2021/08/04 by Whongius, Jack
#FOS: Physical sciences #General Physics (physics.gen-ph)
paper · doi:10.48550/arxiv.2108.01947
Focusing on the algebraical analysis of two various kinds of one-dimensional G-dynamics w( cl ) and w( ri) separately induced by different Hamiltonian operators H are the keypoints. In this work, it's evidently proved that an identity w( cl )u-1/2≡0 always holds for any u>0 based on the formula of one-dimensional G-dynamics w( cl ). We prove that the G-dynamics w( cl ) and w( ri) obey Leibniz identity if and only if w( cl )1=0 and w( ri)1=0, respectively. \par In accordance with the G-dynamics w( cl ), we investigate the unique eigenvalues equation w( cl )L( u,t,λ)=-√(-1)λL( u,t,λ) of the G-dynamics with a precise geometric eigenfunction L( u,t,λ)=u-1/2e^λt,~u>0 as time t∈ [ 0,T ] 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 w( cl ) under coordinate transformation is considered, so that we think of one-dimensional G-dynamics w( cl ) 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.