Geometric Characterization

of Nodal Patterns and

Domains

Y. Elon, S. Gnutzman, C. Joas U. Smilansky

Introduction

• 2002 – Blum, Gnutzman and Smilansky: a “Chaotic billiards can be discerned by counting nodal domains.”

0 jjE

0 Vj

• Research goal – Characterizing billiards by investigating geometrical features of the nodal domains:

• Helmholtz equation on 2d surface (Dirichlet Boundary conditions):

- the total number of nodal domains of . j j

Consider the dimensionless parameter:

jij

ijij E

lA

j

Consider the dimensionless parameter:

jij

ijij E

lA

j

i

• For an energy interval: , define a distribution function:

IEj i

ijjI

Ij

j

NP

1

11

gEEEI ,

• Is there a limiting distribution?

PgEIPE

?

),(,lim

• What can we tell about the distribution?

Rectangle

ynxmNny

Mmx

mn~sin~sinsinsin~

yx

imn

yx

imn

EEmnl

EEmnA

112~1

~12

~~1

)(

2)(

yxmn EEmnE 222 ~~xx

imn ZZmn

mn

1

12~~

~~

2

22)(

EEEEmnmn

yx

imn

1

2~~~~

2

22)(

Rectangle

IEmnI

Imn

mnmn

NP

|~,~

22

~~~~

21

II

dndmdndmmnmn~~~~

2

22

Rectangle

otherwise

PI222

0

84

22

P

Rectangle

222

ddP 1~22

lim0

1. Compact support:

2. Continuous and differentiable

3.

4.

ddP 20

82

lim

Rectangle

mnymnx

imn EEEE

12

)(

• the geometry of the wave function is determined by the energy partition between the two degrees of freedom.

Rectangle

222 ~~ mnEE classmn

quantmn

• can be determined by the classical trajectory alone.

mn

dypm

dxpn

ycl

xcl

21

21

Action-angle variables:

Disc

•

• the nodal lines were estimated using SC method, neglecting terms of order .E1

2)(

)()sin(n

mmn

nmmmn

jE

rjJm

)(

0222

222

)(

0222

222

22

2

1

)tan()21(1)2sin(

84

84

)tan()21(1)2sin(

84

84

8

2)(

C

C

CCdCC

CCdCC

P

n’=1

n’=4

n’=3

n’=2

22)'(

2

'

)'(

arctan1'

)'2sin(11

2

mjmC

C

nm

mn

nmn

Rectangle Disc

Same universal features for the two surfaces:

Disc

222

ddP 1~22

lim0

1. Compact support:

2. Continuous and differentiable

3.

4.

ddP 20

42

lim

n=1m=o

Surfaces of revolution• Simple surfaces of revolution were investigated (Smooth everywhere, single maxima of the profile curve).

• Same approximations were taken as for the Disc.

mxmnmn sin

xf

Surfaces of revolution• Simple surfaces of revolution were investigated (Smooth everywhere, single maxima of the profile curve).

• Same approximations were taken as for the Disc.

mxmnmn sin

2

'2

')'(

2 mxfEm

xfE

nmn

nmnnmn

2,122

2222

2,122

2222

2222

2

2

1

8

84

8

84

84

8)(

i z

i z

i

i

dzzG

dzzG

MP

n’=1

n’=2

n’=3

n’=4

• For the Disc:clcl

rclmn EE

rmrE

2

22

21

• For a surface of revolution:

clcl

xclmn EE

xfmxxfE

2

222'1

21

2

2

2rmrE qm

rEErE qmmn

qmr

xfmxE qm2

2

2 xEExE qm

mnqmx

mnmnrnm ~~~~

2

22Rect

Following those notations:

)'2sin(1

12

Disc

Crmn

2

'2

'SOR

2 mxfEm

xfEr

nmn

nmnmn

mnmn ErEErE

21

12

“Classical Calculation”:

0rnm

“Classical Calculation”:

nmT1. Look at

(Classical

Trajectory)

“Classical Calculation”:

0rr

2. Find a point along the trajectory for which:

“Classical Calculation”:

mnmn

r

EE

EE ,

3. Calculate

mnmnr

mn EEEEr

1

20

Separable surfaces

rmn2. can be deduced (in the SC limit)

knowing the classical trajectory solely.

1. In the SC limit, has a smooth limiting distribution with the universal characteristics: - Same compact support:- diverge like at the lower support- go to finite positive value at the upper support

IP

Random waves

rJmbmar m

N

mmm

0sincos

Random waves

Two properties of the Nodal Domains were investigated:

1.Geometrical:

2. Topological: genus – or: how many holes?

P

G=0 G=

2

G=1

Random waves

22

2

Random waves

Random waves

Random waves

Random waves

2

)1(0j

Random waves

Random waves

2

)1(0j

Model: ellipses with equally distributed eccentricity and area in the interval: 19,

2)1(0j

2

212

2

2

min

2

offcut

avg

knl

knA

kd

Random waves

d

Genus

76.4~# c46.4~# c

3.4~# c

262.4~# c

The genus distributes as a power law!

Genus

In order to find a limiting power law – check it on the sphere

Genus

Power law?

Saturation?

?~~# 2gg

AA ~~#Fisher’s exp:

Random waves

1. The distribution function has different features for separable billiards and for random waves.

2. The topological structure (i.e. genus distribution) shows complicate behavior – decays (at most) as a power-low.

P

Open questions:

• Connection between classical Trajectories and .

• Analytic derivation of for random waves.

• Statistical derivation of the genus distribution

• Chaotic billiards.

P