Born series
The Born series is the expansion of different scattering quantities in quantum scattering theory in the powers of the interaction potential formula_1 (more precisely in powers of formula_2 where formula_3 is the free particle Green's operator). It is closely related to Born approximation, which is the first order term of the Born series. The series can formally be understood as power series introducing the coupling constant by substitution formula_4. The speed of convergence and radius of convergence of the Born series are related to eigenvalues of the operator formula_5. In general the first few terms of the  Born series are good approximation to the expanded quantity for "weak" interaction formula_1 and large collision energy. Born series for scattering states. The Born series for the scattering states reads It can be derived by iterating the Lippmann–Schwinger equation Note that the Green's operator formula_3 for a free particle can be retarded/advanced or standing wave operator for retarded formula_10 advanced formula_11 or standing wave scattering states formula_12. The first iteration is obtained by replacing the full scattering solution formula_13 with free particle wave function formula_14 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 second Born approximation is sometimes used, when the first Born approximation vanishes, but the higher terms are rarely used. The Born series can formally be summed as geometric series with the common ratio equal to the operator formula_5, giving the formal solution to Lippmann-Schwinger equation in the form Born series for T-matrix. The Born series  can also be written for other scattering quantities like the T-matrix which is closely related to the scattering amplitude. Iterating Lippmann-Schwinger equation for the T-matrix we get For the T-matrix formula_3 stands only for retarded Green's operator formula_19. The standing wave Green's operator would give the K-matrix instead. Born series for full Green's operator. 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 formula_21