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

Abstract

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.