Geometric Dynamics of a Harmonic Oscillator, Arbitrary Minimal Uncertainty States and the Smallest Step 3 Nilpotent Lie Group

Abstract

The paper presents a new method of geometric solution of a Schrodinger equation by a construction of 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 shear group (also known as quartic group in literature). 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 the specific fiducial vector to produce a geometric solution. A technical aspect considered in this paper is that some modification of a coherent state transform is required: although representations of the group G square-integrability modulo a subgroup H, the obtained dynamic is transverse to the homogeneous space G/H.

arXiv commentAMS-LaTeX, 23 pp, 5 PDF pictures in two figures; v2-v4: numerous small improvements; v5: journal version

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Cited by in corpus (9)

• New Function Spaces Associated to Representations of Nilpotent Lie Groups and Generalized Time-Frequency Analysis
• Nonlinear Dynamics from Linear Quantum Evolutions
• An Extension of Moebius--Lie Geometry with Conformal Ensembles of Cycles and Its Implementation in a GiNaC Library
• Cross-Toeplitz Operators on the Fock--Segal--Bargmann Spaces and Two-Sided Convolutions on the Heisenberg Group
• Solving the Schrodinger Equation by Reduction to a First-order Differential Operator through a Coherent States Transform
• Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration
• Metamorphism -- an Integral Transform Reducing the Order of a Differential Equation
• Metamorphism as a covariant transform for the SSR group
• The quantum harmonic oscillator and the real Hilbert space