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

TL;DR

This paper investigates the eigenvalue spectrum of a finite discrete 1D Schr"odinger operator with an even potential, revealing polynomial constraints and an effective Coulomb interaction that influence scattering properties.

Contribution

It introduces explicit polynomial constraints on eigenvalues of the finite discrete Schr"odinger operator with even potential, linking them to Coulomb-like interactions and scattering data.

Findings

Eigenvalues satisfy polynomial constraints.

Effective Coulomb interaction between eigenvalues.

Constraints relate to scattering data in the infinite limit.

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.