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

TL;DR

This paper uncovers a new conservation law in non-Hermitian scattering systems, showing that pseudo-Hermiticity ensures energy-difference conservation, with implications for anti-$ ext{PT}$-symmetric systems.

Contribution

It introduces a novel conservation law valid for non-Hermitian scattering centers, linking pseudo-Hermiticity to energy-difference conservation, expanding understanding of symmetry in non-Hermitian physics.

Findings

Conservation law $S^{}H_{c}^{}S(H_{c})=I$ established.

Pseudo-Hermiticity guarantees energy-difference conservation.

Energy-difference conservation varies in anti-$ ext{PT}$-symmetric systems.

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.