The characteristic function for complex doubly infinite Jacobi matrices

Abstract

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.