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 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

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: 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

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

Spectrum and wavefunctions 1 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

Scattering from graphs 1 2 5 6 4 3   Trento, 26 July, 2012

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 Trento, 26 July, 2012

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

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

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

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

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

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

Experimental set-up Trento, 26 July, 2012

Isoscattering microwave networks 4 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

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

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

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

Transplantation relation 3 2b 2c a 4 1 2 b c 1 2a 4 2 3 5 6 Trento, 26 July, 2012

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