  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.
