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.

翻译：我们引入了一类由对角和对角外序列的简单趋同条件决定的双倍无限复杂的雅各基质。 对于属于这一类的每个雅各基质,都与此相关联, 一种分析函数, 称为特性函数, 显示相应的雅各基经营人的点谱限制在合适的域内, 与特性函数的零组合相吻合。 另外, 特性函数零的顺序和相应的等离子值的代数多重的巧合得到了证明。 此外, 提供了一些具体例子, 说明的计算方法。