Resonances and poles in isoscattering microwave networks and graphs

Can one hear the shape of a graph? 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 [O. Hul, M. Ławniczak, S. Bauch, A. Sawicki, M. Kuś, and L. Sirko, Phys. Rev. Lett. 109, 040402 (2012)]. In this paper we consider important local characteristics of graphs such as structures of resonances and poles of the determinant of the scattering matrices. Using these characteristics we study experimentally and theoretically properties of graphs and directly confirm that the pair of graphs considered in the cited paper is isoscattering. The experimental results are compared to the theoretical ones for a broad frequency range from 0.01 to 3 GHz. In the numerical calculations of the resonances of the graphs absorption present in the experimental networks is taken into account.