  In this work we have presented a rather general and easy-to-apply method for
discrete Hilbert space representation of quantum mechanical Green's operators.
We have shown that if in some discrete Hilbert space basis representation the
Hamiltonian takes an infinite symmetric tridiagonal, i.e. Jacobi-matrix form
the corresponding Green's matrix can be calculated on the whole complex energy
plane by a continued fraction. The procedure necessitates only the analytic
calculation of the Hamiltonian matrix elements, which are used to construct the
coefficients of the continued fraction. This continued fraction representation
of the Green's operator was shown to be convergent for the bound state energy
region. The theory of analytic continuation of continued fractions was utilized
to extend the representation to the whole complex energy plane. The presented
method provides a simple, easily applicable and analytically correct recipe for
calculating discrete basis representation of Green's operators.
