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

Abstract

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.