Inverse problem for the two-dimensional discrete Schrödinger equation in a square

Abstract

The potential of the two-dimensional discrete Schrödinger equation can be reconstructed from a part of its spectrum and prescribed symmetry conditions of the basis eigenfunctions. The discrete potential along with the missing eigenvalues is found by solving a polynomial system of equations, which is derived and solved using the REDUCE computer algebra system. To ensure the convergence of the iterative process implemented in the Numeric package in REDUCE, proper initial data must be specified. The prescribed eigenvalues are perturbed original eigenvalues corresponding to the zero discrete potential. The original eigenvalues provide the natural initial data for the corresponding missing eigenvalues. In the case of a square, there are many multiple eigenvalues among the original eigenvalues. The direct application of the variant of the Newton method implemented in the Numeric package in REDUCE is impossible in the case of multiple initial data. A modification of the method proposed earlier for calculating the discrete potential of the two-dimensional discrete Schrödinger equation in a square is illustrated by an example.