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 $\{\bar{r},\bar{t}\}$ satisfy the conservation law ${\bar{r}}^{\ast }r+{\bar{t}}^{\ast }t=1$, 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-$\mathcal{P}\mathcal{T}$-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-$\mathcal{P}\mathcal{T}$-symmetry in non-Hermitian systems.