# Reflection positivity on weighted $L^2$-spaces

Denote by $$(t, x_{1}, \ldots, x_{d-1})$$ the coordinates of $$x \in \mathbb{R}^{d}$$ and set $$\mathbb{R}^{d}_{+}=\left\{t, x_{1}, \ldots, x_{d-1} \in \mathbb{R}^{d}|t > 0\right\}.$$ Write $$\theta$$ for the reflection $$(t, x_{1}, \ldots, x_{d-1}) \mapsto (-t, x_{1}, \ldots, x_{d-1}).$$ as well as for the induced map on $$L^2(\mathbb{R}^{d})$$.

The resolvent $$C=(1-\Delta)^{-1}$$, where $$\Delta$$ is the Laplacian on $$\mathbb{R}^{d}$$, is known to be reflection positive on $$L^2(\mathbb{R}^{d})$$, i.e., one has $$\langle C f ,\theta f \rangle_{L^2} \geq 0$$ for all $$f \in C^{\infty}_{c}(\mathbb{R}^{d}_{+})$$, the compactly supported smooth functions on $$\mathbb{R}^{d}_{+}$$.

Fix $$s \in \mathbb{R}$$ and set $$\rho(x)=(1+\|x\|)^s$$ where $$x \in \mathbb{R}^{d}$$. Is $$C$$ reflection positive on the weighted $$L^2$$-space with weight $$\rho$$ ? In other words, is it true that $$\langle C f ,\rho \hspace{2pt}\theta f \rangle_{L^2} \geq 0$$ for all $$f \in C^{\infty}_{c}(\mathbb{R}^{d}_{+})$$ ?

• Isn't this tantamount to the question whether $\rho C$ is reflection positiive?
– gmvh
Apr 12, 2021 at 5:36
• @gmvh : Certainly it is.
– S.Z.
Apr 12, 2021 at 17:22