Are scattering properties
of graphs uniquely connected
to their shapes?
Leszek Sirko, Oleh Hul
Michał Ławniczak, Szymon Bauch
Institute of Physics
Polish Academy of Sciences, Warszawa, Poland
Adam Sawicki, Marek Kuś
Center for Theoretical Physics, Polish Academy of Sciences,
Warszawa, Poland
EUROPEAN
UNION
Trento, 26 July, 2012
Can one hear the shape of a drum?
M. Kac, Can one hear the shape of a drum?, Am. Math. Mon. (1966)
Is the spectrum of the Laplace operator unique on
the planar domain with Dirichlet boundary conditions?
Is it possible to construct differently shaped drums which
have the same eigenfrequency spectrum (isospectral
drums)?
Trento, 26 July, 2012
One can’t hear the shape of a drum
C. Gordon, D. Webb, S. Wolpert, One can't hear the shape of a drum, Bull.
Am. Math. Soc. (1992)
C. Gordon, D. Webb, S. Wolpert, Isospectral plane domains and surfaces
via Riemannian orbifolds, Invent. Math. (1992)
T. Sunada, Riemannian coverings and isospectral manifolds, Ann. Math.
(1985)
Trento, 26 July, 2012
Isospectral drums
Pairs of isospectral domains could be constructed by
concatenating an elementray „building block” in two different
prescribed ways to form two domains. A building block is joined
to another by reflecting along the common boundary.
C. Gordon and D. Webb
S.J. Chapman, Drums that sound the same, Am. Math. Mon. (1995)
Trento, 26 July, 2012
Transplantation
For a pair of isospectral domains eigenfunctions
corresponding to the same eigenvalue are related to
each other by a transplantation
A
A-B-G
B
-A -B
-C +C
-E -D
C
F
D
A-D-F
E
B-E+F
G
C
-F
-G
D-E+G
Trento, 26 July, 2012
One cannot hear the shape of a drum
S. Sridhar and A. Kudrolli, Experiments on not hearing the shape of drums,
Phys. Rev. Lett. (1994)
Authors used thin microwave cavities shaped in the form
of two different domains known to be isospectral.
They checked experimentally that two billiards have the
same spectrum and confirmed that two non-isometric
transformations connect isospectral eigenfunction pairs.
Trento, 26 July, 2012
Can one hear the shape of a drum?
Isospectral drums could be distinguished by measuring
their scattering poles
Y. Okada, et al., “Can one hear the shape of a drum?”: revisited, J. Phys. A:
Math. Gen. (2005)
Trento, 26 July, 2012
Quantum graphs and microwave
networks
What are quantum graphs?
Scattering from quantum graphs
Microwave networks
Isospectral quantum graphs
Scattering from isospectral graphs
Experimental realization of isoscattering graphs
Experimental and numerical results
Discussion
Trento, 26 July, 2012
Quantum graphs
Quantum graphs were introduced to describe
diamagnetic anisotropy in organic molecules:
L. Pauling, J. Chem. Phys. (1936)
Quantum graphs are excellent paradigms of quantum
chaos:
T. Kottos and U. Smilansky, Phys. Rev. Lett. (1997)
In recent years quantum graphs have attracted much
attention due to their applicability as physical models,
and their interesting mathematical properties
Trento, 26 July, 2012
Quantum graphs, definition
A graph consists of n vertices (nodes) connected by B
bonds (edges)
On each bond of a graph the one-dimensional
Schrödinger equation is defined
d2
 2  i , j ( x)  k 2  i , j ( x)
dx
Topology is defined by n x n connectivity matrix
Ci , j
1, i and j are connected

otherwise
0,
The length matrix Li,j
Vertex scattering matrix ϭ defines boundary conditions
Neumann b. c.
i 
 j, j '
Dirichlet b. c.
 ji,j '   j , j '
2
  j , j ' 
vi
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Spectrum and wavefunctions

3
(3)
L1,3

(1)
1
L1,2
L1,4
4


6
L2,6
2

 i, j ( x )  a
(2)
out
ai, j 
L2,5
(4)
(6)
5

(5)
in
i, j
e
 ikx
 a
out ikx
i, j
e
  (j i, )j ' a inj, j '
j'
Spectral properties of graphs can be written in terms of 2Bx2B
bond scattering matrix U(k)
U
 i , j  j  , m  ( k )
  j , je
ikLi , j
det  I  U  k    0
 j
 
 i ,m

a  U k a
 ki 
Trento, 26 July, 2012
Scattering from graphs

3
in  ikx
c1 e
6
L2,6 
L1,3
out ikx
 c1 e

L1,4
4
(3)

1
(1)

L1,2
(2)
(6)
in  ikx
c2 e
2
L2,5
(4)
Trento, 26 July, 2012
5

(5)
out ikx
 c2 e
Microwave networks
O. Hul et al., Phys. Rev. E (2004)
Quantum graphs can be simulated by microwave networks
Microwave network (graph) consists of coaxial cables connected
by joints
Trento, 26 July, 2012
Hexagonal microwave network
2
1
3
6
4
5
n  6 vertices
B  15 bonds
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Equations for microwave networks
Continuity equation for charge and current:
dqi , j ( x, t )
dt

dJ i , j ( x, t )
dx
Potential difference:
Vi , j ( x, t ) 

qi , j ( x, t )
C
qi , j ( x, t )  eit qi , j ( x)
r1
r2
Vi , j ( x, t )  eitVi , j ( x)
R 0
Trento, 26 July, 2012
Equivalence of equations
Microwave networks
d2
 2
V ( x)  2 Vi , j ( x)  0
2 i, j
dx
c
Quantum graphs
d 2  i , j ( x)
dx 2
Current conservation:
Neumann b. c.
d
d
 Ci , j V j ,i  x 
  Ci , j Vi , j  x   0
x  Li , j
x 0
dx
dx
j i
j i
 i , j ( x)  Vi , j ( x)
 k 2  i , j ( x)  0
k2 
 2
c2
Equations that describe microwave networks with R=0 are
formally equivalent to these for quantum graphs with Neumann
boundary conditions
Trento, 26 July, 2012
Can one hear the shape of a graph?
B. Gutkin and U. Smilansky, Can one hear the shape of a graph?, J. Phys. A:
Math. Gen. (2001)
One can hear the shape of the graph if the graph is
simple and bonds lengths are non-commensurate
Authors showed an example of two isospectral graphs
a
b
a
a
2a+3b
b
2a
2a+2b
a+2b
a
b
2a+b
a+2b
b
Trento, 26 July, 2012
Isospectral quantum graphs
R. Band, O. Parzanchevski, G. Ben-Shach, The isospectral fruits of
representation theory: quantum graphs and drums, J. Phys. A (2009)
Authors presented new method of construction of isospectral
graphs and drums
N
2b
a
a
D
c
N
2c
2a
b
N
c
D
b
D
D
D
N
Trento, 26 July, 2012
N
Isoscattering quantum graphs
R. Band, A. Sawicki, U. Smilansky, Scattering from isospectral quantum
graphs, J. Phys. A (2010)
Authors presented examples of isoscattering graphs
N
N
2b
a
D
a
N
b
2a
c
c
2c
b
D
D
Scattering matrices of those graphs are connected by
transplantation relation
S
( II )
k   T
1
S ( I )  k  T , for k 
Trento, 26 July, 2012
Isoscattering graphs, definition
Two graphs are isoscattering if their scattering phases
coincide




Im log det  S ( I ) ( )    Im log det  S ( II ) ( )  




det  S ( )   A    ei 
 S11
S    
 S21
S12 

S22 
Trento, 26 July, 2012
Experimental set-up
Trento, 26 July, 2012
Isoscattering microwave networks
2b
5
c
b
3
b
4
1
4
a
a
1
2
3
2a
2
6
Network I
2c
c
Network II
Two isoscattering microwave networks were constructed using
microwave cables. Dirichlet boundary conditions were prepared by
soldering of the internal and external leads. In the case of Neumann
boundary conditions, vertices 1 and 2, internal and external leads of
the cables were soldered together, respectively.
Trento, 26 July, 2012
Measurement of the scattering
matrix
5
c
b
3
b
4
1
S   
2a
2
c
6
2b
4
a
a
1
2
3
 S11

 S21
S12 

S22 
2c
Trento, 26 July, 2012
The scattering phase
Two microwave networks are isoscattering if for all values of ν:




Im log det  S ( I ) ( )    Im log det  S ( II ) ( )  




Trento, 26 July, 2012
Importance of the scattering
amplitude
In the case of lossless quantum graphs the scattering
matrix is unitary. For that reason only the scattering
phase is of interest.
However, in the experiment we always have losses. In
such a situation not only scattering phase, but the
amplitude as well gives the insight into resonant
structure of the system
det  S ( )   A    ei 
det  S ( I ) ( )   det  S ( II ) ( ) 
Trento, 26 July, 2012
Scattering amplitudes and phases
O. Hul, M. Ławniczak, S. Bauch,
A. Sawicki, M. Kuś, and L. Sirko,
accepted to Phys. Rev. Lett. 2012
Isoscattering networks
Networks with modified
boundary conditions
Trento, 26 July, 2012
Transplantation relation
4
a
2b
1
a
2
1 1
T 

1
1


3
2c
S
( II )
   T
1
3
5 c
S ( I )   T
Trento, 26 July, 2012
2a
b
1
b
2
4
c 6
Summary
Are scattering properties of graphs uniquely connected
to their shapes? – in general NO!
The concept of isoscattering graphs is not only theoretical
idea but could be also realized experimentally
Scattering amplitudes and phases obtained from the
experiment are the same within the experimental errors
Using transplantation relation it is possible to reconstruct
the scattering matrix of each network using the scattering
matrix of the other one
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Trento, 26 July, 2012