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arXiv:1412.6634v1 [quant-ph] 20 Dec 2014
Multiply Degenerate Exceptional Points
and Quantum Phase Transitions
Denis I. Borisov,
Institute of Mathematics CS USC RAS, Chernyshevskii str., 112, Ufa, Russia, 450008
and
Bashkir State Pedagogical University, October Rev. st., 3a, Ufa, Russia, 450000
e-mail: BorisovDI@yandex.ru
and
Frantiˇsek Ruˇziˇcka and Miloslav Znojil
Nuclear Physics Institute ASCR, Hlavn´ı 130, 250 68 ˇ
Reˇz, Czech Republic
e-mail: fruzicka@gmail.com and znojil@ujf.cas.cz
1
Abstract
The realization of a genuine phase transition in quantum mechanics requires that at least one of
the Kato’s exceptional-point parameters becomes real. A new family of finite-dimensional and
time-parametrized quantum-lattice models with such a property is proposed and studied. All
of them exhibit, at a real exceptional-point time t= 0, the Jordan-block spectral degeneracy
structure of some of their observables sampled by Hamiltonian H(t) and site-position Q(t). The
passes through the critical instant t= 0 are interpreted as schematic simulations of non-equivalent
versions of the Big-Bang-like quantum catastrophes.
2
1 Introduction
Although the concept of phase transition originates from classical thermodynamics, it recently
found a new area of applicability in the context of quantum mechanics where one may decide
to work with the pseudo-Hermitian (the term used in mathematics, see reviews [1, 2]) alias
PT −symmetric (using the terminology of physicists, see review [3]), manifestly non-Hermitian
Hamiltonians which are still capable of generating a stable, unitary evolution of the quantum
system in question.
A key phenomenological novelty is that whenever our quantum Hamiltonian H6=H†varies
with a real parameter (say, H=H(λ)), the “physical intervals” of the acceptability of λare,
typically, different from the whole real axis. In the mathematical language of Kato [4] one can
also say that for the analytic operator functions H(λ) of the parameter, the whole interval (or
rather a union of intervals) lies in an open complex set Dand that the (in general, complex) points
of its boundary ∂Dcoincide with the Kato’s exceptional points (EPs) as introduced in loc. cit..
The main mathematical feature of the EPs λ(EP )
j∈∂Dis that the limiting operators H(EP )=
limλ→λ(EP )
j
H(λ) cease to be tractable as physical Hamiltonians because even if the spectrum of
the energies happens to remain real these operators cease to be diagonalizable. In a suitable basis
they acquire a Jordan-blocks-containing canonical form [1, 5, 6]. Thus, even if one of values of
λ(EP )
jis real, the limiting transition λ→λ(E P )
jmust still be perceived as a process during which
certain eigenvectors of H(λ) “parallelize” and become linearly dependent, resulting in the loss of
the usual probabilistic tractability of the quantum system in question [7] - [13].
It is worth adding that in the vast majority of the standard applications of quantum mechanics
the EP parameters λ(EP )
jare not real so that one cannot “cross” them when a parameter moves
just along the real line. In such a case, as a rule, the boundaries of the domain of unitarity coincide
with the boundaries of the Riemann surface Rof the analyticity of the operator function H(λ)
so that one must speak about an end of the applicability of the Hamiltonian rather than about a
quantum phase transition as realized within the same physical model.
In our present paper we intend to pay attention to the scenarios in which the EP values
λ(EP )
j∈∂Dare real and do not lie on any boundary of R. In such an arrangement (observed
already, in [14], in 1998) one may treat the EPs as the points of a true analytic realization of
quantum phase transitions.
2 Pure quantum states in parallel representations
2.1 Three Hilbert space pattern
Whenever one reveals that the spectrum of a non-Hermitian operator of an observable of a quantum
system (be it a Hamiltonian H6=H†or a position operator Q6=Q†, etc) is all real and discrete, one
feels tempted to expect that a smooth theoretical translation of all predictions into the language
3
of the textbook quantum mechanics may be performed [15].
One of the realizations of such a conceptually appealing project has been proposed in nuclear
physics (cf. [16]). According to our recent summaries [17, 18] of this possibility one simply
has to make use of the simultaneous representation of the quantum states in three alternative
Hilbert spaces H(F,S,P ). Out of the triplet the first space (viz., H(F), often chosen in the form of
L2(R)) is most friendly. Naturally, whenever this space leaves our observables non-Hermitian, it
must be declared unphysical. Fortunately, in the second, standard and physical space H(S)the
Hermitization of the same operators of observables may be comparatively easily achieved via the
mere introduction of an amended, metric-mediated (i.e., often, non-local) inner product.
In the whole scheme, the latter space is finally (and, usually, constructively) declared unitarily
equivalent to the third Hilbert space H(P)with trivial metric which, as a rule, appears prohibitively
complicated and inaccessible to any constructive fructification [16].
2.2 Five Hilbert space pattern
The generic technical nontriviality of the whole three-Hilbert-space (THS) parallel-representation
recipe may force its users to apply the trick twice. More details may be found in [19]. In essence,
the first, preparatory application yields just a tentative, simplified inner-product metric ΘT, the
study of which may be motivated, e.g., by the generic difficulty of the verification of the reality
of the spectrum or of the closed-form construction of any less elementary amendments of the
tentative positive-definite metric. Subsequently, with ΘT= Θ†
T6=Iat our disposal, the final
specification of the second, “sophisticated” metric ΘSis expected to be guided by the redirection
of emphasis from mathematics to phenomenology.
The following diagram characterizes the resulting five-Hilbert-space representation of a given
4
quantum system,
(initial stage)
friendly space H(F)
friendly observable F6=F†
false metric Θ(F)=I
(1)the left map,preparatory
(aim :simplicity)ւ ց
(2)the right,correct Dyson′s map
(aim :real world )
(intermediate result)
test space H(T)
F=F♯= Θ−1
TF†ΘT
trial ΘT= Ω†
TΩT6=I
spectral reality proof
6=
(final result)
standard space H(S)
F=F‡= Θ−1
SF†ΘS
correct ΘS= Ω†
SΩS6=I
experimental predictions
ւր respective unitary equivalences ցտ
(mathematical reference)
auxiliary space H(A)
(math.)
f(math.)= ΩTFΩ−1
T=f†
(math.)
(trivial metric)
nonequivalent outcomes
(physical reference)
prohibited space H(P)
(phys.)
f(phys.)=f†
(phys.)of textbooks,
isospectral to F .
(1)
Typically, one starts from an observable Hamiltonian F=Hor position F=Q(etc) which are
all defined in H(F). One then moves towards the first nontrivial (though still unphysical) positive
definite artificial metric candidate ΘT(say, to prove the reality of the spectrum). In the second
step of construction one proceeds to the ultimate analysis of the standard representation of the
system using a realistic, physical ΘSwhich is often known solely in an approximate form [20].
3 Exceptional points in a schematic model
In the context of our present paper the key purpose of the start of analysis from an operator
of observable Fwhich is defined in an unphysical Hilbert space H(F)(i.e., which is manifestly
non-Hermitian there and which will be mostly sampled by the operator of position Qin what
follows) is that such operators often possess the real exceptional points [21] - [23].
3.1 Traditional studies starting from a Hamiltonian
In the traditional considerations one usually assumes that in the vicinity of a real Kato’s EP value
of parameter λ(EP )
j∈∂D ⊂ Rthe basis in the Hilbert space H(F)(where the superscript (F)stands
5
for “friendly” [18]) is such that the whole diagonalizable part of a pre-determined Hamiltonian
H(λ) is diagonalized. Thus, at λ(E P )
jone may only pay attention to the non-diagonalizable part
of the Hamiltonian. The latter operator may be also assumed to have acquired the standard
canonical form of a direct sum of Jordan blocks. Naturally, one could further restrict attention
just to one of the Jordan blocks (of matrix dimension N), considering its small vicinity in the
following special form of the multiparametric finite- and tridiagonal-matrix toy model with an
additional up-down symmetry of its matrix elements,
H(toy)=
0 1 −α0 0 0 0 ... 0 0 0
α0 1 −β0 0 0 ... 0 0 0
0β0 1 −γ0 0 ... 0 0 0
0 0 γ0 1 −δ0... 0 0 0
0 0 0 δ..........
.
..
.
..
.
.
.
.
..
.
..
.
.......0 1 −δ0 0 0
0 0 0 ... 0δ0 1 −γ0 0
0 0 0 ... 0 0 γ0 1 −β0
0 0 0 ... 000 β0 1 −α
0 0 0 ... 0 0 0 0 α0
.(2)
During our preparatory numerical experiments with the spectra of various Hamiltonian matrices
of the generic tridiagonal form (2) (cf. also Refs. [24, 25]) it became clear that the transitions
of quantum systems through their Jordan-block alias multiple-EP (or degenerate-EP) quantum-
phase-transition points may prove to be a phenomenologically relevant and interesting process.
In fact, there emerged no true surprises in the context of mathematics where one simply
observed differing patterns of behavior (and, in particular, of the complexification) of certain
eigenvalues before and after the EP singularity. Unfortunately, once we started thinking about
time tas parameter (with its EP value, say, at t= 0), we immediately imagined that certain
purely formal nontriviality of the three-Hilbert-space (THS) formalism would force us to accept
the restriction to adiabatically slow changes of the system in general and of the Hamiltonian
operator H(t) in particular (cf. [18] for details).
3.2 An amended strategy of analysis based on a given site operator ˜
Q
The unpleasant necessity of the slowness of changes of H(t) seems hardly compatible with the
abrupt nature of the changes of the system near an EP singularity at t= 0. For methodical reasons
it seem reasonable, therefore, to replace the study of the time-dependent Hamiltonians H(t) (which
combine the role of the operators of an observable energy with a partially independent role of the
6
generators of time evolution) by the formally similar but conceptually less confusing study of
some other operators sharing the same perturbed-Jordan-block matrix form but representing, say,
a non-Hermitian spin Σ(t) [26] - [28] or, better, the observable Q(t) of discretized position alias
site in a finite quantum lattice (cf. [29] - [31]).
Our present strategy of the build-up of the theory initiated by the site operator ˜
Qmight have
been supported not only by the above-mentioned circumvention of the problems with adiabatic
approximation (which are basically technical) but also by a few other, physics-oriented arguments.
One of them is based on the observation [32] that whenever one interprets the one-parametric
family of eigenvalues qn(t) of a site operator ˜
Qchosen in virtually any form of a perturbed
N−dimensional Jordan block, then the unfolding pattern of these eigenvalues (very well sampled
by their particular special case qn(t)∼cn√tas derived, in Refs. [33] - [35], for an exactly solvable
discrete and PT −symmetric anharmonic oscillator) resembles the cosmological phenomenon of
Big Bang. Indeed, in loc. cit., all of these eigenvalues stayed complex (i.e., unobservable) before
the Big-Bang instant t= 0 while all of them became observable (i.e., real and non-degenerate and
even, in absolute value, growing quickly) immediately after the Big-Bang time t= 0.
4 The phase-transition interpretation of the real excep-
tional points
In our present paper let us finalize the definition of our family of models (with Hreplaced by Q
in Eq. (2)) in such a way that all of the components of the input J−plet of variable parameters
~
ξ={α,β,...,ω}={ξ1, ξ2, . . . , ξJ}of Eq. (2) become either proportional to an absolute value of
time (thus, we shall have the even-function-superscripted components ξj=ξ(e)
j(t) = |t|defined at
all real t∈R) or proportional to the plain time (for the remaining, odd-function-superscripted
components ξk=ξ(o)
k(t) = t,t∈R). This means that every eligible quantum model of our family
(with J=entier(N/2) in general) may be naturally characterized by the respective superscripts
in vector ξ, i.e., by a word of length Jin the two-letter alphabet {o, e}(i.e., one has two eligible
words 0= (o) and 1= (e) at J= 1, four candidates 0= (oo), 1= (oe), 2= (eo) and
3= (ee) at J= 2, etc).
Let us now add that in accord with the specific implementations of the three-Hilbert-space
pattern of Refs. [16, 18] the key technical task of its users should be seen to lie in the reconstruction
of a suitable metric ΘSfrom a given site-position matrix Q(N)
()(t). At the not too large and positive
times t > 0 such a construction is, due to our choice of tridiagonal Q(N)
()(t), recurrent and entirely
routine [36].
4.1 Spectra at times close to the Big Bang instant
At an illustrative matrix dimension N= 10, the first nontrivial sample of our present lattice-site
operator (2) will contain five free parameters α,... ,ε alias ξ1, ξ2,...,ξ5which we decided to
7
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
0.1
0.05
0
–0.05
–0.1 t
q(t)
Figure 1: The time-dependence of the real part of the spectrum of the perturbed-Jordan-block
ten-by-ten matrix (3) with index 1= (ooooe).
choose in the first nontrivial concrete form of quintuplet t, t, t, t, |t|. As we indicated above, this
choice is encoded in the word 1= (ooooe) yielding the matrix
Q(10)
(1)(t) =
0 1 −t0 0 0 0 0 0 0 0
t0 1 −t0 0 0 0 0 0 0
0t0 1 −t0 0 0 0 0 0
0 0 t0 1 −t0 0000
0 0 0 t0 1 − |t|0000
0 0 0 0 |t|0 1 −t000
0 0 0 0 0 t0 1 −t0 0
0 0 0 0 0 0 t0 1 −t0
0 0 0 0 0 0 0 t0 1 −t
0 0 0 0 0 0 0 0 t0
.(3)
This matrix represents one of many possible t6= 0 perturbations of the 10 times 10 Jordan block
to which it degenerates at the EP time t= 0.
At the small but non-vanishing times t6= 0 the t−dependence of the real eigenvalues of matrix
(3) is displayed in Fig. 1. Once we accept the physical interpretation of such a matrix as a site-
position operator of a dynamical ten-point quantum lattice near its phase transition instant t= 0
of the Big-Bang type, we may conclude that at the positive times t > 0, i.e., after the Big Bang
instant this operator has the whole spectrum real and, hence, it may be perceived as representing
an observable quantity.
We should emphasize that in contrast to an analogous model with 0= (ooooo) (possessing just
an empty real spectrum to the left from t= 0), there now exists a pair of eigenvalues q(t) which
still remain real also before the Big Bang. Thus, Fig. 1 may be perceived as a schematic sample
of an EP-interrupted evolution in which just a simpler, two-site part of our ten-point quantum
8
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
–0.1
–0.05
0
0.05
0.1 t
q(t)
Figure 2: The time-dependence of the real part of the spectrum of the perturbed-Jordan-block
ten-by-ten matrix with index 7= (ooeee).
lattice may be interpreted as observable at t < 0. In other words, one could read the message
delivered by Fig. 1 as opening the possibility of an innovative, quantum cosmology resembling
scenario in which a subspace-based, simpler, two-level observable Universe evolved from the left,
i.e., during the previous Eon and towards its own t= 0 Big Crunch collapse.
4.2 Alternative changes of physics during the pass of our quantum
lattices through the EP time t= 0
In spite of a truly elementary form of our family of the Big-Bang-simulating toy models Q(N)
()(t),
their capability of simulation and variability of alternative Big-Crunch/Big-Bang quantum phase-
transition scenarios seems truly inspiring. In particular, the N= 10 Fig. 1 obtained at =1=
(ooooe) may be complemented by its analogue of Fig. 2 with 7= (ooeee) where the six levels
remain real before the Big Crunch instant, etc.
Naturally, a fully left-right symmetric picture would be obtained at 31 = (eeeee), representing
a schematic ten-dimensional-Universe counterpart to the well known (though very differently sup-
ported and constructed) Penrose’s cyclic-cosmology pattern in which the structure of the Universe
before and after the Big Bang singular time is not expected to be too different [37]. In contrast,
our present family of models may be understood as supporting a rather non-standard hypothesis
of an “evolutionary” cosmology in which the complexity of the structure of the Universe may
change during the EP singularity and, in principle, grow with time from some more elementary
structures existing during the previous Eons.
Needless to add that within the similar “evolutionary cosmology” phenomenological specu-
lations, the structure of the Universe during the newer Eons (including also, e.g., its spatial
dimensionality, etc) could have been also enriched by certain newly emergent qualities, the ob-
servability (i.e., the reality of the corresponding eigenvalues) of which had been suppressed earlier,
emerging only on a sufficiently sophisticated level of the iterative global cosmic evolution.
Once we now return back to the elementary mathematical level of the concrete study of our
9
schematic quantum models, we still have to address a number of multiple less ambitious questions
like, e.g, the purely technical problem of the necessity of the modification of the definition of the
“old” physical Hilbert spaces which had to exist and describe our quantum lattice before t= 0.
In fact, at the finite dimensions N < ∞such a modification remains mathematically more or
less trivial. Indeed, at t < 0 one merely has to preserve (i.e., project out) just the vector space
which is spanned by the eigenvectors of Q(t) with the real eigenvalues. The resulting algorithmic
pattern will partially resemble Eq. (1) in having the very similar five-Hilbert space form of the
diagram in which the left and right triplet of windows represents our quantum lattice alias discrete
Universe before and after the Big Bang, respectively:
Big −Bang instant plus its vicinity :
the same friendly space H(F)
Q(0) not diagonalizable
before−Big−Bang map
t < 0ւ ց after−Big−Bang map
t > 0
less,N′<N observable sites
reduced space H(R)
reduced QR=Q♯
R= Θ−1
RQ†
RΘR
old −timer ΘR= Ω†
RΩR6=I
△=N−N′ghosts projected out
6=
all N sites
standard space H(S)
Q=Q‡= Θ−1
SQ†ΘS
current metric ΘS= Ω†
SΩS6=I
our Eon
ւր respective unitary equivalences ցտ
Hermitian reference
previous Eon space H(P)
(old)
q(old)= ΩRQRΩ−1
R=q†
(old)
extinct physics
→quantum phase transition→
Hermitian reference
this Eon space H(P)
(now)
“our′′ coordinates q(now)=q†
(now)
contemporary physics .
(4)
Naturally, the extension of such a recipe and diagram to the infinte-dimensional Hilbert space
limit N=∞will lead to multiple further open questions in functional analysis. As long as such
a step already lies fairly beyond the scope of our present paper, let us merely conjecture that in
the infinite-dimensional Hilbert-space limit and in the “previous Eon” with t < 0, the necessary
elimination of the “ghost-supporting” spurious subspace (in which the eigenvalues of Q(t) are not
yet real) might proceed in an analogy with the Mostafazadeh’s elimination trick of Ref. [38].
10
5 Discussion
Our present paper was inspired by the recent enormous popularity of the building of quantum mod-
els in which the unitary evolution is guaranteed and controlled, paradoxically, by non-Hermitian
Hamiltonians H6=H†. On this background our attention was shifted from the traditional
PT −symmetric alias pseudo-Hermitian Hamiltonians (such that one has H†=PHP−16=H
in terms of an indeterminate pseudometric P) to the other, less prominent operators of observ-
ables. We emphasized that the choice of some other observables might enrich, first of all, our
insight in some less understood processes in which the quantum system in question is forced to
move through an exceptional-point singularity.
5.1 Considerations inspired by the open problems in physics
For the sake of definiteness we paid attention to certain specific time-dependent site-position
operators Q(t) which were chosen, for the sake of mathematical simplicity, in the form of finite
matrices. In parallel, in a way emphasizing their phenomenological appeal we choose our Nby N
toy-model matrices Q(N)(t) in such a form that at the EP value of time t= 0 they degenerated to an
N−dimensional and quantum-phase-transiton inducing Jordan-block non-diagonalizable matrix
Q(N)(0).
We emphasized that any operator Qrepresenting an observable quantity admits in fact the
same mathematical treatment as the most often considered energy operator alias Hamiltonian. In
particular, once we wish to declare any operator observable in H(S), the equivalent requirement
of the Hermiticity of its image in H(P)must be imposed, say, in the form
q= ΩSQΩ−1
S=q†.
This condition may be also equivalently reformulated, inside the second physical space H(S), as
follows,
Q†(t)ΘS(t) = ΘS(t)Q(t).(5)
In full analogy with the standard THS recipe, one can start, therefore, from any family of the
benchmark-model forms of the operator Qand perform the reconstruction of the admissible metrics
ΘSvia Eq. (5).
In the context of physics we revealed that after an appropriate further sophistication and de-
velopment our initial idea of studying the phase-transition pass through the Big-Bang-resembling
EP singularity of Q(N)(t) at t= 0 could find multiple future applications even in quantum cos-
mology. We conjectured that the traditional alternative scenarios of “nothing before Big Bang”
and of the “cyclic repetition of Big Bangs” (e.g., in the form as proposed by Penrose [37]) could
be, in this light, complemented by the slightly subtler possibilities and speculations about having
“Darwin-like evolution” and various “physics-structure jumps” at the subsequent Big Bangs.
11
Figure 3: The norm of the resolvent R(z) = [z−Q(10)
(t)]−1with = (eooee) after Big Bang, at
positive t= 0.1.
5.2 Considerations inspired by the open problems in mathematics
Naturally, all of the above-sampled phenomenological speculations suffer from the insufficiently
realistic and oversimplified finite-dimensional alias discrete-lattice nature of our toy-model observ-
ables Q(N)
()(t). In this sense, the main opening mathematical challenge (not to be addressed here
at all) may be now seen in the study of N→ ∞ extensions of the discrete-lattice models.
Another set of the open mathematical problems emerges even at the finite matrix dimensions
N < ∞. One of the most important ones concerns the problem of the influence of perturbations
on any toy-model based qualitative result. In the conclusion of this paper let us now pay more
attention to this fairly important subproblem.
Our basic inspiration has been provided by the experience covered by the Trefethen’s and Em-
bree’s book [39] with its extreme emphasis on the constructive considerations and with its rather
persuasive recommendation that the influence of the perturbations should be always characterized
in the language of the so called pseudospectra [40].
An easy and compact introduction in the concept of the pseudospectrum may be provided here
either by the reference to loc. cit. or to Fig. 3. In the latter picture one sees that the purely real
spectrum (i.e., the real-eigenvalue positions zn∈Rof the infinitely high peaks of the norm |R(z)|
of the resolvent) may be perceived as very well approximated by the 0 < ε ≪1 pseudospectra
(which are defined, according to loc. cit., as the - in general, multiply connected - open complex
domains Jεof zin which |R(z)|>1/ε).
12
Figure 4: The norm of the resolvent of Fig. 3 before Big Bang, at negative t=−0.1.
In contrast to the after Big Bang picture as provided by Fig. 3, a conceptually much more
challenging problem emerges at the negative, pre-Big-Bang times at which some of the eigenvalues
of matrix Q(N)
()(t) remain non-real. As we already mentioned above, we are not going to address
this challenge directly but rather we intend to recall the Trefethen’s and Embree’s advice.
Although, in their book, the pseudospectra are predominantly recommended for an estimate
of the influence of a generic perturbation, the role of their study in our present context will be
different. Indeed, in the light of the diagram of Eq. (4) one of the key tasks of the constructive
approach to the quantitative description of our present versions of the quantum phase transitions
lies in the possibility of a clear separation of the “quantum observables before Big Bang“ (i.e., of
a subspace which is spanned by the real-eigenvalue eigenvectors of Q(t)) from the perpendicular
subspace of the unobservable, non-real-eigenvalue “ghosts”.
In practice, naturally, the separation of this type (which, incidentally, resembles strongly the
Gupta-Bleuler trick known from textbooks on quantum electrodynamics) must be performed nu-
merically. This implies that one must specify the domains of applicability of a feasible separation
of this type.
In this context, our present final recommendation is to use the inspection of the pseudospectra
for the purpose. For illustration, let us first recall Figs. 4 (with its equivalent form 5) in which
one sees an entirely distinct numerical separation of the states with the real eigenvalues from the
ghosts. In such a dynamical scenario (with =19 = (eooee)) one may expect that the projector-
operator elimination of the ghosts from the physical before-the-Big-Bang reduced Hilbert space
13
−0.4 0 Re z
−0.4
0
Im z
Figure 5: The real function of Fig. 4 in its two-dimensional representation. The lines marking the
constant norm are the boundaries of the complex domains called “pseudospectra” [39].
−0.4 Re z 0.4
−0.5
Im z
0.5
Figure 6: A rearrangement of the pseudospectra of Fig. 5 at another index, = (eoooe).
14
−0.5 Re z 0.5
−0.3
Im z
0.3
Figure 7: Another set of the before-the-Big-Bang pseudospectra, with = (eoeee).
H(R)will be numerically feasible and straightforward.
In contrast, the inspection of another, =17 = (eoooe) toy model may be expected to
indicate an emergence of the technical and numerical difficulties because the related Fig. 6 shows
that up to the very small εs, the pseudospectral domains Jεof zare shared by more peaks and do
not sufficiently clearly separate each of the two real eigenvalues from its two complex conjugate
neighbors. At the same time, the four “remote” complex eigenvalues still seem to be separated in
a sufficiently clear manner.
From the perturbation-influence-estimate point of view, by far the worst situation is encoun-
tered at = (eoeee) where our last Fig. 7 indicates that and why the separation of the eight-
dimensional ghost subspace may be expected to be truly difficult.
Acknowledgements
D.B. was partially supported by grant of RFBR, grant of President of Russia for young scientists-
doctors of sciences (MD-183.2014.1) and Dynasty fellowship for young Russian mathematicians.
15
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