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arXiv · 2310.02908

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

H. S. Xu·L. Jin
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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.

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BibTeXRIS

H. S. Xu, L. Jin. 2023-10-04. Pseudo-Hermiticity protects the energy-difference conservation in the scattering. https://doi.org/10.1103/physrevresearch.5.l042005

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