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

Abstract: Symmetry plays a fundamentally important role in physics. In this work, we find a conservation law, $S^{\dagger}(H_{c}^{\dagger})S(H_{c})=I$, which is valid for any non-Hermitian scattering center $H_c$. As a result, the reflections and transmissions of a non-Hermitian system $\left{ r,t\right}$ and its Hermitian conjugation system $\left{ \bar{r},\bar{t}\right} $ 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-$\mathcal{PT}$-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-$\mathcal{PT}$-symmetry in non-Hermitian systems.