Necessary condition for a Hamiltonian to be proper in quantum mechanics

In quantum mechanics, the Hamiltonians are required to be hermitian, since hermiticity guarantees that the energy spectrum is real and the time evolution is unitary. However, some non-hermitian Hamiltonians are also found meeting these requirements. The hermiticity is essentially a sufficient condition. In the current article, we formulate the necessary condition for a Hamiltonian to be proper in quantum mechanics, regarding the quantization condition it follows and the role it plays in the governing equation of dynamic evolution. It can be confirmed that the Hamiltonians adopted in quantum mechanics, even the non-hermitian ones such as$\hat{H}=\hat{p}^2+\mathrm{i} \hat{x}^3$ and $\hat{H}=\hat{p}^2-\hat{x}^4$, meet such a necessary condition. The necessary condition provides the first criterium for the candidate Hamiltonians to be introduced.