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The discrete one-dimensional Schr\"odinger operator is studied in the finiteinterval of length $N=2 M$ with the Dirichlet boundary conditions and anarbitrary 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 ofpolynomial constrains. The most interesting constrain, which is explicitlyobtained, leads to the effective Coulomb interaction between the Hamiltonianeigenvalues. In the limit $M\to\infty$, this constrain induces the requirement,which should satisfy the scattering date in the scattering problem for thediscrete Schr\"odinger operator in the half-line. We obtain such a requirementin the simplest case of the Schr\"odinger operator, which does not have boundand semi-bound states, and which potential has a compact support.

On the spectrum of the discrete $1d$ Schr\"odinger operator with an arbitrary even potential