Uniqueness solution to the inverse spectral problem with distributed parameters on the graph-star

In the space of piecewise smooth functions on a star graph, the question of the uniqueness of the recovery of the differential operator of a boundary value problem from its spectral characteristics is analyzed. The uniqueness of the recovery of the coefficient in a differential expression and the constant in the boundary conditions of a boundary value problem from spectral data is considered — a set of eigenvalues and a set of norms of the operator's eigenfunctions. The operator of the boundary value problem has a singularity generated by the structure of the graph: differential expression is defined on the interior parts of all the edges of the graph, and in the interior node of the graph, where the differential expression loses its meaning, there is a generalized condition of Kirchhoff — the condition of agreement (the condition of conjugating). A spectral approach is used, which is based on the spectral properties of the elliptical operator: the analyticity of Green's function of the boundary value problem on the spectral parameter, spectral completeness and the basis property of the set of its eigenfunctions in the space of square integrable function. The results are the basis for solving inverse problems for evolutionary differential systems of parabolic and hyperbolic types with distributed parameters on the network (graph), the elliptical part of which contains coefficients to be determined. In problems of an applied nature, these are, first of all, parameters that characterize the properties of the transfer of the solid environment and describe the elastic properties of the process of deformation of the environment. The approach mentioned above can also be applied to problems whose space variable is vector-like and varies in a network-like domain.