Anti-$\mathcal{PT}$ symmetry for a non-Hermitian Hamiltonian

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.