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Jack Whongius

Publications and source records attributed to Jack Whongius.

5 recordsLinked to original sources
arXiv2023

The non-Hermitian operators on the Baker-Hausdorff formula

This paper provides a connection to the non-Hermitian operators associated with the geometric potential function $s$ and Baker-Hausdorff formula. The geometric quantum potential is considered in a precise condition. The Ri-operator as a non-Hermitian Hamiltonian to describe the generalized quantum harmonic oscillator can be re-expressed as a more compact quantum formula by using the Baker-Hausdorff formula, more deeply, we prove some results based on the application of this formula.

physics.gen-ph↗
arXiv2022

Eigenvalues of the generalized Laplacian and the G-dynamics of type I

In this paper, we consider the generalized Laplace operator equipped with the G-dynamics operator of type I, the Dirichlet and Neumann eigenvalue problems are extended to associate with the G-dynamics of type I, it is proved that the G-dynamics of type I satisfies an integral identity. The G-dynamics of type II for generalized Laplacian is studied as well. Using the general method related to Dirichlet eigenvalue problem, an estimate analysis for the generalized Laplacian with some conditions is made.

physics.gen-ph↗
arXiv2021

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

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.

physics.gen-ph↗
arXiv2021

On one-dimensional G-dynamics and non-Hermitian Hamiltonian operators

Focusing on the algebraical analysis of two various kinds of one-dimensional G-dynamics ${\hat{w}^{\left( cl \right)}}$ and ${\hat{w}^{\left( ri\right)}}$ separately induced by different Hamiltonian operators $\hat{H} $ are the keypoints. In this work, it's evidently proved that an identity ${\hat{w}^{\left( cl \right)}}{{u}^{-1/2}}\equiv0$ always holds for any $u>0$ based on the formula of one-dimensional G-dynamics ${\hat{w}^{\left( cl \right)}}$. We prove that the G-dynamics ${\hat{w}^{\left( cl \right)}}$ and ${\hat{w}^{\left( ri\right)}}$ obey Leibniz identity if and only if ${\hat{w}^{\left( cl \right)}}1=0$ and ${\hat{w}^{\left( ri\right)}}1=0$, respectively. \par In accordance with the G-dynamics ${\hat{w}^{\left( cl \right)}}$, we investigate the unique eigenvalues equation ${\hat{w}^{\left( cl \right)}}L\left( u,t,λ\right)=-\sqrt{-1}λL\left( u,t,λ\right)$ of the G-dynamics with a precise geometric eigenfunction $L\left( u,t,λ\right)={{u}^{-1/2}}{{e}^{{λ}t}},~u>0$ as time $t\in \left[ 0,T \right]$ develops and the equation of energy spectrum is then induced. The non-Hermitian Hamiltonian operators are studied as well, we obtain a series of ODE with their special solutions, and we prove multiplicative property of the geometric eigenfunction. The coordinate derivative and time evolution of the G-dynamics are respectively considered. Seeking the invariance of G-dynamics ${\hat{w}^{\left( cl \right)}}$ under coordinate transformation is considered, so that we think of one-dimensional G-dynamics ${\hat{w}^{\left( cl \right)}}$ on coordinate transformation, it gives the rule of conversion between two coordinate systems. As a application, some examples are given for such rule of conversion. Meanwhile, we search the conditions that quantum geometric bracket vanishes and a specific case follows.

physics.gen-ph↗