Appropriate Inner Product for PT-Symmetric Hamiltonians

Understand

A Hamiltonian $H$ that is not Hermitian can still have a real and complete energy eigenspectrum if it instead is $PT$ symmetric.

• For such Hamiltonians three possible inner products have been considered in the literature, the $V$ norm, the $PT$ norm, and the $C$ norm.
• Here $V$ is the operator that implements $VHV^{-1}=H^{\dagger}$, the $PT$ norm is the overlap of a state with its $PT$ conjugate, and $C$ is a discrete linear operator that always exists for any Hamiltonian that can be diagonalized.
• Here we show that it is the $V$ norm that is the most fundamental as it is always chosen by the theory itself.