1. 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

2. 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

3. 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

4. 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

5. Transplantation
For a pair of isospectral domains eigenfunctions corresponding to the same eigenvalue are related to each other by a transplantation A B C D E F G
A-B-G
A-D-F
B-E+F
D-E+G -A -C -E -B +C -D C -F -G
Trento, 26 July, 2012

6. 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

7. 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

8. 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

9. Quantum graphs
Quantum graphs were introduced to describe diamagnetic anisotropy in organic molecules:
Quantum graphs are excellent paradigms of quantum chaos:
In recent years quantum graphs have attracted much attention due to their applicability as physical models, and their interesting mathematical properties
L. Pauling, J. Chem. Phys. (1936)
T. Kottos and U. Smilansky, Phys. Rev. Lett. (1997)
Trento, 26 July, 2012

10. 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
Topology is defined by n x n connectivity matrix
The length matrix Li,j
Vertex scattering matrix ϭ defines boundary conditions
Neumann b. c Dirichlet b. c.
Trento, 26 July, 2012

11. Spectrum and wavefunctions1 2 5 6 4 3
Spectral properties of graphs can be written in terms of 2Bx2B bond scattering matrix U(k)
Trento, 26 July, 2012

12. Scattering from graphs1 2 5 6 4 3
Trento, 26 July, 2012

13. 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

14. Hexagonal microwave network
Trento, 26 July, 2012

15. Equations for microwave networks
Continuity equation for charge and current:
Potential difference:
Trento, 26 July, 2012

16. Equivalence of equations
Microwave networks
Quantum graphs
Current conservation:
Neumann b. c.
Equations that describe microwave networks with R=0 are formally equivalent to these for quantum graphs with Neumann boundary conditions
Trento, 26 July, 2012

17. 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
2a+3b 2a
a+2b b a
2a+2b
2a+b
a+2b
Trento, 26 July, 2012

18. 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 2c a D b c 2a D N N D N D
Trento, 26 July, 2012

19. Isoscattering quantum graphs
R. Band, A. Sawicki, U. Smilansky, Scattering from isospectral quantum graphs, J. Phys. A (2010)
Authors presented examples of isoscattering graphs
Scattering matrices of those graphs are connected by transplantation relation b c 2a D N N 2b 2c a D
Trento, 26 July, 2012

20. Isoscattering graphs, definition
Two graphs are isoscattering if their scattering phases coincide
Trento, 26 July, 2012

21. Experimental set-up
Trento, 26 July, 2012

22. Isoscattering microwave networks4 3 2 1 a 2c 2b 2a c b 6 4 3 5 1 2
Network I
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

23. Measurement of the scattering matrix4 3 2 1 a 2c 2b 2a c b 6 4 3 5 1 2
Trento, 26 July, 2012

24. The scattering phase
Two microwave networks are isoscattering if for all values of ν:
Trento, 26 July, 2012

25. 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
Trento, 26 July, 2012

26. Scattering amplitudes and phases
O. Hul, M. Ławniczak, S. Bauch,
Sawicki, M. Kuś, and L. Sirko,
accepted to Phys. Rev. Lett. 2012
Isoscattering networks
Networks with modified
boundary conditions
Trento, 26 July, 2012

27. Transplantation relation3 2b 2c a 4 1 2 b c 1 2a 4 2 3 5 6
Trento, 26 July, 2012

28. 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
EUROPEAN
UNION
Trento, 26 July, 2012