A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry. (16th October 2018) - Rutkevich, S. B.
Abstract : The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the2 M -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 limitM → ∞ 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- 1 . We obtain such requirements in the simplest case of the discrete Schrödinger operator acting inl 2 ( N ), which does not have bound and semibound states and whose potential has a compact support.