A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry

摘要

The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the -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 this identity induces some requirements, which should satisfy the scattering data of the resulting infinite-dimensional Jacobi operator in the half-line, of which super- and subdiagonal matrix elements are equal to . We obtain such requirements in the simplest case of the discrete Schr?dinger operator acting in , which does not have bound and semibound states and whose potential has a compact support.