A formula for eigenvalues of Jacobi matrices with a reflection symmetry

The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the $2M$-dimensional real Jacobi matrices whose entries are symmetric with respect to the secondary diagonal, a new polynomial identity relating the eigenvalues of such matrices with their matrix { entries} is obtained. In the limit $M\to\infty$ this identity induces some requirements, which should satisfy the scattering data of the resulting infinite-dimensional Jacobi operator in the half-line, which super- and sub-diagonal matrix elements are equal to -1. We obtain such requirements in the simplest case of the discrete Schr\"odinger operator acting in ${l}^2( \mathbb{N})$, which does not have bound and semi-bound states, and which potential has a compact support.