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

A formula for eigenvalues of Jacobi matrices with a reflection symmetry