Acta Polytechnica Vol. 51 No. 4/2011

The Horizons of Observability in PT -symmetric
Four-site Quantum Lattices

M. Znojil

Abstract

One of the key merits of PT -symmetric (i.e., parity times time reversal symmetric) quantumHamiltonians H lies in the
existence of a horizon of the stability of the system. Mathematically speaking, this horizon is formed by the boundary
of the domain D(H) ⊂ RD of the (real) coupling strengths for which the spectrum of energies is real and non-degenerate,
i.e., in principle, observable. It is shown here that even in the elementary circular four-site quantum lattices with D =2
or D =3 the domain of hidden Hermiticity D(H) proves multiply connected, i.e., topologically nontrivial.

Keywords: hidden Hermiticity, spectra and exceptional points, horizons, discrete Schrödinger operators, quantum
graphs, loops, four-site lattices, connectedness, strong-coupling anomalies.

1 Introduction
One of the most interesting formulations of the
standard and robust dictum of quantum mechan-
ics emerged in connection with the acceptance of the
so called PT -symmetric operators of observables,
where P means parity while T represents time re-
versal (cf. review papers [1–3] for an exhaustive dis-
cussion). One of the main reasons for the rebirth of
interest in this new paradigm last year may be seen,
paradoxically, in its impact on classical experimental
optics [4].

The latter experimental activities (i.e., basically,
the emergence of a few successful classical-physics
simulations of quantum effects) re-attracted atten-
tion to the innovative theory. We may mention,
pars pro toto, paper [5] which offered an exhaus-
tive constructive classification of all of the PT -
symmetric quantum Hamiltonians H defined in the
finite-dimensional Hilbert spaces H of dimensions
N = 2 and N = 3.

1 3–3 –1

bcb

Fig. 1: Graphical symbol for the straight-line open-
end four-site lattice. The numbers in the small circles
(=sites) are the unperturbed energies while the letters b
and c near the nearest-neighbor-interaction lines repre-
sent the (real) couplings

Inside the most elementary N ≤ 3 family of mod-
els no real surprises and spectral irregularities have
been encountered. In contrast, in Ref. [6] we found
that certain anomalies certainly emerge at N = 8. In
our present brief continuation of these developments
we intend to show that the simplest models exhibiting
similar irregularities in their spectra already occur,
unexpectedly, as early as at the next Hilbert-space
dimension N = 4.

2 Four-site quantum-lattice
models

2.1 The exactly solvable straight-line
case

In Refs. [7] the successful tractability of more-
than-three-dimensional Hamiltonian matrices re-
sulted from a drastic simplification of their structure.
We merely admitted their tridiagonal versions. In the
language of physics this corresponded to the picture
in which the system lived on an N−site straight-line
lattice endowed with the mere nearest-neighbor inter-
actions. At N = 4 this is schematically depicted in
Figure 1. The small circles represent the sites while
their frame-line connections symbolize the interac-
tions.

The left-right symmetric straight-line lattice of
Figure 1 (i.e., of Refs. [7]) is assigned the Hamiltonian
given in the form of two-parametric real matrix

H = H(4)(b, c) =

⎡
⎢⎢⎢⎢⎢⎢⎢⎣

−3 b 0 0

−b −1 c 0

0 −c 1 b

0 0 −b 3

⎤
⎥⎥⎥⎥⎥⎥⎥⎦

. (1)

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Acta Polytechnica Vol. 51 No. 4/2011

A quantitative analysis of the models is more or less
trivial even at larger N > 4. The curious reader may
find many details, say, in review paper [8].

2.2 PT -symmetric circular lattices
and their simplest four-site
example

Once we replace Figure 1 by its circular version of
Figure 2 we may immediately interpret the new dia-
gram as representing the new N = 4 quantum model
which is given by the following three-parametric four-
by-four matrix form of the Hamiltonian studied in
Ref. [9],

H = H(4)(a, b, c) =

⎡
⎢⎢⎢⎢⎢⎢⎢⎣

−3 b 0 −a

−b −1 c 0

0 −c 1 b

a 0 −b 3

⎤
⎥⎥⎥⎥⎥⎥⎥⎦

. (2)

In the new model with one more coupling which con-
nects the “upper two” sites, the method for con-
structing the boundary ∂D(H) does not change. At
any number N of sites along the (circular) lattice the
reality property of the spectrum of the energies will
remain tractable by the standard mathematical tech-
niques. A few representative samples may be found in
Ref. [10]. Interested readers may search for a broader
mathematical context in Refs. [11, 12].

–1 1

–3 3

c

bb

a

Fig. 2: The circular four-site lattice

Once we restrict our attention just to our special
toy model H(4)(a, b, c) it proves sufficient to recall the
entirely elementary considerations of Ref. [10]. This
leads to the conclusion that the spectrum of energies
is real and nondegenerate if and only if the triplet of
parameters (a, b, c) lies inside the domain

D(H) :=
{

(a, b, c) ∈ R3 | W (a, b, c) > 0 , (3)
Q(a, b, c) > 0 , P (a, b, c) > 0 }

where

W (a, b, c) =
(
8 + c2 − a2

)2 − (4)
4
[
16 − (a + c)2

]
b2 ,

Q(a, b, c) =
[
(a + 3)(c − 1) − b2

]
· (5)[

(a − 3)(c + 1) − b2
]

and
P (a, b, c) = 10 − a2 − 2 b2 − c2 . (6)

In other words, for the couplings moving to the two-
dimensional surfaces of D(H) from inside we observe
that the quadruplets of the real bound-state energies
themselves behave in an easily understandable man-
ner. The reason is that we may rewrite the secular
equation in the form S(s, a, b, c) = 0 where the ener-
gies E± = ±

√
s emerge in pairs and where

S(s, a, b, c) := s2 +
(
−10 + c2 + 2 b2 + a2

)
s + 9 +

6 b2 − 9 c2 + b4 − 2 cab2 − a2 + c2a2 .

This recipe generates the two auxiliary roots

4 s = 4 s(±) = 20 − 2 a2 − 2 c2 − 4 b2 ± 2
√

W (a, b, c)

where we already know the function of Eq. (4),

W (a, b, c) = 64 + 16 c2 − 64 b2 − 16 a2 + c4 +
4 c2b2 − 2 c2a2 + 4 b2a2 + a4 + 8 cab2 .

In the spirit of the general results of Ref. [9] we may
summarize that

1. whenever W (a, b, c) → 0+ the two pairs of ener-
gies approach the two distinct values

E
(W=0)
± = ±

√
(10 − a2 − 2 b2 − c2)/2

representing the two limiting doubly-degenerate
energies;

2. whenever Q(a, b, c) → 0+ just the two energies
move to zero while the other two energies do not
vanish in general,

E0,3 = ±
√

10 − a2 − 2 b2 − c2 ;

3. for P (a, b, c) → 0+ we must expect that all of
the four real energies will vanish simultaneously.

3 Two-parametric simplified
versions of the circular
four-site lattice

3.1 The case of a =0
Naturally, in the no-upper-interaction limit
lim
a→0

H(4)(a, b, c) = H(4)(b, c) we return to the ele-

mentary straight-line model of Figure 1. For our
present purposes it is then sufficient to recall that
the main features of such a simplified model were
described in Refs. [7]. In particular, we know that

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Acta Polytechnica Vol. 51 No. 4/2011

at N = 4 the spectrum of energies remains real and
nondegenerate inside the innermost star-shaped do-
main D(H) ⊂ R2 shown, in Figure 3, as lying inside an
auxiliary circumscribed ellipse. The boundary ∂D(H)
(i.e., the physical horizon of the system in question) is
composed of four hyperbola-shaped curves. The key
features of this example (like the triple intersections
of the boundaries, etc.) generalize, mutatis mutan-
dis, to the family of the similar models at all of the
dimensions N < ∞ [8].

2

1

0

–1

–2

–3 –2 –1 0 1 2 3
c

b

Fig. 3: The graphical determination of the innermost
star-shaped domain D(H) as assigned to the quantum lat-
tice of Figure 1 in Refs. [7]

We are now prepared to replace the elementary
and transparent graphical determination of the star-
shaped domain D(H) assigned to the straight-line
quantum lattice and displayed in Figure 3 by its much
more complicated a �= 0 analogue. At a freely vari-
able a the knowledge of the a = 0 section may serve
us and will still serve us as a very useful independent
test of our forthcoming observations and conclusions.

3.2 The case of b =0

Our four-site toy model H(4)(a, b, c) degenerates to
the trivial non-interacting composition (i.e., the di-
rect sum) of the two N = 2 models at b = 0. For this
reason the b = 0 limiting case should be considered
exceptional.

In the b = 0 two-dimensional special case, even
the general definition of the domain D(H) is slightly
misleading. Indeed, Figure 4, which displays the
three sets of boundaries (viz., the two hyperbolas
W (a, 0, c) = 0, the four straight lines Q(a, 0, c) = 0
and the single circle P (a, 0, c) = 0, respectively),
should not be taken too literally. One of the bound-
aries (viz., the doublet of hyperbolas W (a, 0, c) = 0)
describes in fact a sign-non-changing (i.e., the reality-
of-energies non-changing) curve of the doubly degen-
erate (and, hence, irrelevant and removable) zeros of
the function W (a, 0, c) = (8 + c2 − a2)2.

This means that at b = 0 the domain D(H) is
strictly rectangular and strictly simply connected. In
this context one of the key messages of our present

study is the surprising discovery of the loss of both
of these properties in the general case with the freely
variable coupling strength b.

Incidentally, the multinomial W (a, bspec, c) be-
comes factorizable also at bspec = 1,

W (a, 1, c) = (a + c) ·(
a3 − ca2 − 12 a − c2a + 20 c + c3

)
.

This is an artifact which does not carry any immedi-
ate physical meaning. Its manifestation is of a purely
geometrical character, which will only be briefly men-
tioned later.

3.3 The case of c =0

In the third (and last) preparatory step, let us discuss
the vanishing-coupling special case in which c = 0
and

W (a, b, 0) =
(
8 − a2

)2 − 4 [16 − a2] b2 , (7)
Q(a, b, 0) =

(
3 + b2 + a

)(
3 + b2 − a

)
(8)

and
P (a, b, 0) = 10 − a2 − 2 b2 . (9)

The detailed study of precisely this special case re-
veals in fact the possibility of the emergence of a
topological nontriviality in the general case. The de-
tailed form of such a c = 0 hint may be seen in Fi-
gure 5 and in its magnified version 6. As long as the

–3

–2

–1

0

1

2

3

c

–3 –2 –1 0 1 2 3

a

Fig. 4: The degenerate case of the simply connected rec-
tangular domain D(H) at b =0

–2

–1

0

1

2

b

–3 –2 –1 0 1 2 3

a

Fig. 5: The triply connected nature of the triple-overlap
domain D(H) for the quantum lattice of Figure 2 at c =0
(i.e., in the no-central-coupling extreme)

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Acta Polytechnica Vol. 51 No. 4/2011

condition Q(a, b, 0) > 0 degenerates to the elemen-
tary constraint

3 + b2 > a > −3 − b2

just the left and right small horizontal-parabolic seg-
ments (with their extreme at b = 0 and |amax| =
3) should be cut out of the elliptic domain with
P (a, b, 0) > 0 as inadmissible since Q(a, b, 0) < 0
there. As long as we only have 16 > a2, the remain-
ing constraint W (a, b, 0) > 0 acquires the form

|b| <
1
4

∣∣8 − a2∣∣
√

16 − a2
(10)

of the geometric limitation of the admissible range of
b by the two intersecting or rather broken and touch-
ing curves. The nonnegative function W (a, 0, 0) =
(8 − a2)2 solely vanishes at a2 = 8.

–0.4

–0.2

0

0.2

0.4

b

2.8 2.9 3 3.1 3.2

a

Fig. 6: Same as Figure 5 (detail)

–0.4

–0.2

0

0.2

0.4

b

2.8 2.9 3 3.1 3.2

a

Fig. 7: The deformation of Figure 6 at c =0.95

The allowed region decays into three disconnected
open sets (cf. Figures 5–7). The big one is formed
by the eye-shaped vicinity of the origin, with its ex-
tremes at the points (a, b)± = (±

√
8, 0). The other

two smaller open sets are fish-tail-shaped. In the pic-
tures these two domains are easily spotted as contain-
ing the respective b = 0 intervals of |a| ∈ (

√
8, 3). In

this sense they may be expected to support a per-
turbatively inaccessible “strong-coupling” dynamical
regime.

An additional indication of the suspected emer-
gence of topological as well as dynamical nontrivial-
ities is offered by Figure 7, where the same separa-
tion of a strong-coupling piece of the domain D(H) is
shown to survive up to the very extreme of c ≈ 1.

4 The domain of
cryptohermiticity in the
full-fledged
three-parametric dynamical
scenario

4.1 The auxiliary domains and their
boundaries

The domain D(H) of parameters for which Hamilto-
nian H(4)(a, b, c) generates the unitary evolution is
defined as an intersection of the triplet of domains
D(P,Q,W) in R3. Let us now leave all of the three
parameters a, b and c freely variable and recall that
• the domain D(P) is defined by the inequality

P (a, b, c) = 10 − a2 − 2 b2 − c2 > 0 .

It is compact, so that we may restrict our atten-
tion just to the intervals of b2 < 5, a2 < 10 and
c2 < 10. At any fixed b2 < 5 the section of this
first auxiliary domain coincides with the interior
of a central circle in the a − c plane with radius
R =

√
10 − 2b2;

• the allowed interior of the triply connected do-
main D(Q) is defined by the inequality

Q(a, b, c) =
[
(a + 3)(c − 1) − b2

]
·[

(a − 3)(c + 1) − b2
]

.

In the a − c plane the boundaries of this do-
main are two hyperbolas sampled at b2 = 1 in
Figure 8.

–5

0

5

–5 0 5

allowed

allowed

allowed

allowed

allowed

forbidden

forbidden

forbidden

forbidden

a

c

Fig. 8: Boundaries Q(a, b, c) = 0 and forbidden parts of
the a − c plane as sampled at b2 =1

The third auxiliary domain D(W) is defined by the
inequality

W (a, b, c) =
(
8 + c2 − a2

)2 −
4
[
16 − (a + c)2

]
b2 > 0 .

The description of this domain is slightly less trivial.
The interior of this domain covers all the exterior of

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Acta Polytechnica Vol. 51 No. 4/2011

the strip where |a + c| > 4. Then the interior of this
strip may be reparametrized,

c − a = 2 τ (c, a) ∈ (−∞, ∞) ,
c + a = 4 sin ϕ(c, a) , ϕ(c, a) ∈ (−π/2, π/2)

making the rest of the domain D(W) determined by
the elementary inequality

|b| <
|1 + τ (c, a) sin[ϕ(c, a)]|

cos[ϕ(c, a)]
. (11)

This means that within the restricted range of
τ (c, a) ∈ (−

√
10,

√
10) the growth of |b| → ∞ must

be compensated by the decrease of cos[ϕ(c, a)] → 0,
i.e., by the convergence c → ±1 − a. This makes the
strip-restricted part of the domain D(W) very small
but increasing with the decrease of |b| from a suffi-
ciently large initial value.

4.2 The boundaries of the
cryptohermiticity domain

The study of the overlaps of the three auxiliary do-
mains D(P,Q,W) may be started at the maximal ad-
missible plane of b = b(P) =

√
5, which touches

the boundary ∂D(P) at a = c = 0. This point
still lies outside the domains D(W) and D(H) since
W (0,

√
5, 0) = 64 − 320 < 0. In a search for the first

touch between the b−plane and boundary ∂D(H) we
must diminish our b and move into the interior of
D(P).

In the first illustrative example at b =
√

5−1/100,
our Fig. 9 displays the motion of the triplet of bound-
aries ∂D(P,Q,W) projected into the a − c real plane.
This picture shows that the corresponding section
of the first domain D(P) becomes nonempty. Still,
it just occupies the interior of a very small circle
C(b) = ∂D(P)|b=f ixed with the center at the origin.

The interior of the second domain D(Q) is per-
ceivably bigger since, in the manner indicated by
Fig. 8 above, it occupies the large domain between
the two outermost, b − dependent hyperbolic curves
H1,2(b) ⊂ ∂D(Q)|b=f ixed. The overlap D(H) itself
remains empty because the third domain D(W) is lo-
calized behind the two remaining and less trivially
parametrized curves G1,2(b) ⊂ ∂D(W)|b=f ixed.

During the subsequent decrease of b sampled by
Figure 9, the two curves Gj (b) and Hj (b) (assigned
the same subscript j = 1 or j = 2) get closer to each
other while the internal circle C(b) gets larger. At
each j and at the same value of b both the curves
Gj ,Hj touch the circle C(b). At a still smaller
b =

√
5 − 1/2 = 1.736 067 977 they already move

inside, sharing their two separate intersections with
the circle. This situation is illustrated in Figure 10.

The formation of the first two non-empty com-
ponents of the physical domain D(H) emerges dur-
ing the further decrease of b. Due to the fact that

the two triple-intersection points between C(b), Gj
and Hj move apart at any j, these two compo-
nents remain disconnected, extremely narrow and
eye-shaped. These “eyes” look like “almost closed”
and “slowly opening” with the further decrease of
b. Graphically, the generic situation in illustrated by
Figures 11 and 12.

–3

–2

–1

0

1

2

3

c

–3 –2 –1 0 1 2 3

a

Fig. 9: b =
√
5 − 1/100 =2.226067977

–3

–2

–1

0

1

2

3

c

–3 –2 –1 0 1 2 3

a

Fig. 10: b =
√
5 −1/2 =1.736067977

–3

–2

–1

0

1

2

3

c

–3 –2 –1 0 1 2 3

a

Fig. 11: b =
√
5 − 1= 1.236067977

–3

–2

–1

0

1

2

3

c

–3 –2 –1 0 1 2 3

a

Fig. 12: b =1.01

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Acta Polytechnica Vol. 51 No. 4/2011

–3

–2

–1

0

1

2

3

c

–3 –2 –1 0 1 2 3

a

Fig. 13: b =1

–3

–2

–1

0

1

2

3

c

–3 –2 –1 0 1 2 3

a

Fig. 14: b =0.999

–3

–2

–1

0

1

2

3

c

–3 –2 –1 0 1 2 3

a

Fig. 15: b =0.6

–3

–2

–1

0

1

2

3

c

–3 –2 –1 0 1 2 3

a

Fig. 16: b =0.4

The next qualitative change of the pattern oc-
curs at the above-mentioned special value of b = 1,
at which we touch the saddle of the surface ∂D(W).
Slightly before this happens, we encounter the situ-

ation depicted in Figure 12 where the two separate
subdomains of the physical domain D(H)|b=f ixed al-
ready almost touch. Next they do touch (cf. Fi-
gure 13) and, subsequently, get connected (cf. the
next Figure 14).

Surprisingly enough, below the saddle point b = 1
the topological surprises are still not at the end.
There is no real news even at b = 0.6 (cf. Figure 15).
Nevertheless, in the latter picture we must already
pay attention to the two subdomains with the maxi-
mal a2s.

Having selected just the right end of the (sym-
metric) picture at the positive a ≈ 3, we re-
veal the emergence of a tendency towards a new
intersection between the (hitherto, safely exter-
nal and non-interfering) second branches of the

Q(a, b, c, )−related hyperbolas H(second)j and of the
back-bending boundaries ∂D(W)|b=f ixed. For exam-
ple, these curves get very close to each other but still
do not intersect yet at b = 0.4, also staying outside
the central circular domain D(P) (cf. Figure 16).

A change in the pattern is finally achieved slightly
below b = 0.4, at the moment when both of the
new intersection candidates touch the circle C(b) =
∂D(P)|b=f ixed in a single point. Subsequently, this
point splits into the pair of triple intersections. The
further decrease of |b| forms the pattern which is sam-
pled in Figure 17 at b = 0.2 and in Figure 18 at
b = 0.1.

–3

–2

–1

0

1

2

3

c

–3 –2 –1 0 1 2 3

a

Fig. 17: b =0.2

–3

–2

–1

0

1

2

3

c

–3 –2 –1 0 1 2 3

a

Fig. 18: A return to the triply connected D(H) at b =0.1

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Acta Polytechnica Vol. 51 No. 4/2011

–1.5

–1

–0.5

0

0.5

1

c

2.6 2.7 2.8 2.9 3 3.1 3.2

a

Fig. 19: A magnified detail of Figure 18

0.9

0.92

0.94

0.96

0.98

1

1.02

c

2.98 2.99 3 3.01 3.02

a

Fig. 20: A magnified detail of Figure 19

Certainly, the key features of the new situation
are better visible, at the illustrative b = 0.1, in its
magnified presentation, as mediated by Figures 19
and 20. Now we may return to the limiting pattern
of Figure 4, where we witness the abrupt change of
the topology caused by the final confluence of the
straight and backbanding branch of the boundary
∂D(W)|b=f ixed in the limit b → 0.

Obviously, in a way confirmed by the complemen-
tary results of section 3.3, such a confluence of the
boundaries only occurs in the limit, so that the thee-
dimensional version of the open set D(H) is ultimately
confirmed to be triply connected.

5 Summary

In the history of pure mathematics the specifica-
tion of the horizons ∂D(H) (called, often, “discrim-
inant surfaces” in this context) has been perceived
as a challenging and rather difficult problem even
in its first “unsolvable” case characterized, in our
present notation, by the Hilbert-space dimensions
N = 5 [11]. In certain mathematically natural direc-
tions real progress is of amazingly recent date [12].
Remarkable parallel developments also occurred in
several applied-mathematics oriented studies paying
attention to the natural presence of more symme-

tries in the Hamiltonian [7] and/or to the introduc-
tion of more observable quantities within a given phe-
nomenological quantum model [13].

In a constructive, more pragmatic setting as sam-
pled by our recent paper [6], we restricted our at-
tention to the topological problem of horizons. The
most obvious motivation for such an effort has been
given by the fact that the disconnectedness of do-
main D(H) immediately requires the transition from
its traditional perturbation-theory descriptions (with
a recommended recent compact sample given in [14])
to non-perturbative methods, or to strong-coupling
perturbation techniques [15]. In Ref. [6] the paral-
lel and less formal motivation has been emphasized
to lie in a systematic search for the possible physical
origin of the dynamical anomalies in a kinematical
nontriviality of the topology of phase space.

The conclusions of our present paper are encour-
aging. Firstly we have demonstrated that for many
purposes it may be sufficient to use the matrices with
a not too large N . Secondly, we have shown an in-
crease in the feasibility describing models H with ad-
ditional symmetries. At the first nontrivial Hilbert-
space dimension N = 4 we encountered, for example,
the decrease of the minimal necessary number of pa-
rameters to D = 3 or even to D = 2 .

Thirdly, we clarified that once we work with a
tridiagonal N by N Hamiltonian H

(N)
0 complemented

by a computationally suitable specific perturbation,
the existence of the disconnected subdomains in D(H)
opens direct access to the strong-coupling dynamical
regime.

Fourthly, on the mathematical side, we are now
able to recommend the use of auxiliary symmetries in
the Hamiltonians (e.g., of the ones of Ref. [16]). In
such cases, the algebraic secular equations pertain-
ing to the model often happen to factorize, leading
to polynomial equations of perceivably lower orders.
The latter fact rendered our toy model easily solv-
able.

Last but not least, it seems worth emphasizing
that on the background given by Refs. [6, 17] it took
some time for us to imagine that the anomalies of
spectra could also be sought at dimensions as small
as N = 4. In this sense the message of our present
study is encouraging. Several specific spectral ir-
regularities as observed at N = 8 in Ref. [6] were
found also for matrices with the dimension as low as
N = 4. Our model reconfirmed the hypothesis of a
very close, topology-related connection between the
loop-shaping of the lattices (i.e., presumably, Betti
numbers in continuous limit) and the existence of
strong-coupling dynamical anomalies in the spectra
of the energy levels.

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Appendix A. The
three-Hilbert-space formulation
of quantum mechanics

In section 2 of Ref. [5], one of the most compact intro-
ductions to the abstract formalism of PT -symmetric
quantum mechanics (PTSQM) is given. Thus, we
may shorten the introductory discussion and restrict
ourselves to a few key comments on the general the-
oretical framework.

In such a compression, the PTSQM formalism
may be characterized as such a version of entirely
standard quantum mechanics in which, in princi-
ple, the system in question is defined in a certain
prohibitively complicated physical Hilbert space of
states H(P), where the superscript may be read as
abbreviating “prohibited” as well as “physical” [3].

Typical illustrative realistic examples may be
sought in the physics of heavy nuclei, where the cor-
responding fermionic states are truly extremely com-
plicated. In the latter exemplification the first half
of the PTSQM recipe lies in the transition to a suit-
able, unitary equivalent Hilbert space, H(P) → H(S),
where the superscript “(S)” may stand for “suitable”
or “simpler” [3].

In the above-mentioned realistic-system illustra-
tion, for example, the new space H(S) coincided with
a suitable “interacting boson model” (IBM). In the
warmly recommended review paper of this field [18]
it has been emphasized that the requirement of uni-
tary equivalence between the two Hilbert spaces H(P)
and H(S) may only be achieved in two ways. Either
the corresponding boson-fermion-like mapping Ω be-
tween these two Hilbert spaces (known, in this con-
text, as the Dyson’s mapping) remains unitary (and
the mathematical simplification of the problem re-
mains inessential) or is admitted to be non-unitary (a
less restrictive option which may enable us to achieve
a really significant simplification, say, of the compu-
tational determination of the spectra).

What remains for us to perform and explain now
is the second half of the general PTSQM recipe. Its
essence lies in weakening the most common unitarity
requirement imposed upon the Dyson mapping,

Ω† = Ω−1

to the mere quasi-unitarity requirement

Ω† = Θ Ω−1 .

The symbol Θ �= I represents here the so-called
metric operator which defines the inner product in
Hilbert space H(S).

More details using the present notation may be
found in [3]. Just a few of them have to be recalled

here. Firstly, the main source of the purely techni-
cal simplifications of the efficient numerical calcula-
tions (say, of the spectra of energies) is to be seen in
the introduction of the third, purely auxiliary Hilbert
space H(F), where the superscript “(F)” combines the
meaning of “friendlier” with “falsified” [3].

By definition, the two Hilbert spaces H(S) and
H(F) coincide as the mathematical vector spaces (“of
ket vectors” in the Dirac terminology). We only re-
place the nontrivial metric Θ(S) ≡ Ω†Ω of the former
space by its trivial simplification Θ(F) ≡ I in the
latter Hilbert space. As an immediate consequence,
the latter space acquires the status of an auxiliary,
manifestly unphysical space which does not carry any
immediate physical information or probabilistic in-
terpretation of its trivial though, at the same time,
maximally mathematically friendly inner products.

Appendix B. The role of
PT -symmetry
In its most widely accepted final form, described in
Ref. [1], the PTSQM recipe complements the lat-
ter general scheme by another assumption. It may
be given the mathematical form of the introduction
of the second auxiliary, manifestly unphysical vec-
tor space K(P) which is, by definition, not even the
Hilbert space. In fact, this fourth vectors space is
assumed endowed with the formal structure of Krein
space [19].

Ref. [14] may be consulted for more details. Here,
let us only remind the readers that the symbol P in
the superscript carries a double meaning and com-
bines the mathematical role of the indefinite metric
P (defining in fact the Krein space) with an input
physical interpretation (usually, of the operator of
parity). In addition, the theoretical pattern

H(S) ↔ K(P) ↔ H(F) . (12)

is complemented by the requirement that there exists
a “charge” operator C such that the (by assumption,
non-trivial, sophisticated) metric Θ(S) �= I which de-
fines the inner product in the second Hilbert space
H(S) coincides with the product of the two above-
mentioned operators,

Θ(S) = PC . (13)

The contrast between the feasibility of the N �= 3
constructions presented in Ref. [5] and the discour-
aging complexity and incompleteness of the next-step
N = 4 constructions as performed in paper [17] and
in its sequels [7] was also thoroughly discussed in our
review [8]. In our present text we do not deviate
from the notation and conventions accepted for this
review. We pay attention solely to the class of models

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Acta Polytechnica Vol. 51 No. 4/2011

where the N−dimensional matrix of parity is unique
and given, in advance, in the following form,

P =

⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 . . . . . . 0

0 −1 0 . . . . . . 0
0 0 1 0 . . . 0

0 0 0 −1
. . .

...
...

. . .
. . .

. . .
. . . 0

0 . . . 0 0 0 ∓1

⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (14)

In parallel, we made use of the time-reversal operator
T of the form presented, e.g., in Ref. [5] as mediat-
ing just the transposition plus complex conjugation
of vectors and/or matrices. We should add that once
we work with real vectors and matrices, we are even
allowed to perceive T as a mere transposition.

Acknowledgement

Work supported by GAČR grant Nr. P203/11/1433,
by MŠMT “Doppler Institute” project Nr. LC06002,
and by the Inst. Res. Plan AV0Z10480505.

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Miloslav Znojil
E-mail: znojil@ujf.cas.cz
Nuclear Physics Institute ASCR
250 68 Řež, Czech Republic

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