which the practical calculations remain feasible but
in which it should still be possible to enhance the flexibility of the picture
of the EP-related dynamics. Let us now mention a few hints for the future
projects oriented in this direction.

###  6.1 Parallels between harmonic oscillator and our $N<\infty$ models.

The turn of attention to the more general classes of models might open new
ways towards an immediate further development of the theory itself. In order
to be more specific let us recall, once more, the harmonic oscillator results
as sampled in Fig. 1 and in Eq. (3) above. In place of the canonical Jordan-
block limit of Eq. (7) one obtains, for them, an alternative, infinite-
dimensional but partitioned Jordan-block limit

$\lim_{g\to
g^{(EP)}}\,H^{(HO)}(g)\sim\left(\begin{array}[]{cc|cc|cc}2&1&0&0&0&\ldots\\\
0&2&0&0&0&\ldots\\\ \hline\cr 0&0&6&1&0&\ldots\\\ 0&0&0&6&0&\ldots\\\
\hline\cr 0&0&0&0&10&\ldots\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\ddots\end{array}\right)\,$ (28)

of the form of Eq. (15) with infinite sequence of the energy mergers available
in closed form, $\eta_{j}=4j-2$ [44].

The EP singularities of our present $N<\infty$ models (12) lead to an
analogous canonical-representation limit with the known values of $\eta_{j}$.
In particular, for our $N=10$ model (18) characterized by the secular
polynomial of Eq. (26) and by the canonical form (15) of the EP10 limit with
$K=5$, it would be easy to construct the so called transition matrices
$Q^{{(10)}}$ and to evaluate the canonical-representation form of the
Hamiltonian,

$H^{{(canonical)}}(w)\,=[Q^{{(10)}}]^{-1}\,H^{{(10)}}(w)\,.Q^{{(10)}}\,$ (29)

In the EP limit we would get

$\lim_{w\to w^{(EP)}}\,H^{(10)}(w)\sim
H^{{(canonical)}}(1)=\left(\begin{array}[]{cc|cc|cc|cc|cc}-\sqrt{3}&1&&&&&&&&\\\
0&-\sqrt{3}&&&&&&&&\\\ \hline\cr&&-1&1&&&&&&\\\ &&0&-1&&&&&&\\\
\hline\cr&&&&0&1&&&&\\\ &&&&0&0&&&&\\\ \hline\cr&&&&&&1&1&&\\\ &&&&&&0&1&&\\\
\hline\cr&&&&&&&&\sqrt{3}&1\\\ &&&&&&&&0&\sqrt{3}\\\ \end{array}\right)\,.$
(30)

Such a canonical-Hamiltonian matrix is block-diagonal. At a fixed algebraic
multiplicity of EPN (i.e., at $N=10$ in this case) the number $K$ of its
independent eigenvectors (called the geometric multiplicity of EPN) is maximal
(here, we have $K=5$).

###  6.2 Asymmetric real-matrix models.

One of the next-step model-building strategies could be inspired by the less
explored non-numerical constructions of Refs. [69, 70]. The necessary
simplification of the technicalities has been achieved there by the reduction
of the class of the eligible Hamiltonians to the mere tridiagonal real and
real-parameter-dependent asymmetric matrices admitting off-diagonal
interaction terms. In contrast to our preceding models, the weakly non-local
interactions of such a type proved useful, e.g., in the pseudo-Hermitian
models of scattering [71, 72, 73, 74, 75, 76].

For an illustration of their specific merits let us recall now the six-by-six-
dimensional special case of the $N$ by $N$ matrices of Ref. [69], and let us
complement it by an ${\cal O}(g)$ perturbation. This leads to one of the most
user-friendly real-matrix two-parametric Hamiltonians, viz.,

$H^{(toy)}(g,\lambda)=\left[\begin{array}[]{cccccc}-5+g&\sqrt{5+5\,{\it\lambda}}&0&0&0&0\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr-\sqrt{5+5\,{\it\lambda}}&-3&2\,\sqrt{2+2\,{\it\lambda}}&0&0&0\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&-2\,\sqrt{2+2\,{\it\lambda}}&-1&3\,\sqrt{1+{\it\lambda}}&0&0\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr
0&0&-3\,\sqrt{1+{\it\lambda}}&1&2\,\sqrt{2+2\,{\it\lambda}}&0\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr
0&0&0&-2\,\sqrt{2+2\,{\it\lambda}}&3&\sqrt{5+5\,{\it\lambda}}\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr
0&0&0&0&-\sqrt{5+5\,{\it\lambda}}&5-g\end{array}\right]\,.$ (31)

The real, non-degenerate and equidistant spectrum is obtained in a
$g-$dependent unitarity-compatible interval ${\cal D}^{(toy)}(g)$ of
parameters $\lambda$. The simplest proof becomes available in the unperturbed
case with $g=0$. One merely has to recall the closed formulae of Ref. [69]
yielding the admissibility interval of $\lambda\in(-\infty,0)$ or, in the
real-matrix case, the narrower range of $\lambda\in(-1,0)$.

###  6.3 The parameter-controlled change of the geometric multiplicity.

In model (31) with small $g>0$ we may omit, as trivial, the half-line of
parameters $\lambda<-1$ at which the matrix becomes complex but Hermitian. We
are left with the variability of the single unitarity-supporting parameter
$\lambda\in(-1,\mu(g))={\cal D}^{(toy)}(g)$ with $\mu(g)\leq 0$, and we notice
that the parameter-dependence of the spectrum is entirely different from our
preceding models. At the left boundary $\lambda=-1$ the matrix
$H^{(toy)}(g,-1)$ becomes diagonal, i.e., Hermitian and tractable as a
truncated conventional harmonic oscillator with equidistant spectrum. In
contrast, the spectral pattern is very different at the right boundary of
${\cal D}^{(toy)}$: See Fig. 5 for illustration.

Figure 5: The $\lambda-$dependent spectra of Hamiltonian (31) at $g=0$
[spectrum (a)], $g=1/500$ [spectrum (b)], $g=1/40$ [spectrum (c)] and $g=1/5$
[spectrum (d)].

The comparatively elementary nature of the model facilitates a detailed
interpretation of the shapes of the spectra. At the smallest $g$s the value of
the upper bound $\mu(g)$ is determined by the central energy merger
representing an EP of order two (EP2). We may set $\mu(g)=\lambda^{(EP)}_{1}$.
In the opposite extreme of a large shift $g$ one gets another, different
behavior and formula for $\mu(g)=\lambda^{(EP)}_{0}=\lambda^{(EP)}_{2}$. The
change of the pattern clearly reflects the fragility of the off-cental states
exhibiting, at larger $g$s, the confluence of their EP2 singularities (in the
notation of Ref. [77] one would write EP=EP2$\oplus$EP2).

The main qualitative  difference from the dynamics of the “local” models (12)
can be now formulated as the following observation.

###### Conjecture 5

In model (31) the separate generic $g-$dependent EP2 coordinates
$\lambda=\lambda^{(EP2)}_{j}({g})$ of the pairwise mergers of the neighboring
energies $E_{2j}^{(10)}({\lambda},g)$ and $E_{2j+1}^{(10)}({\varrho},g)$ with
$j=0,1,2$ are all strictly decreasing functions of ${g}\in(0,1/5)$ such that

$\lambda^{(EP2)}_{1}({g})<\lambda^{(EP2)}_{0}({g})=\lambda^{(EP2)}_{2}({g})<0\,$
(32)

near the origin (i.e., for $g\ll 1/40$), and such that

$0>\lambda^{(EP2)}_{1}({g})>\lambda^{(EP2)}_{0}({g})=\lambda^{(EP2)}_{2}({g})\,$
(33)

at the larger $g\gg 1/40$.

From this bracketing feature one can immediately deduce the following obvious
result which we will give here without a formal proof.

###### Proposition 6

In model (31) there exists an EP6=EP2$\oplus$EP2$\oplus$EP2 singularity
$g^{(EP6)}\in(0,1/5)$ with the geometric multiplicity $K=3$.

The approximate numerical estimate of $g^{(EP6)}\approx 1/40$ has been used to
display the corresponding $\lambda-$dependence of the energies in Fig. 5
[marked as “spectrum (c)”]. Obviously, the local boundary of the physical
interval ${\cal D}^{(toy)}$ is given by the function
$\lambda^{(EP2)}_{1}({g})$ at $g<g^{(EP6)}$, and by the function
$\lambda^{(EP2)}_{0}({g})=\lambda^{(EP2)}_{2}({g})$ at $g>g^{(EP6)}$.

We may conclude that our toy model (31) admits a smooth transition from the
dynamical regime with the minimal geometric multiplicity $K=1$ of the EP6 at
$g=g^{(EP6)}_{(K=1)}=0$ to its maximal-geometric-multiplicity alternative with
$K=3$ at $g=g^{(EP6)}_{(K=3)}\approx 1/40$. In the former case all of the
bound-state energies converge, in a way prescribed by Eq. (6), to the single
EP value $\eta=E^{(EPN)}$ with $N=6$ at
$\lambda^{(EP)}=\lambda^{(EP)}(g)=\lambda^{(EP)}(0)$,

$\lim_{{\lambda}\to{\lambda}^{(EP)}}E_{j}({\lambda})=E^{(EP6)}\,,\ \ \ \
j=0,1,\ldots,5\,.$ (34)

In [69], via solvable tridiagonal real-matrix models we managed to simulate
such a minimal geometric multiplicity behavior of the energies for an
arbitrary preselected finite Hilbert-space dimension $N<\infty$. Using a
brute-force numerical search such a type of construction with minimal $K=1$
remains feasible even in the models which are realistic [78].

In the opposite extreme of the dynamical scenario near
EPN=EP2$\oplus$EP2$\oplus\ldots\oplus$EP2 with the even algebraic EP
multiplicity $N=2K$ (such that $K$ now represents the maximal geometric
multiplicity) the energy degeneracy is “maximally incomplete”, having
proceeded merely pairwise,

$\lim_{{\lambda}\to{\lambda}^{(EP)}}E_{n_{j}}({\lambda})=\eta_{j}\,,\ \ \ \
j=1,2,\ldots,K\,,\ \ \ \ n_{1}=0,1\,,\ \ n_{2}=2,3\,,\ \ \ldots\,\ \
n_{K}=2K-2,2K-1\,.$ (35)

For our model (31) we just have to insert $K=3$ and specify
$\lambda^{(EP)}=\lambda^{(EP)}\left(g^{(EP6)}_{(K=3)}\right)$.

Along similar lines one can simulate the genuine quantum phase transition
phenomena with an optional geometric multiplicity $K$. The first applications
of such an approach may already be found in the elementary methodical toy
models [79], with the next stage of developments to be aimed at the topical
realistic applications of the theory, say, in the descriptions of the
mechanism of the Bose-Einstein condensation using the multi-bosonic pseudo-
Hermitian Bose-Hubbard Hamiltonians [77, 80, 81, 82].

Multiple related mathematical questions remain open. Nevertheless, using the
standard Kato’s terminology [18], we certainly will have to distinguish, at a
fixed algebraic EPN multiplicity $N$, between the occurrence of a minimal
geometric multiplicity $K=1$ [leading to the canonical-representation limit of
Eq. (7)], of a maximal geometric multiplicity $K=N/2$ [yielding the
alternative canonical-representation limit of Eq. (15)], and of all of the
other possibilities in between these two extremes. This leads us to our final
methodical conclusion.

###### Proposition 7

Any given EPN-supporting quantum closed-system Hamiltonian may be
characterized, in its EPN limit, by its canonical $N$ by $N$ matrix form
$H^{{(canonical)}}$ with the most general direct-sum alias block-diagonal-
matrix structure

$H^{{(canonical)}}=\bigoplus_{j=1}^{K}\,J^{(N_{j})}(\eta_{j})\,,\ \ \ \
N_{1}+N_{2}+\ldots N_{K}=N\,$ (36)

containing nontrivial partitions $N_{j}\geq 2$.

The latter operator EPN limit is fully characterized by the partitioning of
$N$ (check some of its number-theory aspects in [83]) and by the $K-$plet of
the EPN energies $\eta_{j}$. Thus, every classification of phase transitions
should refer to the pair of the multiplicites $N$ (algebraic) and $K$
(geometric). The above-studied minimal- and maximal$-K$ models also become
reclassified as the two extreme special cases which are, in some sense, just
most elementary.

###  6.4 Outlook.

The phenomenology-oriented core of our present message is that one of the most
promising innovative means of the control of unitary quantum dynamics should
be sought in a purposeful manipulation with the Kato’s exceptional points
$g^{(EP)}$.

At the first sight such a statement sounds like an oxymoron because the
unitarity of the evolution requires, in Schrödinger picture, the self-
adjointness of the Hamiltonian, while such a requirement is manifestly
violated by $H(g)$ at $g=g^{(EP)}$. For this reason it is necessary to
emphasize the existence of the two tacit assumptions behind the PHQM SP
theory. The first one is that we really exclude  the singularity $g=g^{(EP)}$,
and that we only work in its vicinity ${\cal D}_{(physical)}$, with the
parameter $g$ admitted to lie arbitrarily close to $g^{(EP)}$ (i.e., formally,
$g^{(EP)}\in\partial{\cal D}_{(physical)}$).

The second tacit assumption is more standard and means the acceptance of the
currently very popular PHQM update of quantum theory. In this framework the
self-adjointness of $H(g)$ is considered $g-$dependent or, more precisely,
metric-operator-dependent, $\Theta(g)-$dependent. Precisely due to this
freedom, the desirable limiting transition $g\to g^{(EP)}$ can always be
performed in a mathematically consistent and unitarity-compatible manner.

In the current literature, unfortunately, the EP-related field of
phenomenology is predominantly developed in its applications to the open
(i.e., in other words, manifestly non-unitary) quantum systems [31] and/or to
various non-quantum or even non-linear systems [26, 27, 84, 85]. One of the
reasons is that in such a setup the model-building process is technically
easier, being still allowed to work with the trivial choice of $\Theta(g)=I$.

In this sense, we tried to lower here the related psychological barriers.
Having emphasized the fundamental aspect of the strictly unitary theory
(requiring, typically, nontrivial metrics $\Theta(g)\neq I$), we illustrated
its user-friendly nature by a detailed analysis of certain finite-dimensional
$N$ by $N$ matrix benchmark Hamiltonians $H^{(N)}(g)$.

The validity of our conclusions remains model-independent of course. In their
brief summary let us emphasize that one of the key benefits of the PHQM
formulation of the theory lies in its capability of covering multiple
apparently exotic system-evolution scenarios in which $g$ is not too far from
$g^{(EP)}$. Then, the metric $\Theta(g)$ becomes very different from the
conventional choice of $\Theta(g)=I$ of textbooks. Besides an undeniable
phenomenological appeal of the anisotropy $\Theta(g)\neq I$ the second deep
merit of the scenario lies in the one-to-one correspondence between the
geometry of Hilbert space and the characteristics of the EP. In this manner,
the behavior of dynamics becomes directly controlled by the characteristics of
the EP, i.e., by its algebraic multiplicity $N$ and by its geometric
multiplicity $K$. In the vicinity of a given EP, the latter two integers will
characterize the dynamics in a unified qualitative manner tractable as a
certain quantum analogue of the classical Thom’s catastrophe theory.

##  7 Summary

In Introduction we formulated our present project as a transfer of the Thom’s
classical concept of catastrophes to quantum theory. We reminded the readers
that the geometric nature of the Thom’s theory (in which the stability of a
long-time equilibrium of the system in question is mimicked and simulated by
the local stability of a local minimum of the so called Lyapunov function
$V(x)$) cannot easily be transferred to quantum mechanics, i.a., due to the
phenomenon of tunneling (see also [9]).

Now, let us add that there still exist multiple parallels between the present
considerations and the Thom’s theory. Indeed, in the latter case, a
classification of classical catastrophes was achieved via the reduction of
arbitrary $V(x)$s to its “canonical” form. The resulting bifurcation scenarios
were given the intuitively appealing names (like the “fold catastrophe” with
“canonical” one-parametric $V(x)=x^{3}+ax$, or the “cusp catastrophe” with the
two-parametric but still one-dimensional $V(x)=x^{4}+ax^{2}+bx$, etc [5]).

All this made the classical Thom’s theory popular. On this background we
pointed out, in [86], that many of the standard Lyapunov functions $V(x)$
could rather easily be reinterpreted as mimicking certain strictly quantum
analogues of the classical elementary catastrophes. Indeed, once we decided to
define the catastrophes, qualitatively, as the “sudden shifts in behavior
arising from small changes in circumstances” [3], we were immediately able to
reinterpret many (i.e., not all!) Lyapunov functions $V(x)$ as the “benchmark”
quantum potentials in Schrödinger Eq. (1) with $H=-\triangle+V(x)$.

The latter idea found its constructive applications even in more dimensions,
with $x\in\mathbb{R}^{d}$ at nontrivial $d=2$ in [87], or at the more
realistic $d=3$ in [88]. Nevertheless, the price had to be paid for the
strictly shared locality of the benchmark potentials $V(x)$. This made the
fairly close classical-quantum analogy incomplete and, unfortunately, just
approximative. Indeed, a key weakness of the approach lied in the nature of
the assignment of a suitable EP parameter $g^{(EP)}$ to the corresponding
quantum catastrophe. The reason was that in a way motivated by the above-cited
Stone theorem, the Hamiltonians were chosen self-adjoint. This implied that
Im$(g^{(EP)})\neq 0$. Thus, the unavoidable presence of a small imaginary
components in the parameters made the process of reaching the phase transition
non-unitary. In other words, the simulation of the quantum “energy-level-
degeneracy” catastrophe (achieved, in our preceding sections, due to the
hidden Hermiticity of $H$) would require an analytic continuation. Without
such a modification of the model, the “shifts in behavior” would not be
“sudden”, and the well known “avoided level mergers” would be experimentally
observed. In comparison, our present, EP-related simulation of the energy-
level mergers proved more successful, exact and “unavoided” (see also, in this
context, the exactly solvable non-Hermitian differential-operator model in
[43]).

In any case, a word of warning must be added. The point is that the domain of
the realistic physics in which one deals with the concept of the quantum phase
transition (for reference, the readers should consult, e.g., the Sachdev’s
classical monograph [24]) is, naturally, much larger than its EP-based
subdomain as studied and clarified in our present paper. In parallel, the
source of optimism concerning the future developments of our present approach
could be sought in the possibility of a partial return to the open-system
theory. Indeed, in its more ambitious forms one could use, typically, Lindblad
operators [14] or Liouvillians [89, 90, 91]. In such a framework, the present
classification of some of the EP-based phase-transition processes could also
be, in a next-step development, included.

Many open question emerge in such an open-system setting at present. They are
mostly connected with the specific, Liouvillean-picture-related phenomena like
quantum jumps [89], in a way moving beyond the limitations characterizing
various standard quantum master equation descriptions [92]. In our preceding
text we only stressed that when speaking about Hamiltonians with EPs, one
usually deals with the information about the environment which is incomplete.
Now, let us add that some more sophisticated open-system Hamiltonians might
remain Hermitian and, thus, fundamental. In the constructions of this type
(see, e.g., [93, 94]) the consistency of the theory is achieved, via
introduction of the so called Langevin force, in the Heisenberg picture, i.e.,
not in the present SP framework.

## Acknowledgements

The author is grateful to Excellence project PřF UHK 2211/2022-2023 for the
financial support.

## Data Availability

No datasets were generated or analysed during the current study.

## Author Contribution statement

The author is the single author of the paper.

## Competing interests

The author declares no competing interests.

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