CAN ONE HEAR THE SHAPES OF GRAPHS AND NETWORKS?

Can one hear the shapes of graphs and networks?" - this is a modification of the famous question of Mark Kac "Can one hear the shape of a drum?" which can be asked in the case of scattering systems such as quantum graphs and microwave networks. It addresses an important mathematical problem whether scattering properties of such systems are uniquely connected to their shapes? Recent experimental results based on a characteristics of graphs such as the cumulative phase of the determinant of the scattering matrices indicate a negative answer to this question [1,2]. In this presentation we will consider also important local characteristics of graphs and networks such as structures of resonances and poles of the determinant of the scattering matrices. Using the analytical formulas for the elements of the scattering matrices we show that it is possible to link the structure of the scattering poles of the determinant of the scattering matrices with the experimental spectra of the microwave networks. Furthermore, we show that theoretically reconstructed spectra of the networks are in good agreement with the experimental ones.

[1]    O. Hul, M. Ławniczak, S. Bauch, A. Sawicki, M. Kuś, L. Sirko, Phys. Rev. Lett 109,  040402
        (2012). 
[2]    M. Ławniczak, A. Sawicki, S. Bauch, M. Kuś, L. Sirko, Phys. Rev. E 89, 032911 (2014).
                        

All interested are invited.
M. Kowal, W. Piechocki, L. Roszkowski, J. Skalski