Potential reconstruction for two-dimensional discrete Schrِdinger equation

The potential reconstruction problem for the two-dimensional discrete Schrِdinger equation is reduced to reconstruction of a symmetric five-diagonal matrix. We developed algorithms of reconstructing the symmetric five-diagonal matrices of a given spectrum and first k components for each of the basic eigenvectors, and the symmetric nine-diagonal matrix of a given spectrum and prescribed symmetry of the basic eigenvectors. These matrices correspond to the nonlocal potential with five-point and nine-point supports. Main difficulty lies in solving the cumbersome polynomial systems developed and solved by using CAS REDUCE. The algorithm of reconstruction of the symmetric nine-diagonal matrix proved to be more stable.