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 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 𝔾 (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 𝔾 is square-integrable modulo a subgroup H, the obtained dynamic is transverse to the homogeneous space 𝔾/H.