ap-5-10.dvi


Acta Polytechnica Vol. 50 No. 5/2010

Rectifiable P T -symmetric Quantum Toboggans
with Two Branch Points

M. Znojil

Abstract

Certain complex-contour (a.k.a. quantum-toboggan) generalizations of Schrödinger’s bound-state problem are reviewed
and studied in detail. Our key message is that the practical numerical solution of these atypical eigenvalue problems
may perceivably be facilitated via an appropriate complex change of variables which maps their multi-sheeted complex
domain of definition to a suitable single-sheeted complex plane.

Keywords: quantum bound-state models, wave-functions with branch-points, complex-contour coordinates, PT-sym-
metry, tobogganic Hamiltonians, winding descriptors, single-sheet maps, Sturm-Schrödinger equations.

1 Introduction
The one-dimensional Schrödinger equation for bound
states

−
h̄2

2m
d2

dx2
ψn(x)+ V (x)ψn(x) = En ψn(x) , (1)

ψn(x) ∈ L2(R)

is one of the most friendly phenomenological models
inquantummechanics [1]. Forvirtuallyall of the rea-
sonable phenomenological confining potentials V (x)
the numerical treatment of this eigenvalue problem
remains entirely routine.
During certain recent numerical experiments [2]

it became clear that many standard (e.g., Runge-
Kutta [3]) computational methods may still en-
counter new challenges when one follows the advice
by Bender and Turbiner [4], by Buslaev and Grec-
chi [5], by Bender et al [6] or by Znojil [7], and when
one replaces the most common real line of coordi-
nates x ∈ R in ordinary differential Eq. (1) by some
less trivial complex contour of x ∈ C(s)whichmaybe
conveniently parametrized, whenever necessary, by a
suitable real pseudocoordinate s ∈ R,

−
h̄2

2m
d2

dx2
ψn(x)+ V (x)ψn(x) = En ψn(x) , (2)

ψn(x) ∈ L2(C) .

Temporarily, the scepticism has been suppressed by
Weideman [8] who showed that many standard nu-
merical algorithmsmay be reconfirmed to lead to re-
liable results even for many specific analytic samples
of complex interactions V (x) giving real spectra via
Eq. (2).
Unfortunately, the scepticism reemergedwhenwe

proposed, in Ref. [7], to study so-called quantum to-
boggans characterized by the relaxation of the most

common tacit assumption that the above-mentioned
integrationcontours C(s)must always lie just inside a
single complex plane R0 equipped by suitable cuts.
Subsequently, the reemergence of certain numerical
difficulties accompanying the evaluation of the spec-
tra of quantum toboggans has been reported by Bı́la
[9] and by Wessels [10]. Their empirical detection of
the presence of instabilities in their numerical results
may be recognized as one of the key motivations for
our present considerations.

2 Illustrative tobogganic
Schrödinger equations

2.1 Assumptions

Whenever the complex integration contour C(s)
used in Eq. (2) becomes topologically nontrivial
(cf. Figures 1–4 for illustration), it may be inter-
preted as connecting several sheets of the Riemann
surface R(multisheeted) supporting the general solu-
tion ψ(general)(x) of the underlying complex ordi-
nary differential equation. It is well known that
these solutions ψ(general)(x) are non-unique (i.e.,
two-parametric – cf. [9]). From the point of view
of physics this means that they may be restricted
by some suitable (i.e., typically, asymptotic [4, 5])
boundary conditions (cf. also Ref. [7]). In what fol-
lows we shall assume that
(A1) these general solutions ψ(general)(x) live on un-

bounded contours called “tobogganic”, with the
name coined and with the details explained in
Ref. [7];

(A2) ourparticular choiceof the tobogganic contours

C(s)= C(tobogganic)(s) ∈ R(multisheeted)

84



Acta Polytechnica Vol. 50 No. 5/2010

–1

–1.5 –0.5 0.5
Re x

Im x

Fig. 1: The central segment of the typical PT -symmetric
double-circle tobogganic curve of x ∈ C(LR)(s)withwind-
ing parameter κ = 3 in Eq. (10). This curve is obtained
as the imageof the straight line of z ∈ C(0)(s) at ε =0.250

Re x

-Im x

–1

–2 –1 1

Fig. 2: An alternative version of the double-circle curve
of Figure 1 obtained at the “almost maximal” ε =
ε
(critical) − 0.0005 (note that ε(critical) ∼ 0.34062502)

Re x

-Im x

–1

–2 –1 1

Fig. 3: The extreme version of the double-circle curve

C(LR)(s) at ε
<
≈ ε(critical)

Re x

-Im x

–1

–2 –1 1

Fig. 4: The change of topology at ε
>
≈ ε(critical) when

Eq. (10) starts giving the single-circle tobogganic curves
C(RL)(s) at κ =3

will be specified by certain multiindex � so that
C(tobogganic)(s) ≡ C(�)(s);

(A3) for the sake of brevity our attentionmay be re-
stricted to the tobogganicmodelswhere themul-
tiindices � are nontrivial but still not too com-
plicated. For this reasonwe shall study just the
subclass of the tobogganic models

−
h̄2

2m
d2

dx2
ψn(x) + V

(2)
(j) (x)ψn(x)=

En ψn(x) , (3)

ψn(x) ∈ L2(C(�))

containing, typically, potentials

V
(2)
(1) (x) = V(HO)(x)=

x2 +

[
F

(x − 1)2
+

F

(x +1)2

]
, (4)

F 0 1

or

V
(2)
(2) (x) = V(ICO)(x)=

ix3 +

[
G

(x − 1)2
+

G

(x +1)2

]
,(5)

G 0 1

with two strong singularities inducing branch
points in the wave functions.

In this manner we shall have to deal with the two
branch points x

(BP)
(±) = ±1 in ψ

(general)(x). In
the language of mathematics the obvious topologi-
cal structure of the correspondingmulti-sheetedRie-
mann surface R(multisheeted) will be “punctured” at
x
(BP)
(±) = ±1. In the vicinity of these two “spikes” we
shall assume the generic, “logarithmic” [11] structure
of R(multisheeted).

85



Acta Polytechnica Vol. 50 No. 5/2010

2.2 Winding descriptors �

The multiindex � will be called a “winding descrip-
tor” in what follows. It will be used here in the
form introduced inRef. [12], where each curve C(�)(s)
has been assumedmoving from its “left asymptotics”
(where s � −1) to a point which lies below one of
the branch points x

(BP)
(±) = ±1. During the further

increase of s one simply selects one of the following
four alternative possibilities:

• one moves counterclockwise around the left
branch point x

(BP)
(−) (this move is represented by

the first letter L in the “word” �),
• one moves counterclockwise around the right
branch point x

(BP)
(+) (this move is represented by

letter R),
• onemovesclockwisearoundthe left branchpoint

x
(BP)
(−) (this move is represented by letter Q or

symbol L−1 ≡ Q),
• one moves clockwise around the right branch
point x

(BP)
(+) (this move is represented by letter

P or symbol R−1 ≡ P).
In thismannerwemay compose themoves and char-
acterize each contour by a word � composed of the
sequenceof letters selected fromthe four-letteralpha-
bet R, L, Q and P . Once we add the requirement of
P T -symmetry (i.e., of a left-right symmetry of con-
tours) we arrive at the sequence of eligible words �
of even length 2N.
At N =0 wemay assign the empty symbol � = ∅

or � =0 to the one-parametric family of the straight
lines of Ref. [5],

C(0)(s) ≡ s − iε , ε > 0. (6)

Thus, one encounters precisely four possible arrange-
ments of the descriptor, viz,

� ∈
{
LR , L−1R−1 , RL , R−1L−1

}
, N =1 (7)

in the first nontrivial case. In the more complicated
caseswhere N > 1 itmakes sense to re-express the re-
quirement of P T -symmetry in the formof the string-
decomposition � = Ω

⋃
ΩT where the superscript T

marks an ad hoc transposition, i.e., the reverse read-
ing accompanied by the L ↔ R interchange of sym-
bols. Thus, besides the illustrative Eq. (7) we may
immediately complement the first nontrivial list

Ω ∈
{
L , L−1 , R , R−1

}
, N =1 ,

by its N =2 descendant{
LL, LR, RL, RR, L−1R, R−1L, LR−1 , RL−1 ,

L−1L−1, L−1R−1, R−1L−1, R−1R−1
} (8)

etc. The four “missing” words LL−1 , L−1L , RR−1

and R−1R had to be omitted as trivial here because

they cancel each other when interpreted as wind-
ings [12].

3 Rectifications

3.1 Formula
The core of our present message lies in the idea that
the non-tobogganic straight lines (6)maybemapped
on their specific (called “rectifiable”) tobogganic de-
scendants. For this purpose one may use the follow-
ing closed-form recipe of Ref. [12],

M :
(
z ∈ C(0)(s)

)
→
(
x ∈ C(�)(s)

)
(9)

where one defines

x = −i
√
(1 − z2)κ − 1 . (10)

This formula guarantees the P T symmetry of the re-
sulting contour aswell as the stability of the position
of our pair of branch points. Another consequence
of this choice is that the negative imaginary axis of
z = −i|z| is mapped upon itself.
Some purely numerical features of the mapping

(10)mayalso be checkedvia the freely available soft-
ware of Ref. [13]. On this empirical basis we shall
require exponent κ to be chosen here as an odd pos-
itive integer, κ =2M +1, M =1,2, . . .. In this case
theasymptoticsof the resultingnontrivial tobogganic
contours (with M 
= 0) will still parallel the κ = 1
real line C(0)(s) in the leading-order approximation.

3.2 Sequences of critical points

An inspection of Figures 2 and 3 and a comparison
with Figures 4 and 5 reveals that one should expect
the emergence of sudden changes of the winding de-
scriptors � during a smooth variation of the shift

Re x
Im x

–6

–4

–2

–4 –2 2

Fig. 5: The fully developed version of the single-circle to-
bogganic curve C(RL)(s) obtained at κ =3 and ε =0.400

86



Acta Polytechnica Vol. 50 No. 5/2010

ε > 0 of the initial straight line of z introduced via
Eq. (6). Formally we may set � = �(ε) and mark the
set of corresponding points of changes of �(ε) by the

sub- and superscript in ε
(critical)
j .

A quantitative analysis of these critical points
is not difficult since it is perceptibly simplified by
the graphical insight gained via Figures 2–4 and via
their appropriately selected more complicated de-
scendants. Trial and error constructions enable us to
formulate (and, subsequently, to prove) the very use-
ful hypothesis that the transition between different
descriptors �(ε) always proceeds via the same mech-
anism. Its essence is characterized by the confluence
and “flip” of the curve at any j = 1,2, . . . , M in
ε = ε

(critical)
j . At this point two specific branches of

the curve C(�)(s) touch and reconnect in the manner
sampled by the transition fromFigure 2 to Figure 4.
The key characteristics of this flip is that it takes

place in the origin so that we can determine the
point x

(critical)
j = 0 which carries the obvious geo-

metric meaning mediated by the complex mapping
(10). Thus, the vanishing x

(critical)
j =0 is to be per-

ceived as an image of some doublet of z = z
(critical)
j

or, due to the left-right symmetry of the picture, as
an image of a symmetric pair of the pseudocoordi-

nates s
(critical)
j = ±

∣∣∣s(critical)j ∣∣∣.
At any κ =2M +1 the latter observations reduce

Eq. (6) to the elementary relation

1=
{
1+[i(s − iε)]2

}κ
(11)

which may be analyzed in the equivalent form of the
following 2M +1 independent relations

e2πim/(2M+1) =1+(is+ε)2 =1+ε2−s2+2is ε . (12)

These relations numbered by m =0, ±1, . . . , M may
further be simplified via the two known elementary
trigonometric real and non-negative constants A and
B such that[

1 − e2πim/(2M+1)
]
= A ± iB .

In terms of these constants we separate Eq. (12) into
it real and imaginary parts yielding the pair of rela-
tions

s2 − ε2 − A =0 , 2s ε = B . (13)
As longas ε > 0wemayrestrict ourattention tonon-
negative s and eliminate s = B/(2ε). The remaining
quadratic equation

B2/(2ε)2 − ε2 − A =0

finally leads to the following unique solution of the
problem,

ε =
1

√
2

√
−A +

√
A2 + B2 . (14)

This formula perfectly confirms the validity and pre-
cision of our illustrative graphical constructions.

4 Samples of countours of
complex coordinates

For the most elementary toboggans characterized
by the single branching point the winding descrip-
tor � becomes trivial because it is being formed by
the words in a one-letter alphabet. This means
that all the information about windings degener-
ates just to the length of the word � represented by
an (arbitrary) integer [14]. Obviously, these mod-
els would be too trivial from our present point of
view.
In an opposite direction one could also contem-

plate tobogganic models where a larger number of
branch points would have to be taken into account.
An interesting series of exactly solvable models of
this form may be found, e.g., in Ref. [15]. Natu-
rally, the study of all of these far reaching general-
izations would still proceed along the lines which are
tested here on the first nontrivial family character-
ized by the presence of the mere two branch points
in ψ(x).
From the pedagogical point of view the merits

of the two-branch-point scenario comprise not only
the simplicity of the formulae (cf., e.g., Eq. (10) in
the preceding section) but also the feasibility and
transparency of the graphical presentation of the in-
tegrationcontours C(�) of the tobogganicSchrödinger
equations. This assertionmay easilybe supported by
a few explicit illustrative pictures.

4.1 Rectifiable tobogganic contours
with κ =3

The change of variables (10) generating the rectifi-
able tobogganic Schrödinger equations must be im-
plemented with due care because the knot-shaped
curves C(�)(s) may happen to run quite close to the
points of singularities at certain values of s. This is
well illustrated by Figure 1 or, even better, by Fig-
ure 6. At the same time all our Figures clearly show
thatonecancontrol theproximity to the singularities
by means of the choice of the shift ε of the (conven-
tionally chosen) straight line of the auxiliary variable
z ∈ C(0) given by Eq. (6).
Once we fix the distance ε of the complex line

C(0) from the real line R we may still vary the odd
integers κ. Vice versa, even at the smallest κ = 3
the recipe enables us to generate certain mutually
non-equivalent tobogganic contours C(�)(s) in the
ε−dependentmanner. This confirms the existence of
discontinuities. Their emergence and form are best
illustrated by the pair of Figures 3 and 4.

87



Acta Polytechnica Vol. 50 No. 5/2010

Re x
Im x

–1

–0.5 0.5

Fig. 6: The quadruple-circle tobogganic curve of x ∈
C(LLRR)(s). With winding parameter κ = 5 in Eq. (10)
this sample is obtained at ε = ε

(critical)
1 − 0.0005, i.e.,

just slightly below the first critical value of ε
(critical)
1 ∼

0.21574990

Re x
Im x

–1

–0.5 0.5

Fig. 7: The topologically different, triple-circle curve
C(RRLL)(s) obtained at κ =5 and ε = ε(critical)1 +0.0005

We may conclude that in general one has to deal
here with the very high sensitivity of the results to
the precision of the numerical input or to the preci-
sion of the computer arithmetics. This confirms the
expectations expressed in our older paper [12] where
we emphasized that the descriptor � is not necessar-
ily easily inferred from a nontrivial, detailed analysis
of the mapping M.

4.2 Rectifiable tobogganic contours
with κ ≥ 5

Oncewe select the next odd integer κ =5 inEq. (10)
the study of the knot-shaped structure of the result-
ing integration contours C(�)(s) becomes even more
involved because in the generic case sampled by Fig-
ure 6 the size of the internal loops proves unexpect-
edly small in comparison. As a consequence, their
very existence may in principle escape our attention.

Thus, one might even mistakenly perceive the curve
ofFigure 6asan inessential deformationof the curves
in Figures 1 or 2.
Naturally, not all of the features of our toboganic

integration contours will change during the transi-
tion from κ = 3 to κ = 5. In particular, the partial
parallelism between Figures 2 and 6 survives as the
similar global-shape partial parallelism between Fig-
ures 4 (where κ = 3) and 7 (where κ = 5). More-
over, a certain local-shape partial parallelismmaybe
also foundbetweenFigure 2 (where the twoupwards-
oriented loops almost touch at κ = 3) and Figure 8
(where the two downwards-oriented “inner” loops al-
most touch at κ = 5). The latter parallels seem to
sample a certainmore generalmechanism, since Fig-
ure 4 also finds its replica inside the upper part of
Figure 9, etc. Obviously, the next-step transition
from κ = 5 to κ = 7 (etc.) may also be expected
to proceed along similar lines.

Re x
Im x

–10

–4 4

Fig. 8: The other extreme triple-circled κ = 5 curve
C(RRLL)(s) as emerging at ε = ε(critical)2 − 0.005, i.e.,
close to the second boundary ε

(critical)
2 ∼ 0.49223343

Re x
Im x

–10

–4 4

Fig. 9: The twice-circling tobogganic κ = 5 curve
C(RLRL)(s) as emerging slightly above the second criti-
cal shift-parameter, viz, at ε = ε

(critical)
2 +0.005

88



Acta Polytechnica Vol. 50 No. 5/2010

Table 1: Transition parameters for κ =2M +1 with M =1,2, . . . ,6

M ε
(critical)
(m) pseudocoordinate angle

m B [critical shift in C(0)(s)
] ∣∣∣s(critical)(m) ∣∣∣ ϕ(critical)(m)

1 1 0.8660 0.34062501931660664017 1.2712 0.2618

2 1 0.9510 0.49223342986833679823 0.96606 0.4712

2 0.5878 0.21574989943840034163 1.3622 0.1571

3 1 0.7818 0.49560936234793313854 0.78876 0.5610

2 0.9749 0.41300244005317039597 1.1803 0.3366

3 0.4339 0.15634410200136762402 1.3876 0.1122

4 1 0.6428 0.47438630343334929661 0.67749 0.6109

2 0.9848 0.47917814904271720218 1.0276 0.4363

3 0.8660 0.34062501931660664017 1.2712 0.2618

4 0.3420 0.12231697600600608108 1.3981 0.08727

5 1 0.5406 0.44984366535166445772 0.60092 0.6426

2 0.9096 0.49834558687374848153 0.91265 0.4998

3 0.9898 0.42964189183273983152 1.1519 0.3570

4 0.7557 0.28670826353957054964 1.3180 0.2142

5 0.2817 0.10037407570525388131 1.4034 0.071400

6 1 0.4647 0.42666576745054519911 0.54460 0.6646

2 0.8230 0.49875399287559237235 0.82504 0.5437

3 0.9927 0.47264256935707423545 1.0502 0.4229

4 0.9350 0.38168235795277279438 1.2249 0.3021

5 0.6631 0.24649719795540125795 1.3451 0.1812

6 0.2393 0.085076232785825555735 1.4065 0.06042

For computer-assisted drawing of the graphical
representationof the curves C(�) the formulaeofpara-
graph3.2 should be recalledas the source of themost
useful informationabout the criticalparameters. The
extended-precision values of the underlying coordi-
nates of the points of instability are needed in such
an application. Their M ≤ 6 sample is listed here in
Table 1.
On this basiswemay summarize that at a generic

κ the variation (i.e., in all of our examples, the
growth)of the shift ε makes certain subspirals of con-
tours C(�) larger andmoving closer and closer to each
other. In this context ourTable 1 could, in principle,
serve as a certain systematic guide towards a less in-
tuitive classification of our present graphical pictures
characterizing transitions between different winding
descriptors � and, hence, between the topologically
non-equivalent rectifiable tobogganic contours C(�).
During such phase-transition-like processes [4] the
value of ε crosses a critical point beyond which the
asymptotics of the contours change. As a conse-
quence, also the spectra of the underlying tobogganic

quantum bound-state Hamiltonians will, in general,
be changed [16].

5 Conclusions
Wehave confirmed the viability of an innovated, “to-
bogganic” version of P T -symmetric Quantum Me-
chanics of bound states in models where the gen-
eral solutions of the underlying ordinary differential
Schrödinger equation exhibit two branch-point sin-
gularities located, conveniently, at x(BP) = ±1.
In particular we have clarified that many topo-

logically complicated complex integrations contours
which spiral around the branch points x(BP) in var-
ious ways may be rectified. This means that one
can apply an elementary change of variables z(s) →
x(s) and replace the complicated original tobogganic
quantumbound-state problem by an equivalent sim-
plifieddifferential equation defined along the straight
line of complex pseudocoordinates z = s − iε.
In detail a few illustrative rectificationshavebeen

described where we have succeeded in assigning the

89



Acta Polytechnica Vol. 50 No. 5/2010

differentwindingdescriptors � to the tobogganic con-
tours controlledsolelyby thevariationof the “initial”
complex shift ε. An interesting supplementary result
of our present considerationsmaybe seen in the con-
structive demonstration of the feasibility of an ex-
plicit description of these transitions between topo-
logically non-equivalent quantum toboggans char-
acterized by non-equivalent winding descriptors �.
Nevertheless, a full understanding of these structures
remains an open problem recommended for deeper
analysis in the nearest future.
In summary we have to emphasize that our

present rectification-mediated reconstruction of the
ordinary-differential-equation representation of qu-
antum toboggans can be perceived as an important
step towards their rigorous mathematical analysis
and, in particular, towards an extension of the ex-
isting rigorous proofs of the reality/observability of
the energy spectra to these promising innovativephe-
nomenological models.

Acknowledgement

Support from Institutional Research Plan AV0Z
10480505 and from MŠSMT Doppler Institute
project LC06002 is acknowledged.

References

[1] Flügge, S.: Practical Quantum Mechanics I, II.
Berlin, Springer, 1971.

[2] Znojil,M.: Experiments inPT-symmetric quan-
tum mechanics, Czech. J. Phys. Vol. 54 (2004),
p. 151–156 (quant-ph/0309100v2).

[3] Znojil, M.: One-dimensional Schrödinger equa-
tion and its “exact” representation on a dis-
crete lattice, Phys. Lett. Vol. A 223 (1996),
p. 411–416.

[4] Bender, C. M., Turbiner, A. V.: Analytic Con-
tinuation of Eigenvalue Problems, Phys. Lett.
Vol. A 173 (1993), p. 442–445.

[5] Buslaev, V., Grechi, V.: Equivalence of un-
stable anharmonic oscillators and double wells,
J. Phys. A: Math. Gen. Vol. 26 (1993),
p. 5541–5549.

[6] Bender, C. M., Boettcher, S.: Real spec-
tra in non-Hermitian Hamiltonians having PT
symmetry, Phys. Rev. Lett. Vol. 80 (1998),
p. 5243–5246;

Bender, C. M., Boettcher, S., Meisinger, P. N.:
PT-symmetric quantum mechanics, J. Math.
Phys. Vol. 40 (1999), p. 2201.

[7] Znojil, M.: PT-symmetric quantum toboggans,
Phys. Lett. Vol. A 342 (2005), p. 36–47.

[8] Weideman, J.A.C.: Spectraldifferentiationma-
trices for the numerical solutions of Schrödinger
equation,J.Phys.A:Math.Gen.Vol.39 (2006),
p. 10229–10238.

[9] Bı́la,H.: Non-Hermitian Operators in Quantum
Physics (PhD thesis supervised by M. Znojil).
Prague, Charles University, 2008;

Bı́la, H.: Pramana, J. Phys. Vol. 73 (2009),
p. 307–314.

[10] Wessels, G. J. C.: A numerical and analytical
investigation into non-Hermitian Hamiltonians
(Master-degree thesis supervised byH.B.Geyer
and J. A. C. Weideman). Stellenbosch, Univer-
sity of Stellenbosch, 2008.

[11] see, e.g., “branch point” in
http://eom.springer.de.

[12] Znojil, M.: Quantum toboggans with two
branch points. Phys. Lett. Vol. A 372 (2008),
p. 584–590 (arXiv: 0708.0087).

[13] Novotný, J.:
http://demonstrations.wolfram.com/
TheQuantumTobogganicPaths.

[14] Znojil, M.: Spiked potentials and quantum to-
boggans.J. Phys. A:Math. Gen.Vol.39 (2006),
p. 13325–13336 (quant-ph/0606166v2);

Znojil,M.: Identification of observables in quan-
tum toboggans. J. Phys. A: Math. Theor. Vol.
41 (2008), p. 215304 (arXiv:0803.0403);

Znojil, M.: Quantum toboggans: models ex-
hibiting a multisheeted PT symmetry. J. Phys.:
Conference Series, Vol. 128 (2008), p. 012046;

Znojil, M., Geyer, H. B.: Sturm-Schroedinger
equations: formula formetric.Pramana J.Phys.
Vol. 73 (2009), p. 299–306 (arXiv:0904.2293).

[15] Sinha, A., Roy, P.: Generation of exactly solv-
ablenon-Hermitianpotentialswith realenergies.
Czech. J. Phys. Vol. 54 (2004), p. 129–138.

[16] Znojil,M.: Topology-controlled spectra of imag-
inary cubic oscillators in the large-L approach.
Phys. Lett. Vol. A 374 (2010), p. 807812
(arXiv:0912.1176v1).

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

90