Inverse scattering at a fixed energy for Discrete Schrödinger Operators on the square lattice

Spectral Properties of Schrödinger Operators on Perturbed Lattices

“…Inverse potential scattering for discrete Schrödinger operators have already been considered in [24] on the square lattice (see also [13]) and in [4] on the hexagonal lattice, where knowledge of the S-matrix for all energies is used for the reconstruction of the potential by the method of complex Born approximation. In [27], inverse potential scattering on the square lattice from the S-matrix of one fixed energy is studied. The main idea of [27] is to reduce the issue to an inverse boundary value problem on a bounded domain, and the reconstruction is done through the Dirichlet-Neumann map.…”

“…In [27], inverse potential scattering on the square lattice from the S-matrix of one fixed energy is studied. The main idea of [27] is to reduce the issue to an inverse boundary value problem on a bounded domain, and the reconstruction is done through the Dirichlet-Neumann map. To relate the scattering matrix with the D-N map for the bounded domain, the discrete analogue of the Rellich type uniqueness theorem plays an important role ( [26]).…”

Inverse Scattering for Schrödinger Operators on Perturbed Lattices

“…Once we have proven Lemma 3.2, the following Theorem 3.3 can be derived in the same way as in the whole space [4], as was done in Theorem 6.3 of [35] for the square lattice. We do not repeat the details.…”

“…Parallelogram in the hexagonal lattice. In this section, we reconstruct a scalar potential from the D-N map on a bounded domain following the works [14], [17], [51], [35]. This method depends strongly on the geometry of the lattice.…”

Embedded eigenvalues of generalized Schrödinger operators

“…and is well known (see e.g. [7,Section 4.2]) that in d ≥ 3 dimensions its Fermi surface M λ has points of vanishing Gaussian curvature, i.e. k < d − 1 in Theorem 1.2.…”