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

Abstract

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.

Inverse spectral problem for the Schrödinger operator on the square lattice

“…By virtue of lemma 3.1, if a function u = {u e } e∈EΩ solves the edge boundary value problem (2), then its values {u(v)} v∈Ω in the vertices satisfy (11).…”