  The discrete one-dimensional Schr\"odinger operator is studied in the finite
interval of length $N=2 M$ with the Dirichlet boundary conditions and an
arbitrary potential even with respect to the spacial reflections. It is shown,
that the eigenvalues of such a discrete Schr\"odinger operator (Hamiltonian),
which is represented by the $2M\times2M$ tridiagonal matrix, satisfy a set of
polynomial constrains. The most interesting constrain, which is explicitly
obtained, leads to the effective Coulomb interaction between the Hamiltonian
eigenvalues. In the limit $M\to\infty$, this constrain induces the requirement,
which should satisfy the scattering date in the scattering problem for the
discrete Schr\"odinger operator in the half-line. We obtain such a requirement
in the simplest case of the Schr\"odinger operator, which does not have bound
and semi-bound states, and which potential has a compact support.
