We study the validity of the Neumann or Born series approach in solving the Helmholtz equation and coefficient identification in related inverse scattering problems. Precisely, we derive a sufficient and necessary condition under which the series is strongly convergent. We also investigate the rate of convergence of the series. The obtained condition is optimal and it can be much weaker than the traditional requirement for the convergence of the series. Our approach makes use of reduction space techniques proposed by Suzuki \cite{Suzuki-1976}. Furthermore we propose an interpolation method that allows the use of the Neumann series in all cases. Finally, we provide several numerical tests with different medium functions and frequency values to validate our theoretical results.

翻译：我们研究了在求解亥姆霍兹方程及相关逆散射问题中系数识别时，诺伊曼或玻恩级数方法的有效性。具体而言，我们推导了级数强收敛的一个充分必要条件，并探讨了级数的收敛速率。所获条件是最优的，且可能远弱于传统上对级数收敛的要求。我们的方法利用了Suzuki提出的约化空间技术 \cite{Suzuki-1976}。此外，我们提出了一种插值方法，使得诺伊曼级数在所有情况下均可使用。最后，我们通过多个不同介质函数和频率值的数值测试，验证了我们的理论结果。