Perturbation foundation of \(q\)-deformed dynamics
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Publication:1810389
DOI10.1140/EPJC/S2002-01086-1zbMATH Open1030.81504arXivhep-th/0211051OpenAlexW3106010200MaRDI QIDQ1810389
Publication date: 4 June 2003
Published in: The European Physical Journal C. Particles and Fields (Search for Journal in Brave)
Abstract: In the q-deformed theory the perturbation approach can be expressed in terms of two pairs of undeformed position and momentum operators. There are two configuration spaces. Correspondingly there are two q-perturbation Hamiltonians, one originates from the perturbation expansion of the potential in one configuration space, the other one originates from the perturbation expansion of the kinetic energy in another configuration space. In order to establish a general foundation of the q-perturbation theory, two perturbation equivalence theorems are proved: (I) Equivalence theorem {it I}: Perturbation expressions of the q-deformed uncertainty relations calculated by two pairs of undeformed operators are the same, and the two q-deformed uncertainty relations undercut Heisenberg's minimal one in the same style. (II) The general equivalence theorem {it II}: for {it any} potential (regular or singular) the expectation values of two q-perturbation Hamiltonians in the eigenstates of the undeformed Hamiltonian are equivalent to all orders of the perturbation expansion. As an example of singular potentials the perturbation energy spectra of the q-deformed Coulomb potential are studied.
Full work available at URL: https://arxiv.org/abs/hep-th/0211051
Quantum groups and related algebraic methods applied to problems in quantum theory (81R50) Perturbation theories for operators and differential equations in quantum theory (81Q15)
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