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arXiv:2302.10116

The non-Hermitian operators on the Baker-Hausdorff formula

Jack Whongius

Published Jan 29, 2023• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2302.10116

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This paper provides a connection to the non-Hermitian operators associated with the geometric potential function sss 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.
The family of metric operators, constructed by Musumbu et al ( 2007 J. Phys. A: Math. Theor. 40 F75 ), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian -symmetric part, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized non-Hermitian (not necessarily -symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed.
The family of metric operators, constructed by Musumbu et al (2007 J. Phys. A: Math. Theor. 40 F75), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian PT-symmetric part, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized non-Hermitian (not necessarily PT-symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed. Short title: Non-Hermitian oscillator Hamiltonian and su(1,1)
In this paper, we further study the G-dynamics newly emerged in the covariant dynamics defined by the QCPB theory. Then we want to seek the precise formulations of the G-dynamics to calculate the practical quantum problems. We explicitly verify how the role of G-dynamics plays in quantum mechanics, and then we propose three new operators based on the G-dynamics: D-operator, T-operator and G-operator, we find that the non-Hermitian operators are inevitable to appear in the framework of the QCPB. Then the eigenvalue equation of them are discussed accordingly by using the Schrodinger equation and geometrinetic energy operator (GEO) which leads to the Ri-operator, respectively, we obtain some useful results between two quantum situations. Meanwhile, we present generalized covariant wave equation based on the G-operator. The GEO for quantum harmonic oscillator as an application is further calculated.
After a few generalities we compare fundamental quantum mechanics applied to the harmonic oscillator with unusual oscillatorlike developments dealing with non-hermitian operators, this latter aspect exploiting in particular the property of subnormality. In this last context we can also restore the hermiticity of the Hamiltonian operator and discover a nice property of the new scalar product. General constructions of oscillatorlike Hamiltonians are also considered and new ideas connected with Heisenberg relations are presented.
Examples are given of non-Hermitian Hamiltonian operators which have a real spectrum. Some of the investigated operators are expressed in terms of the generators of the Weil-Heisenberg algebra. It is argued that the existence of an involutive operator J^\hat JJ^ which renders the Hamiltonian J^\hat JJ^-Hermitian leads to the unambiguous definition of an associated positive definite norm allowing for the standard probabilistic interpretation of quantum mechanics. Non-Hermitian extensions of the Poeschl-Teller Hamiltonian are also considered. Hermitian counterparts obtained by similarity transformations are constructed.