On onedimensional Gdynamics and nonHermitian Hamiltonian operators
Abstract
Focusing on the algebraical analysis of two various kinds of onedimensional Gdynamics ${\hat{w}^{\left( cl \right)}}$ 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 onedimensional Gdynamics ${\hat{w}^{\left( cl \right)}}$. We prove that the Gdynamics ${\hat{w}^{\left( cl \right)}}$ 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 Gdynamics ${\hat{w}^{\left( cl \right)}}$, we investigate the unique eigenvalues equation ${\hat{w}^{\left( cl \right)}}L\left( u,t,\lambda \right)=\sqrt{1}\lambda L\left( u,t,\lambda \right)$ of the Gdynamics with a precise geometric eigenfunction $L\left( u,t,\lambda \right)={{u}^{1/2}}{{e}^{{\lambda }t}},~u>0$ as time $t\in \left[ 0,T \right]$ develops and the equation of energy spectrum is then induced. The nonHermitian 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 Gdynamics are respectively considered. Seeking the invariance of Gdynamics ${\hat{w}^{\left( cl \right)}}$ under coordinate transformation is considered, so that we think of onedimensional Gdynamics ${\hat{w}^{\left( cl \right)}}$ 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.01947
 Bibcode:
 2021arXiv210801947W
 Keywords:

 Physics  General Physics
 EPrint:
 32 pages, submitted to the JMAA, comments are welcome