On the eigenvalue problem for a particular class of finite Jacobi matrices

✍ Scribed by F. Štampach; P. Šťovı´ček

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 302 KB
• Volume: 434
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2010.11.010

✦ Synopsis

A function F with simple and nice algebraic properties is defined on a subset of the space of complex sequences. Some special functions are expressible in terms of F, first of all the Bessel functions of first kind. A compact formula in terms of the function F is given for the determinant of a Jacobi matrix. Further we focus on the particular class of Jacobi matrices of odd dimension whose parallels to the diagonal are constant and whose diagonal depends linearly on the index. A formula is derived for the characteristic function. Yet another formula is presented in which the characteristic function is expressed in terms of the function F in a simple and compact manner. A special basis is constructed in which the Jacobi matrix becomes a sum of a diagonal matrix and a rank-one matrix operator. A vector-valued function on the complex plain is constructed having the property that its values on spectral points of the Jacobi matrix are equal to corresponding eigenvectors.