Almalki, F and Kisil, VV orcid.org/0000-0002-6593-6147 (2018) Geometric Dynamics of a Harmonic Oscillator, Arbitrary Minimal Uncertainty States and the Smallest Step 3 Nilpotent Lie Group. Journal of Physics A: Mathematical and Theoretical, 52. 025301. ISSN 1751-8113
Abstract
The paper presents a new method of geometric solution of a Schrödinger equation by constructing an equivalent first-order partial differential equation with a bigger number of variables. The equivalent equation shall be restricted to a specific subspace with auxiliary conditions which are obtained from a coherent state transform. The method is applied to the fundamental case of the harmonic oscillator and coherent state transform generated by the minimal nilpotent step three Lie group—the group (also known under many names, e.g. quartic group). We obtain a geometric solution for an arbitrary minimal uncertainty state used as a fiducial vector. In contrast, it is shown that the well-known Fock–Segal–Bargmann transform and the Heisenberg group require a specific fiducial vector to produce a geometric solution. A technical aspect considered in this paper is that a certain modification of a coherent state transform is required: although the irreducible representation of the group is square-integrable modulo a subgroup , the obtained dynamic is transverse to the homogeneous space .
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2018 IOP Publishing Ltd. This is an author produced version of a paper published in Journal of Physics A: Mathematical and Theoretical. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | math-ph; math-ph; math.MP; math.RT; quant-ph; 81R30, 20C35, 22E70, 35Q70, 35A25 |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 04 Jan 2019 12:53 |
Last Modified: | 01 Nov 2019 01:39 |
Status: | Published |
Publisher: | IOP Publishing |
Identification Number: | 10.1088/1751-8121/aaed4d |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:132456 |