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Are scattering properties of graphs uniquely connected to their shapes ?

Are scattering properties of graphs uniquely connected to their shapes ?. Leszek Sirko , Oleh Hul Michał Ławniczak, Szymon Bauch I nst itute of Physics Polish Academy of Sciences , Warszawa , Poland Adam Sawicki, Marek Kuś

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Are scattering properties of graphs uniquely connected to their shapes ?

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  1. Are scattering properties of graphs uniquely connected to their shapes? Leszek Sirko, OlehHul Michał Ławniczak, Szymon Bauch Institute of Physics PolishAcademy 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. Is the spectrum of the Laplace operatorunique on the planar domain withDirichlet boundary conditions? Is it possible to construct differently shaped drums which have the same eigenfrequency spectrum (isospectral drums)? Can one hear the shape of a drum? M. Kac, Can one hear the shape of a drum?, Am. Math. Mon. (1966) 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 A B A-B-G -A -C -E -B +C -D C D A-D-F B-E+F E C -F -G F G D-E+G For a pair of isospectral domains eigenfunctions corresponding to the same eigenvalue are related to each other by a transplantation Trento, 26 July, 2012

  6. One cannot hear the shape of a drum 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. S. Sridhar and A. Kudrolli, Experiments on not hearing the shapeof drums,Phys. Rev. Lett. (1994) 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. 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 Quantum graphs and microwave networks 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 Neumann b. c. Dirichlet b. c. A graph consists of nvertices (nodes) connected by B bonds (edges) On each bond of a graph the one-dimensional Schrödinger equation is defined Topology is defined by nxn connectivity matrix The length matrixLi,j Vertex scattering matrixϭdefines boundary conditions Trento, 26 July, 2012

  11. Spectrum and wavefunctions 3 6 1 2 4 5 Spectral properties of graphs can be written in termsof 2Bx2Bbond scattering matrixU(k) Trento, 26 July, 2012

  12. Scattering from graphs 6 3 1 2 4 5 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? 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 b a+2b 2a+b a+2b b B. Gutkin and U. Smilansky, Can one hear the shape of a graph?, J. Phys. A: Math. Gen. (2001) Trento, 26 July, 2012

  18. Isospectral quantum graphs R. Band, O. Parzanchevski, G. Ben-Shach, The isospectral fruitsof representation theory: quantum graphs and drums, J. Phys. A (2009) Authors presented new method of construction of isospectral graphs and drums 2a b N 2b c N N D a a c b 2c D D D D N N Trento, 26 July, 2012

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

  20. Isoscatteringgraphs, 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 networks 2b b 3 5 c 1 a a 4 3 1 2 2a 2 Network I Network II c b 6 4 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. 2c Trento, 26 July, 2012

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

  28. Summary EUROPEAN UNION Are scatteringpropertiesof graphs uniquely connected to their shapes? – in generalNO! The concept of isoscatteringgraphs 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 Trento, 26 July, 2012

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