Energy conservation in approximated solutions of multi-beam scattering problems

Scattering of particles by time-independent potentials is describable by wave solutions in the form of series expansion. As mathematically demonstrated here, such form of solutions when applied to multi-beam scattering phenomena can lead to energy violation where the incident and scattered numbers of particles are not preserved. In this work, we outline a general basic requirement for developing consistent solutions of time-dependent multi-beam scattering problems by extended potentials.