The method of continued fractions is a method developed specifically for solution of integral equations of quantum scattering theory like Lippmann–Schwinger equation or Faddeev equations. It was invented by Horáček and Sasakawa  in 1983. The goal of the method is to solve the integral equation
iteratively and to construct convergent continued fraction for the T-matrix
The method has two variants. In the first one (denoted as MCFV) we construct approximations of the potential energy operator  in the form of separable function of rank 1, 2, 3 ... The second variant (MCFG method) constructs the finite rank approximations to Green's operator. The approximations are constructed within Krylov subspace constructed from vector  with action of the operator . The method can thus be understood as resummation of (in general divergent) Born series by Padé approximants. It is also closely related to Schwinger variational principle.
In general the method requires similar amount of numerical work as calculation of terms of Born series, but it provides much faster convergence of the results.