The method of continued fractions is a method developed specifically for solution of integral equations of
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
[2]) 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.
Algorithm of MCFV
The derivation of the method proceeds as follows. First we introduce rank-one (separable)
approximation to the potential
The integral equation for the rank-one part of potential is easily soluble. The full solution of the original problem can therefore be expressed as
in terms of new function . This function is solution of modified Lippmann–Schwinger equation
with
The remainder potential term is transparent for incoming wave
i. e. it is weaker operator than the original one.
The new problem thus obtained for is of the same form as the original one and we can repeat the procedure.
This leads to recurrent relations
It is possible to show that the T-matrix of the original problem can be expressed in the form of chain fraction
where we defined
In practical calculation the infinite chain fraction is replaced by finite one assuming that
This is equivalent to assuming that the remainder solution
is negligible. This is plausible assumption, since the remainder potential has all vectors
in its
null space
and it can be shown that this potential converges to zero and the chain fraction converges to the exact T-matrix.
Algorithm of MCFG
The second variant[2] of the method construct the approximations to the Green's operator
now with vectors
The chain fraction for T-matrix now also holds, with little bit different definition of coefficients .[2]
Properties and relation to other methods
The expressions for the T-matrix resulting from both methods can be related to certain class of variational principles. In the case of first iteration of MCFV method we get the same result as from Schwinger variational principle with trial function . The higher iterations with N-terms in the continuous fraction reproduce exactly 2N terms (2N + 1) of
Lippmann-Schwinger equation with the potential given by
finite-rank operator. The number of iterations is then equal to the rank of the potential. The method has been successfully used for solution of problems in both
nuclear[3] and
molecular physics.
[4]
References
- .
- ^ .
- ^ Sasakawa T. "Models and methods in few body physics", edited by Ferreira, Fonseca, Sterit, Springer-Verlag, Berlin, Heidelberg 1987
- .