Inverse problem for the two-dimensional discrete Schrodinger equation

We solve a boundary value problem for the two-dimensional discrete Schrodinger equation in the rectangle M x N, M less than or equal to N, with zero boundary conditions. In this work it is found that the inverse problem reduces to the reconstruction of the symmetric five-diagonal matrix C by the given spectrum and the first k(M, N), 1 less than or equal to k < N, components for each of basis eigenvectors. The matrix C has a lacuna between the second and (N + 1)-th diagonals. As a result, the first N components of the basis eigenvectors must satisfy (N - 1)(2)(M - 1) additional conditions and N conditions of compatibility. The elements of C together with 'lacking' (N-k) components can be determined by solving the system of polynomial equations which involve the additional conditions, the compatibility conditions, and the orthonormality conditions as well as the relations defining the elements of the matrix C by the eigenvalues and components of the basis eigenvectors. We could completely clarify the statement of the problem in concrete calculations. The derivation and the solution of the cumbersome polynomial systems were performed on SPP by using CAS REDUCE 3.6.