A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry. (16th October 2018)

• Record Type: Journal Article
• Authors: Rutkevich, S. B.
• Other Names: Kurasov Pavel Academic Editor.
• Is Part Of: Advances in mathematical physics. Volume 2018(2018)
• Publication Date: 2018-10-16
• Subjects: Mathematical physics -- Periodicals; Mathematical physics; Periodicals; 530.15
• DOI: 10.1155/2018/9784091
• Languages: English
• ISSNs: 1687-9120

Abstract

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.