Pseudo-Hermiticity protects the energy-difference conservation in the scattering

Symmetry plays a fundamentally important role in physics. In this work, we find a conservation law, S^{†}(H_{c}^{†})S(H_{c})=I, which is valid for any non-Hermitian scattering center H_{c}. As a result, the reflections and transmissions of a non-Hermitian system {r,t} and its Hermitian conjugation system {r[over ¯],t[over ¯]} satisfy the conservation law r[over ¯]^{*}r+t[over ¯]^{*}t=1, instead of the energy conservation law that applies to incoming and outgoing waves in a Hermitian system. Consequently, the pseudo-Hermiticity of a non-Hermitian system ensures an energy-difference conservation. Furthermore, we demonstrate that the energy-difference conservation is respectively valid and invalid in two prototypical anti-PT-symmetric systems, where the energy-difference conservation is protected by the pseudo-Hermiticity. Our findings provide profound insight into the conservation law, the pseudo-Hermiticity, and the anti-PT-symmetry in non-Hermitian systems.