Abstract

Anti-$\mathcal{PT}$ symmetry, $(\mathcal{PT})H=-H(\mathcal{PT})$, is a plausible variant of $\mathcal{PT}$ symmetry. Of particular interest is the situation when all the eigenstates of an anti-$\mathcal{PT}$-symmetric non-Hermitian Hamiltonian $H$ are also eigenstates of the $\mathcal{PT}$ operator; then, the quasi-energies are purely imaginary, which implies that the Hermitian conjugate $H^{+}=-H$, and thus they are connected via the relation $(\mathcal{PT})H=H^{+}\mathcal{PT}$, similar to the quasi-Hermiticity relation. Therefore, the eigenfunctions of the anti-$\mathcal{PT}$-symmetric $H$ form a complete orthonormal set with positive definite norms, and moreover the time evolution is unitary.

中文翻译：

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非Hermitian哈密顿量的反$ \ mathcal {PT} $对称性

摘要

反$ \ mathcal {PT} $对称性，$（\ mathcal {PT}）H = -H（\ mathcal {PT}）$是$ \ mathcal {PT} $对称性的合理变体。当反$ \ mathcal {PT} $对称非Hermitian哈密顿量$ H $的所有本征态也是$ \ mathcal {PT} $算子的本征态时，这种情况尤其令人感兴趣。那么，准能量是纯虚数的，这意味着厄米共轭$ H ^ {+} =-H $，因此它们通过关系$（\ mathcal {PT}）H = H ^ {+}连接\ mathcal {PT} $，类似于准Hermiticity关系。因此，反$ \ mathcal {PT} $对称$ H $的本征函数形成具有正定范数的完整正交集，而且时间演化是统一的。