On the eigenvalue problem for a particular class of finite Jacobi matrices
Abstract
A function $\mathfrak{F}$ 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 $\mathfrak{F}$, first of all the Bessel functions of first kind. A compact formula in terms of the function $\mathfrak{F}$ 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 $\mathfrak{F}$ 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 rankone matrix operator. A vectorvalued 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2010
 DOI:
 10.48550/arXiv.1011.1241
 arXiv:
 arXiv:1011.1241
 Bibcode:
 2010arXiv1011.1241S
 Keywords:

 Mathematical Physics;
 Mathematics  Commutative Algebra;
 47B36;
 15A18;
 33C10