The Born series[1] is the expansion of different scattering quantities in quantum scattering theory in the powers of the interaction potential
is the free particle Green's operator).
In general the first few terms of the Born series are good approximation to the expanded quantity for "weak" interaction
The Born series for the scattering states reads It can be derived by iterating the Lippmann–Schwinger equation Note that the Green's operator
for a free particle can be retarded/advanced or standing wave operator for retarded
or standing wave scattering states
The first iteration is obtained by replacing the full scattering solution
with free particle wave function
on the right hand side of the Lippmann-Schwinger equation and it gives the first Born approximation.
The second iteration substitutes the first Born approximation in the right hand side and the result is called the second Born approximation.
In general the n-th Born approximation takes n-terms of the series into account.
The Born series can formally be summed as geometric series with the common ratio equal to the operator
, giving the formal solution to Lippmann-Schwinger equation in the form The Born series can also be written for other scattering quantities like the T-matrix which is closely related to the scattering amplitude.
stands only for retarded Green's operator
The standing wave Green's operator would give the K-matrix instead.
The Lippmann-Schwinger equation for Green's operator is called the resolvent identity, Its solution by iteration leads to the Born series for the full Green's operator