Tensor Product Decomposition of Operators

Question

I read recently that Laplacian on $L^2(\mathbb{R}^2)$ can be written as closure of $-\Delta_x\otimes Id+Id\otimes(-\Delta_y)$ on $L^2(\mathbb{R})\otimes L^2(\mathbb{R})$.

Reed, Michael; Simon, Barry, Methods of modern mathematical physics. I: Functional analysis. Rev. and enl. ed, New York etc.: Academic Press, XV, 400 p. (1980).

Let $A$ be an unbounded operator on $L^2(\mathbb{R}^d)$.

1. Is there a natural way to write an arbitrary unbounded operator as a tensor product of operators on $L^2(\mathbb{R}^{d-1})\otimes L^2(\mathbb{R})$, that also preserves the symmetries of the original operator?

2. As an example, one of the ways in which this could work for the operator $-\Delta+V$ on $L^2(\mathbb{R}^d)$ is if $V$ could be written as a sum of functions of respective variables, i.e., $V(x_1,x_2)=V(x_1)+V(x_2)$.

3. Is this also a necessary condition?