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The Characteristic Function for Complex Doubly Infinite Jacobi Matrices
Stockholm University, Faculty of Science, Department of Mathematics. Czech Technical University in Prague, Czech Republic.
Number of Authors: 12017 (English)In: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 88, no 4, p. 501-534Article in journal (Refereed) Published
Abstract [en]

We introduce a class of doubly infinite complex Jacobi matrices determined by a simple convergence condition imposed on the diagonal and off-diagonal sequences. For each Jacobi matrix belonging to this class, an analytic function, called a characteristic function, is associated with it. It is shown that the point spectrum of the corresponding Jacobi operator restricted to a suitable domain coincides with the zero set of the characteristic function. Also, coincidence regarding the order of a zero of the characteristic function and the algebraic multiplicity of the corresponding eigenvalue is proved. Further, formulas for the entries of eigenvectors, generalized eigenvectors, a summation identity for eigenvectors, and matrix elements of the resolvent operator are provided. The presented method is illustrated by several concrete examples.

Place, publisher, year, edition, pages
2017. Vol. 88, no 4, p. 501-534
Keywords [en]
Doubly infinite Jacobi matrix, Spectral analysis, Characteristic function
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-147942DOI: 10.1007/s00020-017-2357-yISI: 000409890100004OAI: oai:DiVA.org:su-147942DiVA, id: diva2:1149927
Available from: 2017-10-17 Created: 2017-10-17 Last updated: 2017-10-17Bibliographically approved

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