A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry
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 thei...
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| Format: | Article |
| Language: | English |
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Wiley
2018-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2018/9784091 |
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| author | S. B. Rutkevich |
| author_facet | S. B. Rutkevich |
| author_sort | S. B. Rutkevich |
| collection | DOAJ |
| description | 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→∞ 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 in l2(N), which does not have bound and semibound states and whose potential has a compact support. |
| format | Article |
| id | doaj-art-e0e13472219644a49f778b112920010c |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-e0e13472219644a49f778b112920010c2025-02-03T06:46:26ZengWileyAdvances in Mathematical Physics1687-91201687-91392018-01-01201810.1155/2018/97840919784091A Formula for Eigenvalues of Jacobi Matrices with a Reflection SymmetryS. B. Rutkevich0Fakultät für Physik, Universität Duisburg-Essen, Duisburg 47048, GermanyThe 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→∞ 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 in l2(N), which does not have bound and semibound states and whose potential has a compact support.http://dx.doi.org/10.1155/2018/9784091 |
| spellingShingle | S. B. Rutkevich A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry Advances in Mathematical Physics |
| title | A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry |
| title_full | A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry |
| title_fullStr | A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry |
| title_full_unstemmed | A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry |
| title_short | A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry |
| title_sort | formula for eigenvalues of jacobi matrices with a reflection symmetry |
| url | http://dx.doi.org/10.1155/2018/9784091 |
| work_keys_str_mv | AT sbrutkevich aformulaforeigenvaluesofjacobimatriceswithareflectionsymmetry AT sbrutkevich formulaforeigenvaluesofjacobimatriceswithareflectionsymmetry |