Using normal operators as a common footing, the basic properties of Hermitian operators, unitary operators, and anti-Hermitian operators are treated from a unified viewpoint. The interlinks among the eigenvalues of these operators are elucidated with the help of suitable prescriptions for mapping in the plane of complex eigenvalues. Further, we show that some of the characteristics of Hermitian operators hold good under a perspective which is broader than the defining criterion for Hermitian operators.Using normal operators as a common footing, the basic properties of Hermitian operators, unitary operators, and anti-Hermitian operators are treated from a unified viewpoint. The interlinks among the eigenvalues of these operators are elucidated with the help of suitable prescriptions for mapping in the plane of complex eigenvalues. Further, we show that some of the characteristics of Hermitian operators hold good under a perspective which is broader than the defining criterion for Hermitian operators.

American Journal of PhysicsApr 1, 19780 citations

A rigorous Hermitian proof about the G-dynamics and analogy with Berry-Keating's Hamiltonian

Jack Whongius

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 $\left( {\hat{w}^{\left( cl \right)}}ϕ,φ\right)=\left( ϕ,{\hat{w}^{\left( cl \right)}}φ\right)$ for any two states $ϕ$ and $φ$ , 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 ${\hat{H}^{\left( \text{bk}\right)}}=-\sqrt{-1}\hbar \hat{θ}/2\in Her$ 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 $\hat{θ}/2=x\frac{d}{dx}+1/2\in SHer$ , etc.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 $\left( {\hat{w}^{\left( cl \right)}}ϕ,φ\right)=\left( ϕ,{\hat{w}^{\left( cl \right)}}φ\right)$ for any two states $ϕ$ and $φ$, 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 ${\hat{H}^{\left( \text{bk}\right)}}=-\sqrt{-1}\hbar \hat{θ}/2\in Her$ 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 $\hat{θ}/2=x\frac{d}{dx}+1/2\in SHer$, etc.

arXiv:2109.03068Aug 24, 20210 citations

{\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)}}ϕ,φ\right)=\left( ϕ,{\hat{w}^{\left( cl \right)}}φ\right)

for any two states

ϕ

and

φ

, 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{θ}/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{θ}/2=x\frac{d}{dx}+1/2\in SHer

, etc.