Isoscattering non-isospectral quantum graphs

Using the formalism of the Titchmarsh-Weyl M-function we study a pair of isoscattering non-isospectral quantum graphs. These graphs have the same total length L, but different topologies. The isoscattering but non-isospectral quantum graphs are difficult to identify and very complex to analyze. However, failing to recognize their existence will lead to incorrect interpretation of the spectra. We show that a potential experimental solution to this problem could involve constructing a pair of parameter-dependent non-isoscattering and non-isospectral graphs with the same total length L. Subsequently, by implementing the transition of the parameter towards zero, it can be demonstrated that the graphs transform into a pair of isoscattering graphs that remain non-isospectral. These theoretical predictions are subjected to experimental verification using microwave networks.