PT symmetry as a necessary and sufficient condition for unitary time evolution

Understand

While Hermiticity of a time-independent Hamiltonian leads to unitary time evolution, in and of itself, the requirement of Hermiticity is only sufficient for unitary time evolution.

• In this paper we provide conditions that are both necessary and sufficient.
• We show that $PT$ symmetry of a time-independent Hamiltonian, or equivalently, reality of the secular equation that determines its eigenvalues, is both necessary and sufficient for unitary time evolution.
• For any $PT$-symmetric Hamiltonian $H$ there always exists an operator $V$ that relates $H$ to its Hermitian adjoint according to $VHV^{-1}=H^{\dagger}$.