On the spectrum of the discrete $1d$ Schrödinger operator with an arbitrary even potential

Abstract

The discrete one-dimensional Schrödinger operator is studied in the finite interval of length $N=2M$ 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 Schrödinger operator (Hamiltonian), which is represented by the $2M\times 2M$ 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 Schrödinger operator in the half-line. We obtain such a requirement in the simplest case of the Schrödinger operator, which does not have bound and semi-bound states, and which potential has a compact support.

14 pages, no figures