  Quantum covariant Hamiltonian system theory provides a coherent framework for
modelling the complex dynamics of quantum systems. In this paper, we centrally
deal with the Hermiticity of quantum operators that directly links to the
physical observable, thusly, we give a rigorous proof to verify one-dimensional
G-dynamics ${{\hat{w}}^{\left( cl \right)}}={{\hat{w}}^{\left( cl
\right)\dagger }}\in Her$ that is a Hermitian operator satisfying $\left(
{{{\hat{w}}}^{\left( cl \right)}}\phi ,\varphi \right)=\left( \phi
,{{{\hat{w}}}^{\left( cl \right)}}\varphi \right)$ for any two states $\phi$
and $\varphi$, and its eigenvalues are real. We also prove that curvature
operator is a skew-Hermitian operator as well. The act of finishing this
Hermitian proof valuably enables us to ensure the non-Hermitian Hamiltonian
operator ${{\hat{H}}^{\left( ri \right)}} ={{\hat{H}}^{\left( g \right)}}
-{{\hat{H}}^{\left( \operatorname{clm} \right)}}\in NHer$ that is divided into
the Hermitian operator ${{\hat{H}}^{\left( g \right)}} ={{\hat{H}}^{\left( cl
\right)}}-{{E}^{\left( s \right)}}/2\in Her$ and the skew-Hermitian operator
${{\hat{H}}^{\left( \operatorname{clm} \right)}}=\sqrt{-1}\hbar
{{\hat{w}}^{\left( cl \right)}}\in SHer$ generally, and ${{\hat{H}}^{\left( ri
\right)}}$ always has the complex eigenvalues. We use the formula of the
G-dynamics to evaluate the Berry-Keating's Hamiltonian operator
${{\hat{H}}^{\left( \text{bk}\right)}}=-\sqrt{-1}\hbar \hat{\theta }/2\in Her$
and its extensive version $\hat{H}^{\left( \text{gbk}\right)}\in NHer$ as the
applications of the G-dynamics, to see how the similarity appears in the light
of obvious factor $\hat{\theta }/2=x\frac{d}{dx}+1/2\in SHer$, etc.
