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We introduce a class of doubly infinite complex Jacobi matrices determined bya simple convergence condition imposed on the diagonal and off-diagonalsequences. For each Jacobi matrix belonging to this class, an analyticfunction, called a characteristic function, is associated with it. It is shownthat the point spectrum of the corresponding Jacobi operator restricted to asuitable domain coincides with the zero set of the characteristic function.Also, coincidence regarding the order of a zero of the characteristic functionand the algebraic multiplicity of the corresponding eigenvalue is proved.Further, formulas for the entries of eigenvectors, generalized eigenvectors, asummation identity for eigenvectors, and matrix elements of the resolventoperator are provided. The presented method is illustrated by several concreteexamples.

The characteristic function for complex doubly infinite Jacobi matrices