Spectral properties of the two-dimensional multiwell potential
Two-dimensional multiwell Hamiltonian system with four local minima is considered. The motion of the system
 shifts from regular to chaotic through “mixed state”, i.e. the state, when regular and irregular regimes of motion 
 coexist in different local minima. Three regimes of motion...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2007 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2007
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| Цитувати: | Spectral properties of the two-dimensional multiwell potential / N.A. Chekanov, E.V. Shevchenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 270-264. — Бібліогр.: 10 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860264156945448960 |
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| author | Chekanov, N.A. Shevchenko, E.V. |
| author_facet | Chekanov, N.A. Shevchenko, E.V. |
| citation_txt | Spectral properties of the two-dimensional multiwell potential / N.A. Chekanov, E.V. Shevchenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 270-264. — Бібліогр.: 10 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | Two-dimensional multiwell Hamiltonian system with four local minima is considered. The motion of the system
shifts from regular to chaotic through “mixed state”, i.e. the state, when regular and irregular regimes of motion 
coexist in different local minima. Three regimes of motion – regular ( R), mixed state (RC), and chaotic (C) – are 
considered. For each energy region the spectrum is calculated by direct diagonalization in polar coordinates, the 
eigenstates are classified according to the irreducible representations of C3v -point group, and the spectral
statistical properties are analyzed and compared to the theoretical predictions for integrable, chaotic and generic 
(neither regular nor chaotic) systems.
Розглянуто квантову гамільтонову систему, поверхня потенційної енергії якої має чотири локальних мінімуми, і яка в класичній межі допускає в деякому інтервалі енергій змішаний стан. Для даної системи методом диагоналізації обчислений енергетичний спектр, розподіл відстаней між сусідніми рівнями й ∆₃-жорсткість Дайсона. Отримані результати зіставлені з теоретичними передбаченнями для регулярних, хаотичних систем і систем, в яких регулярні й хаотичні траєкторії співіснують.
Рассмотрена квантовая гамильтонова система, поверхность потенциальной энергии которой имеет четыре локальных минимума и которая в классическом пределе допускает в некотором интервале энергий смешанное состояние. Для данной системы методом диагонализации вычислен энергетический спектр, распределение расстояний между соседними уровнями и ∆₃-жесткость Дайсона. Полученные результаты сопоставлены с теоретическими предсказаниями для регулярных, хаотических систем и систем, в которых регулярные и хаотические траектории сосуществуют.
|
| first_indexed | 2025-12-07T18:58:34Z |
| format | Article |
| fulltext |
SPECTRAL PROPERTIES OF THE TWO-DIMENSIONAL
MULTIWELL POTENTIAL
N.A. Chekanov and E.V. Shevchenko
Belgorod State University, Belgorod, Russian Federation;
e-mail: chekanov@bsu.edu.ru
Two-dimensional multiwell Hamiltonian system with four local minima is considered. The motion of the system
shifts from regular to chaotic through “mixed state”, i.e. the state, when regular and irregular regimes of motion
coexist in different local minima. Three regimes of motion – regular ( ), mixed state ( ), and chaotic ( C ) – are
considered. For each energy region the spectrum is calculated by direct diagonalization in polar coordinates, the
eigenstates are classified according to the irreducible representations of vC3 -point group, and the spectral
statistical properties are analyzed and compared to the theoretical predictions for integrable, chaotic and generic
(neither regular nor chaotic) sy
R RC
the
stems.
PACS: 05.45-а, 05.45.Ac
1. INTRODUCTION
Searching for quantum manifestations of classical
chaos has been the subject of series of investigations for
the past decades [e.g. 1, 2]. The one way to do it is to
analyze statistical properties of spectrum. Such investi-
gations have been done for model systems or systems
with simple topology [e.g. 3, 4]. The present research is
concerned to the two-dimensional multiwell system
with the Hamiltonian in the form
( ) ),(
2
1 22 yxVppH yx ++= , (1а)
( ) ( 2223222
3
1
2
1
),(
yxcyyxbyx
yxV
++
−++= ) . (1b)
The system (1) describes surface quadrupole oscilla-
tions of a spherical drop of some matter. This system
has been previously studied numerically by
V. Berezovoj et al. [5]. We present here research con-
cerning the so called “exact” eigenvalues, derived via
the diagonalization procedure.
The number of critical points of the potential (1b)
depends on the parameter cb2=
min =
W . We consider the
case , so that the system (1) is multiwell with
four local minima and three saddle points (cf. Fig. 1).
All the minima are of the value V .
18=W
0
For the system (1) the transition regularity-chaos-
regularity has been shown [6]. That means that the sys-
tem (1) moves regularly in the region ; invari-
ant tori are destroyed as approaches and the
motion becomes chaotic in the region ;
in the region the regularity of the motion is
recovered. In particular, for the case W critical
energies are found to be for the central
minima and for the peripheral minima. Thus
in the region
1crEE <
1crE
1cr EE <
18=
2
E
E
2crE<
2crEE >
scr VE ≈1
/1 scr V≈
sVs E<2V regular and irregular re-
gimes of motion coexist in different local minima. Such
a phenomenon is called mixed state.
<
Fig. 1. Level lines of the potential part of the Hamil-
tonian (1) with
(
048.0=b , 000128.0=c
182 == cbW ). Dashed lines denote the zero Gaus-
sian curvature
We investigated the motion of the system (1) in
three different regions: regular ( R ) with energies
, mixed ( ) with energies in the range 2/sVE < RC
sV
b
s EV <<2 and chaotic ( ) with V .
Parameters and were chosen independently for
each energy region in order to obtain enough energy
levels for further statistical analysis (cf. Table 1).
C 2crs EE <<
c
Table 1. The choice of parameters and for investi-
gating system (1) in energy regions , , C
b c
R RC
Parameters
Type of
motion b c 1crE sE 2crE
R 0.018 0.000018 385.8 868.1 1617451.8
RC 0.048 0.000128 54.3 122.1 227454.2
C 0.42 0.0098 0.7 1.5 2970.8
2. CLASSICAL ORBITS
The one of the commonly used ways for classifying
the classical motion of the system is computation of the
Poincaré surface of section pictures.
To obtain surface of section pictures shown on
Fig. 2 we used the classical equations of motion for the
Hamiltonian (1)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (2), p. 270-274. 270
)(42 22 yxcxbxyxx +−−−= , (2) H
)(4)( 2222 yxcyyxbyy +−−−−= (3)
and the fourth-order Runge-Kutta step method for com-
puting the classical trajectories.
Fig. 2,a. Type of motion . . Shown is
the surface of section for central (left) and peripheral
(right) local minima
R 0.175=E
Fig. 2,b. Type of motion . . Shown is
the surface of section for central (left) and peripheral
(right) local minima
RC 0.80=E
Fig. 2,c (left). Type of motion . RC
0703125.122== sVE
Fig. 2,d (right). Type of motion C . 0.80=E
Fig. 2 demonstrates the type of classical motion in
three energy regions of interest ( , , and C ). Par-
ticularly, mixed state is seen on Fig. 2b, 2c.
R RC
3. COMPUTATION OF QUANTUM ENERGY
SPECTRA
For the purpose of quantum mechanical calculations
we consider the Hamiltonian (1) in polar form
4
3
2
2
2
22
2
3sin
3
1
2
1ˆ crrbr
rr
H +ϕ+
−
ϕ∂
∂
+
∂
∂
−= .(4)
In order to calculate the quantum energy spectra the
eigenvalue problem
),( ),(),(ˆ ϕψ=ϕψϕ rErr (5)
should be solved. Eigenvalues of the Hamiltonian (4)
were calculated by diagonalizing the Hamiltonian ma-
trix lNHlN ,ˆ, ′′ . The basis functions lN , were
chosen in the form
( )),(),(
2
1),(~ *
,,, ϕ+ϕ=ϕ rujruru lNlNlN , (6)
where ),(, ϕru lN are basis functions of the unperturbed
harmonic oscillator:
),(
2
1),(, lNReru il
lN
ϕ−
π
=ϕ , . (7) 1±=j
The value in (7) is defined
as
),( lNR
)8(,(
!
2
!2),(
2
2
l
lN
l
N
rLr
lN
ilNR
ωω
ω
−×
+
=
)
2
2 2re
lN
ω−
−
( )
where )(tLl
n is the Laguerre polynomial, is the fit-
ting parameter, l is the angular momentum. The
following recursion relations can be obtained
ω
:
)9(,)1,1(
2
2
)1,1(
2
),(
−+
+−
+
−−
+
=
lNRlN
lNRlNilNrR
)10(.)1,1(
2
2
)1,1(
2
),(
++
++
+
+−
−
−=
lNRlN
lNRlNilNrR
It is important to take the full symmetry of the Ham-
iltonian (4) into account for two reasons: 1) the matrix
lNHlN ,ˆ, ′′ can be divided into submatrices, corre-
sponding to the different irreducible representations of
the symmetry group of the Hamiltonian (4) (that allows
to calculate the eigenvalues of each type separately);
2) it is necessary to distinguish energy levels of each
symmetry type to perform spectral statistical analysis
correctly [7].
The full symmetry of the Hamiltonian (4) is the
-point group, which is the symmetry group of an
equilateral triangle. It has three irreducible representa-
tions: , , and . The eigenvalues corresponding
to and symmetries are generally non-
degenerate, while those of symmetry are doubly
degenerate. Basis functions (6) are classified according
vC3
1A
1
2A E
A 2A
E
271
to the irreducible representations of the -point
group as shown in Table 2.
vC3
)3
),3
)3
)(
2
2
( 0
++
−
′ll
l
N
l
δ
δ
.
8
)6)(4)(2
8
)4)(2)((3
8
)2)(2)((3
8
)4)(2)((),(
)3()1(
)1()3
+′+′
−′
+−+−+−+−+−+
+
−++−+
+
−+−++
=
NNNN
NN
lNlNlNlNlNlN
lNlNlNlNlNlNlNF
δδ
δδ
(
( −′
+
NN
Table 2. Classification of the basis functions (5)
ccording to the irreducible representations of the
-point group
a
vC3
jSymmetry l
1A 1 (mod0=l
2A 1−
>
=
0
(mod0
l
l
1E 1
E :
2E 1−
(mod0≠l
Using the orthogonality relation
together with the re-
cursion relations (9), (10) we get the explicit formula
for matrix elements is the Kronecker delta symbol
and depending on the type of the symmetry (cf.
Table 2).
'0
( , ) ( , ) N NR N l R N l rdr δ
∞
′ =∫
nmδ
1±=j
( )
,)2)(2(
2
1
)1())((
2
1
2
1)4)(4(
)2)(2(
4
1)2)(2()2(23
22
3
))(()2)(2)((
4
1),(
),(1
6
)1(,ˆ,
)2(
)2()4(
)2(
2
2
)2()4(2)3(
)3()3(2)23 0
+−+++
++−+
−
+
+−++×
+−++++−++
++−+
−++−−−++
+
−+
+⋅−+=′′
+′
′−′+′
+′′
−′−′′+′
+−′−′+′′
′
NN
NNNNNN
NNNN
NNNNllll
llllNNll
lNlN
NlNlNlNlN
lNlNlNNNNlN
lNlNNlNlNllNclNF
lNFjbNlNHlN
l
δ
δδ
ω
ω
δ
δδ
δδ
ω
δδ
δδ
ω
δδω
δ
(11)
where
(12)
In practical calculations the elements of the Hamil-
tonian matrix were ordered by the value of N ,
. We calculated matrix elements
choosing , except for -symmetry in the
region , where calculations were performed for
. The matrix
max...,,1,0 NN =
max =N
C
175max =N
250 E
lN ,HlN ˆ, ′′ is banded with
the band , where m depends on the quantum
number as
12 +m
N
≥+−−
<
=
,4,1)4(dim)(dim
,4),(dim
)(
NNN
NN
Nm
HH
H (13)
where is dimension of the Hamiltonian ma-
trix for particular .
)(dim NH
N
Diagonalization of the Hamiltonian matrix
lNHlN ,ˆ, ′′ was performed using the reduction to the
tridiagonal form of symmetric band matrix via Jacobi
rotations [8, p. 244] followed by the procedure for cal-
culating specific eigenvalues in given interval of sym-
metric tridiagonal matrix via the method of bisection [8,
p. 367]. We obtained 5334 energy levels of -type,
5208 levels of -type in each energy region consid-
ered, and we obtained 10542 levels of -type in the
regions and , and 5192 levels in the region .
The accuracy of the results was examined by changing
the size of the basis and by varying the fitting parameter
in expressions (8) and (11). The energy levels, reli-
able with an accuracy ( s is the mini-
mum spacing between nearest-neighbor levels) were
accepted for further statistical analyses. The error is
defined as
1A
2A
RC
1E
min
R C
∆
ω
min1,0 s⋅<∆
( )
− −1maxN
ia
)1−
maxN N
maxN
i
ia
N
max
i
N
ia N
max= N
( )
=∆ a , (14)
where , are the i-th energy levels,
obtained via the diagonalization of Hamiltonian matri-
ces computed for and respec-
tively.
( )max ( max
= 1−
4. STATISTICAL PROPERTIES
OF SPECTRAL FLUCTUATIONS
Spectral statistical analysis is applied to spectral
fluctuations, i.e. spectrum deviations from its smooth
(locally uniform) behavior [1,9]. The distribution func-
272
tion (the staircase function) for a discrete spec-
trum can be written as
)(EN
)()()( ENENEN fluctav += , (15)
where is the average part and is the
fluctuation part of the staircase function. Since the
smooth behavior is not universal, it is removed by the
spectrum “unfolding” procedure via the mapping
)(EN av )(EN fluct
)(21 navn ENx =+ . (16)
We calculated in terms of the few lower-
order spectral moments by using a truncated Gram-
Charlier expansion [10] for the distribution function
of the normalized quantity
( is the expectation, is the standard deviation of
the spectrum ).
)(EN av
)(xF σ−= /)( EmEx
Em σ
{ }iE
The following statistical properties of the system (4)
were investigated: 1) the distribution of spac-
ing
)(sp
s between nearest-neighbor levels of the spectrum;
and 2) the Dyson’s -statistic which measures the
spectral rigidity and is defined by
3Δ
[∫
+α
α
+−≡αΔ
L
BA
dxBAxxN
L
L 2
,
3 )()(min1);( ] , (17)
),()( 33 LL iαΔ=Δ . (18)
The following theoretical predictions are
known [1,2]:
(1) (Poisson distribution) and
for integrable classical systems;
)exp()( ssp −=
15/~)(3 LLΔ
(2) ( )2exp)( BsAssp −≈ β and
for chaotic systems. Particularly,
δ+γΔ LL ln~)(3
( )4/exp2)( 2sssp π−π≈ (Wigner distribution) and
00695.0ln1~)(3 −πΔ LL for systems with quantum
spectrum well described by random matrix theory,
namely by statistical properties of the Gaussian or-
thogonal ensemble (GOE);
(3) In case of a generic system where regular and
chaotic trajectories coexist, distribution may well
be fitted by Brody distribution
)(sp
( )qq
q sssp +β−α= 1exp)( , (19)
([ qqqq +++Γ=ββ+=α 1)1()2(,)1( )] , (20)
Note that with approaches Poisson dis-
tribution, and with approaches Wigner
distribution.
0→q )(spq
1→q )(spq
5. RESULTS AND CONCLUSIONS
Since the results obtained for different symmetry
types are qualitatively the same, only the results for
E -symmetry type are given. The solid line on pictures
below is used to display the predictions for regular sys-
tem, the dashed line is for the predictions for chaotic
system.
As can be seen from Fig. 3, regularity of the classi-
cal motion in the region is approved by the statistical
properties of the fluctuations of spectra. Namely,
is close to the Poissonian distribution, while
in the range , and
“saturates” after that – i.e. all the classical trajectories
complement to the value of , and moves
away from straight line and fluctuates about some value
, which is not universal. Figs. 4 and 5 present result
the results for mixed RC region.
R
)(sp
15/~)(3 LLΔ max0 LL ≤≤ )(3 LΔ
)(3 LΔ )(3 LΔ
∞Δ
REGION R
0 1 2 3 4
0
0.2
0.4
0.6
0.8
1.0
s
P
Hs
L
HaL
0 20 40
0
1.0
L
D
3
HL
L
HbL
Fig. 3. -type. Total number of levels – 701 (lev-
els from 5000 to 5700, , ),
number of bins in the histogram – 15; is the
ensemble average (levels from 4950 to 5650, from 5000
to 5700, and from 5050 to 5750)
1E
7.1715000 ≈E 2.1835700 ≈E
)(3 LΔ
REGION RC
0 1 2 3
0
0.2
0.4
0.6
0.8
1.0
s
P
Hs
L
HaL
0 20 40
0
1.0
L
D
3
HL
L
HbL
Fig. 4. -type. Total number of levels – 501 (lev-
els from 1380 to 1880, , ),
number of bins in the histogram – 12; is the
ensemble average (levels from 1320 to 1820, from 1350
to 1850, and from 1380 to 1880). Dash-dotted line is
the Brody distribution (19)-(20) with
1E
7.871380 ≈E 5.1011880 ≈E
)(3 LΔ
6262.0≈q
0 1 2 3
0
0.2
0.4
0.6
0.8
1.0
s
P
Hs
L
HaL
0 20 40
0
1.0
L
D
3
HL
L
HbL
Fig. 5. -type. Total number of levels – 501 (lev-
els from 1680 to 2180, , ),
number of bins in the histogram – 12; is the
ensemble average (levels from 1620 to 2120, from 1650
to 2150, and from 1680 to 2180). Dash-dotted line is
the Brody distribution (22) with
1E
2.961680 ≈E 7.1082180 ≈E
)(3 LΔ
7858.0≈q
The more the levels in the analysis get close to the
saddle energy V (particularly, V ), the s 0703125.122=s
273
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HbL
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СВОЙСТВА СПЕКТРА ДВУМЕРНОГО МНОГОЯМНОГО С3V СИММЕТРИЧНОГО
ГАМИЛЬТОНИАНА
Н.А. Чеканов, Е.В. Шевченко
Рассмотрена квантовая гамильтонова система, поверхность потенциальной энергии которой имеет четы-
ре локальных минимума и которая в классическом пределе допускает в некотором интервале энергий сме-
шанное состояние. Для данной системы методом диагонализации вычислен энергетический спектр, распре-
деление расстояний между соседними уровнями и Δ3-жесткость Дайсона. Полученные результаты сопостав-
лены с теоретическими предсказаниями для регулярных, хаотических систем и систем, в которых регуляр-
ные и хаотические траектории сосуществуют.
ВЛАСТИВОСТІ СПЕКТРУ ДВОВИМІРНОГО БАГАТОЯМНОГО С3V СИМЕТРИЧНОГО
ГАМIЛЬТОНИАНУ
М.О. Чєканов, Є.В. Шевченко
Розглянуто квантову гамільтонову систему, поверхня потенційної енергії якої має чотири локальних
мінімуми, і яка в класичній межі допускає в деякому інтервалі енергій змішаний стан. Для даної системи
методом диагоналізації обчислений енергетичний спектр, розподіл відстаней між сусідніми рівнями й
Δ3-жорсткість Дайсона. Отримані результати зіставлені з теоретичними передбаченнями для регулярних,
хаотичних систем і систем, в яких регулярні й хаотичні траєкторії співіснують.
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| id | nasplib_isofts_kiev_ua-123456789-110965 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:58:34Z |
| publishDate | 2007 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Chekanov, N.A. Shevchenko, E.V. 2017-01-07T15:14:08Z 2017-01-07T15:14:08Z 2007 Spectral properties of the two-dimensional multiwell potential / N.A. Chekanov, E.V. Shevchenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 270-264. — Бібліогр.: 10 назв. — англ. 1562-6016 PACS: 05.45-а, 05.45.Ac https://nasplib.isofts.kiev.ua/handle/123456789/110965 Two-dimensional multiwell Hamiltonian system with four local minima is considered. The motion of the system
 shifts from regular to chaotic through “mixed state”, i.e. the state, when regular and irregular regimes of motion 
 coexist in different local minima. Three regimes of motion – regular ( R), mixed state (RC), and chaotic (C) – are 
 considered. For each energy region the spectrum is calculated by direct diagonalization in polar coordinates, the 
 eigenstates are classified according to the irreducible representations of C3v -point group, and the spectral
 statistical properties are analyzed and compared to the theoretical predictions for integrable, chaotic and generic 
 (neither regular nor chaotic) systems. Розглянуто квантову гамільтонову систему, поверхня потенційної енергії якої має чотири локальних мінімуми, і яка в класичній межі допускає в деякому інтервалі енергій змішаний стан. Для даної системи методом диагоналізації обчислений енергетичний спектр, розподіл відстаней між сусідніми рівнями й ∆₃-жорсткість Дайсона. Отримані результати зіставлені з теоретичними передбаченнями для регулярних, хаотичних систем і систем, в яких регулярні й хаотичні траєкторії співіснують. Рассмотрена квантовая гамильтонова система, поверхность потенциальной энергии которой имеет четыре локальных минимума и которая в классическом пределе допускает в некотором интервале энергий смешанное состояние. Для данной системы методом диагонализации вычислен энергетический спектр, распределение расстояний между соседними уровнями и ∆₃-жесткость Дайсона. Полученные результаты сопоставлены с теоретическими предсказаниями для регулярных, хаотических систем и систем, в которых регулярные и хаотические траектории сосуществуют. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Spectral properties of the two-dimensional multiwell potential Властивості спектру двовимірного багатоямного С3V симетричного гамiльтониану Свойства спектра двумерного многоямного С3V симметричного гамильтониана Article published earlier |
| spellingShingle | Spectral properties of the two-dimensional multiwell potential Chekanov, N.A. Shevchenko, E.V. |
| title | Spectral properties of the two-dimensional multiwell potential |
| title_alt | Властивості спектру двовимірного багатоямного С3V симетричного гамiльтониану Свойства спектра двумерного многоямного С3V симметричного гамильтониана |
| title_full | Spectral properties of the two-dimensional multiwell potential |
| title_fullStr | Spectral properties of the two-dimensional multiwell potential |
| title_full_unstemmed | Spectral properties of the two-dimensional multiwell potential |
| title_short | Spectral properties of the two-dimensional multiwell potential |
| title_sort | spectral properties of the two-dimensional multiwell potential |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/110965 |
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