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

TL;DR

This paper investigates the continuous representation of certain operators on a function space, deriving the action of a specific exponential operator and its unitary equivalent related to multiplication operators in a direct sum of Hilbert spaces.

Contribution

It explicitly characterizes the action of the exponential of the multiplication operator on a specific function space and its unitary equivalent in a direct sum of Hilbert spaces.

Findings

Derived the action of $ ext{exp}(PQ+QP)$ on $ ext{S}( ext{R})$

Established the unitary equivalence with a multiplication operator

Provided explicit formulas for the operators involved

Abstract

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 $U[p,q]=\exp(-\frac{iqQ^2}{2}) \exp(i\log p\frac{PQ+QP}{4}).$ 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.