Consequences of the fundamental conjecture for the motion on the Siegel-Jacobi disk

We find the homogenous K\"ahler isomorphism $FC$ which expresses the K\"ahler two-form on the Siegel-Jacobi domain $\mathcal{D}^J_1=\mathbb{C}\times\mathcal{D}_1$ as the sum of the K\"ahler two-form on $\mathbb{C}$ and the one on the Siegel ball $\mathcal{D}_1$. The classical motion and quantum evolution on $\mathcal{D}^J_1$ determined by a linear Hamiltonian in the generators of the Jacobi group $G^J_1=H_1\rtimes\text{SU}(1,1)$ is described by a Riccati equation on $\mathcal{D}_1$ and a linear first order differential equation in $z\in\mathbb{C}$, where $H_1$ denotes the real 3-dimensional Heisenberg group. When the transformation $FC$ is applied, the first order differential equation for the variable $z\in \mathbb{C}$ decouples of the motion on the Siegel disk. Similar considerations are presented for the Siegel-Jacobi space $\mathcal{X}^J_1=\mathbb{C}\times\mathcal{X}_1$, where $\mathcal{X}_1$ denotes the Siegel upper half plane.