An inverse spectral problem for Sturm–Liouville operators with singular potentials on graphs with a cycle

This paper is devoted to the solution of inverse spectral problems for Sturm–Liouville operators with singular potentials from class $W^{-1}_2$ on graphs with a cycle. We consider the lengths of the edges of investigated graphs as commensurable quantities. For the spectral characteristics, we take the spectra of specific boundary value problems and special signs, how it is done in the case of classical Sturm–Liouville operators on graphs with a cycle. From the spectra, we recover the characteristic functions using Hadamard's theorem. Using characteristic functions and specific signs from the spectral characteristics, we construct Weyl functions (m-function) on the edges of the investigated graph. We show that the specification of Weyl functions uniquely determines the coefficients of differential equation on a graph and we obtain a constructive procedure for the solution of an inverse problem from the given spectral characteristics. In order to study this inverse problem, the ideas of spectral mappings method are applied. The obtained results are natural generalizations of the well-known results of on solving inverse problems for classical differential operators.