Study of Multiplication Operator on ${\mathcal{H}}_{\frac{1}{2}}\oplus {\mathcal{H}}_{-\frac{1}{2}}$

In this paper we focus on the continuous representation on $\mathcal{S}(\mathbb{R})\subset L^2(\mathbb{R})$ with the operators $\frac{(PQ+QP)}{4}$ and $\frac{Q^2}{2}$ as generators given by $U[p,q]=\exp (-\frac{iq{Q}^2}{2})\exp (i\log p\frac{PQ+QP}{4})\mathrm{.}$ Action of the operator $\exp (PQ+QP)$ and the unitary equivalent operator on $\mathcal{S}(\mathbb{R})\subseteq L^2(\mathbb{R})$ of the multiplication operator in ${\mathcal{H}}_{\frac{1}{2}}\oplus {\mathcal{H}}_{-\frac{1}{2}}$ is obtained.