Energy conservation in approximated solutions of multiple beam   scattering problems
  Solutions in the form of series expansion, as the Born approximation, are very useful for describing time-independent scattering of quantum particles. In this work, it is mathematically demonstred that such solutions, when applied to multiple beam scattering phenomena, can lead to energy violation where the number of incident and scattered particles is not preserved. General basic conditions for developing consistent solutions of time-dependent multiple beam scattering problems are outlined. 
S\'ergio L. Morelh\~ao, Luis H. Avanci, Stefan Kycia