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Inverse problems for locally perturbed lattices -- Discrete Hamiltonian and quantum graph

We consider the inverse scattering problems for two types of Schrödinger operators on locally perturbed periodic lattices. For the discrete Hamiltonian, the knowledge of the S-matrix for all energies determines the graph structure and the coefficients of the Hamiltonian. For locally perturbed equilateral metric graphs, the knowledge of the S-matrix for all energies determines the graph structure.