How to hear the shape of isoscattering networks

This paper addresses the problem of isospectrality of quantum graphs considered as scattering systems. It was shown in our recent experiment [O. Hul et al., Phys. Rev. Lett. 109, 040402 (2012).] that it is possible to create isoscattering microwave networks, i.e., networks that have identical amplitudes and phases of determinants of their scattering matrices. We will extend our previous study by considering the scattering process in the time domain instead of the frequency domain. We will show that it is possible to distinguish isoscattering microwave networks from their scattering data, to reconstruct the lengths of their bonds, and to find valuable information about the boundary conditions. We will also demonstrate that boundary conditions at the vertices of microwave networks are frequency dependent.