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Tracing periodic orbits and cycles in the nodal count Amit Aronovitch, Ram Band, Yehonatan Elon,...
1
Tracing periodic orbits and cycles in the nodal count Amit Aronovitch, Ram Band, Yehonatan Elon, Idan Oren, Uzy Smilansky xz ( stands for the integral value of x) Fourier transforming the oscillating part of the nodal count of the following graph (Boundary conditions and lengths of bonds are indicated) gives the result: Introduction - billiards Introduction – metric graphs • Classical – A particle which moves freely inside the billiard and is reflected from the walls • Quantum – The eigenfunctions and the eigenvalues of the laplacian on the billiard: • Classical – A particle which moves freely on the bonds and is probabilistically transmitted to any connected bond when it reaches a vertex • Quantum - The eigenfunctions and the eigenvalues of the second derivative on the graph: 1 n n Introduction – discrete graphs 1 2 n n k The connection between quantum and classical descriptions – trace formula • The quantum spectral counting function is: . It can be presented as a sum of a smooth part and an oscillating part. [1,2] } . s.t of {# ) ( k k i k N i • Quantum - The eigenvectors and the eigenvalues of the Laplacian matrix L: L i,j =L j,i <0 if vertices i and j are adjacent; L i,j =0 otherwise. • The trace formula: ; . is the length of the classical periodic orbit. is a weight factor. Counting nodal domains … • Nodal domains count – the number of connected domains where the n th wavefunction is of constant sign. • For billiards we can also count the number of intersections of the nodal lines with the boundary (marked with ). … on billiards … on metric graphs … on discrete graphs n 4 4 11 11 The classical information stored in the nodal count – recent results ) ( ) ( ) ( osc smooth k N k N k N po A po l i i i k 2 2 1 9 201 1 n n i i i k 2 dx d 2 2 1 2 n n k 10 24 V n n f 1 V n n 1 - - 0.12 0.12 0.2 0.2 4 4 - - 0.1 0.1 2 2 0.3 0.3 9 9 - - 0.7 0.7 8 8 0.3 0.3 9 9 3 f - - 0.41 0.41 - - 0.4 0.4 1 1 - - 0.4 0.4 1 1 0.41 0.41 0.4 0.4 1 1 0.4 0.4 1 1 - - + + - - - - + + + + + + + + + + - - - - - - ) cos( ) ( po orbits periodic po osc l k A k N Fourier transforming to obtain - the power spectrum of ) ( osc k N po po i osc osc ) ( d ) ( ) ( ˆ po kl l l A k e k N l N Few classical periodic orbits of the Sinai billiard ) ( ˆ l N osc 5 f 8 5 12 12 10 10 23 24 2 3 4 5 n [1] M. G. Gutzwiller, in ”Chaos in Classical and Quantum Mechanics”, Vol. 1, Springer-Verlag, New York, 1990. [2] T.Kottos T and U. Smilansky 1999 Ann. Phys., NY 274 76 [3] G. Blum and U. Smilansky, Nodal Domains Statistics: A Criterion for Quantum Chaos, Phys Rev Letters, vol 88 (11), 2002. [4] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I ,Interscience a a 2b 2c D N Courant’s Theorem: Nodal domains count of the n th eigenfunction is smaller or equal to n. [4] Define: and examine the distribution: The average taken over an ensemble of Laplacians. ] 1 , 0 ( n n n n ] ) ( 1 [ E ) ( 1 V n n V p A priory, , where the three dots represents other parameters of the graph, such as: diameter, average degree, etc. For large enough graphs, we observed a data collapse: ,.. , ( ) ( r V f p ) ( ) ( V r f p V = # of vertices B = # of bonds r = cycle dimension Example: 1 V B r 3 1 6 8 r The power spectrum contains lengths of periodic orbits as well as differences of lengths of bonds. Numerically, it was found that the formula for the nodal count sequence of this graph is: c b a c b n n 1 5 . 0 5 . 0 x The nodal count and nodal intersection count of generic billiards are quantum mechanical properties. Current knowledge about them amounts to numerics and heuristic models for their smooth part. [3] n Nevertheless , lengths of classical periodic orbits appear in the power spectrum of their oscillating part - as demonstrated below: c 0 . 2 a l 4 . 2 b l 12 . 3 c l 47 . 6 l 35 . 7 l b a l a l b l c 2l a 2l b 3l a 2l c 3l b 4l a 3l c 4l b 5l a 3l b 2l c 4l a 4l b 5l a l a l c 2l a l l l ) ( osc k N 68 . 6 l n n 1 . 0 V r 1 V r 5 . 0 V r 10 V r ) ( p ) ( p 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 400 300 400 0 0.5 1 0 0.05 0.1 0.15 0.2 350 300 350 600 0 0.5 1 0 0.01 0.02 0.03 0.04 0.05 0.06 300 350 350 350 ) 1 ( ) ( p ) ( p 0 0.5 1 0 0.2 0.4 0.6 0.8 300 300 400 ) ( p 2 ) 1 ( 1 ) ( V p Thus, the lengths of the classical periodic orbits can be deduced from the quantum spectral counting function.
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Transcript of Tracing periodic orbits and cycles in the nodal count Amit Aronovitch, Ram Band, Yehonatan Elon,...
Tracing periodic orbits and cycles in the nodal count
Amit Aronovitch, Ram Band, Yehonatan Elon, Idan Oren, Uzy Smilansky
xz
( stands for the integral value of x)
Fourier transforming the oscillating part of the nodal count of the following graph(Boundary conditions and lengths of bonds are indicated) gives the result:
Introduction - billiards Introduction – metric graphs
• Classical – A particle which moves freely inside the billiard and is reflected from the walls
• Quantum – The eigenfunctions and the eigenvalues of the laplacian on the billiard:
• Classical – A particle which moves freely on the bonds and is probabilistically transmitted to any connected bond when it reaches a vertex
• Quantum - The eigenfunctions and the eigenvalues of the second derivative on the graph:
1nn
Introduction – discrete graphs
1
2
nnk
The connection between quantum and classical descriptions – trace formula• The quantum spectral counting function is: . It can be presented as a sum of a smooth part and an oscillating part. [1,2]}.s.tof{#)( kkikN i
• Quantum - The eigenvectors and the eigenvalues of the Laplacian matrix L: Li,j=Lj,i<0 if vertices i and j are adjacent;Li,j=0 otherwise.
• The trace formula: ; . is the length of the classical periodic orbit. is a weight factor.
Counting nodal domains …• Nodal domains count – the number of connected domains where the nth wavefunction is of constant sign.• For billiards we can also count the number of intersections of the nodal lines with the boundary (marked with ).
… on billiards … on metric graphs … on discrete graphs
n
4
4
11
11
The classical information stored in the nodal count – recent results
)()()( oscsmooth kNkNkN poApol
iii k 22
1 9 201
1nn
iii k 2
dxd
2
2 1
2
nnk
10 24
Vnnf 1 Vnn 1
--0.120.12
0.20.244
--0.10.1
22
0.30.399
--0.70.7
88
0.30.399
3f--
0.410.41
--0.40.4
11
--0.40.4
11
0.410.41
0.40.411
0.40.411
--
++
-- --
++++ ++
++
++ --
----
)cos()( poorbitsperiodic
poosc lkAkN
Fourier transforming
to obtain- the power spectrum of
)(osc kN
popo
ioscosc
)(
d)()(ˆ
po
kl
llA
kekNlN
Few classical periodic orbits of the Sinai billiard
)(ˆ lNosc
5f
8
5
12
12
1010 2324 23 45
n
[1] M. G. Gutzwiller, in ”Chaos in Classical and Quantum Mechanics”, Vol. 1, Springer-Verlag, New York, 1990.[2] T.Kottos T and U. Smilansky 1999 Ann. Phys., NY 274 76
[3] G. Blum and U. Smilansky, Nodal Domains Statistics: A Criterion for Quantum Chaos, Phys Rev Letters, vol 88 (11), 2002.[4] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I ,Interscience Publishers, Inc., New York, N.Y., 1953.
a
a2b
2cD
NCourant’s Theorem: Nodal domains count of the nth eigenfunction is smaller or equal to n. [4]
Define:
and examine the distribution:
The average taken over an ensemble of Laplacians.
]1,0( nn
n n
])(1
[E)(1
V
nnV
p
A priory, , where the three dots represents other parameters of the graph, such as: diameter, average degree, etc.
For large enough graphs, we observed a data collapse:
,...),()( rVfp
)()( Vrfp
V = # of verticesB = # of bondsr = cycle dimension
Example:
1 VBr
3168 r
The power spectrum contains lengths of periodic orbits as well as differences of lengths of bonds.Numerically, it was found that the formula for the nodal count sequence of this graph is:
cba
cb
nn 15.05.0 x
The nodal count and nodal intersection count of generic billiards are quantum mechanical properties. Current knowledge about them amounts to numerics and heuristic models for their smooth part. [3]
n
Nevertheless,lengths of classical periodic orbits appear in the power spectrum of their oscillating part -as demonstrated below:
c
0.2al
4.2bl
12.3cl
47.6l 35.7l
ba
lalb
lc
2la
2lb
3la
2lc 3lb
4la
3lc
4lb
5la
3lb
2lc 4la
4lb
5la
la lc 2la
l
l
l
)(osc kN
68.6l
n
n
2)1(1)(
Vp
1.0Vr
1Vr
5.0Vr
10Vr
)(p
)(p
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
400 300 400
0 0.5 10
0.05
0.1
0.15
0.2
350 300 350 600
0 0.5 10
0.01
0.02
0.03
0.04
0.05
0.06
300 350350 350
)1()( p )(p
0 0.5 10
0.2
0.4
0.6
0.8
300300400
)(p2)1(
1)(
V
p
Thus, the lengths of the classical periodic orbits can be deduced from the quantum spectral counting function.
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