Some classes of inverse spectral problems for quantum graphs

Abstract

Inverse spectral problems for quantum graphs are analyzed. Under hypothesis of rational independence of lengths of edges it is possible, thanks to trace formulas, to reconstruct every information of compact and not compact graphs from the knowledge, respectively, of the spectrum of Laplacian and of the scattering phase. In the case of Sturm-Liouville operators defined on the graphs and more in general for differential operators, unknown potentials can be recovered from the knowledge of the spectrum of operators.