Let $N\in\mathbb{N}$, and consider the formal $N$-body Schrodinger operator $$\sum_{j=1}^{N}-\partial_{x_{j}}^{2}+2c\sum_{1\leq j_{1}<j_{2}\leq N}\delta(X_{j_{1}}-X_{j_{2}}), \tag{1}$$ where $c\in\mathbb{R}$ is some constant. Above, $\delta(X_{j_{1}}-X_{j_{2}})$ denotes multiplication by the distribution $\delta(x_{j_{1}}-x_{j_{2}})$ (i.e. the Lebesgue measure on the hyperplane in $\{x_{j_{1}}=x_{j_{2}}\}$ in $\mathbb{R}^{N}$). By using the KLMN theorem (see [Reed-Simon, MMP V. II] Theorem X.17), one can realize a self-adjoint operator $H_{N}:Dom(H_{N})\rightarrow L_{s}^{2}(\mathbb{R}^{N})$ on the space of wave functions invariant under permutation (i.e. bosonic) corresponding to the formal expression (1), which is associated to the quadratic form
$$q(\Phi,\Psi) := \langle{\nabla\Phi,\nabla\Psi}\rangle_{L^{2}(\mathbb{R}^{N})} + 2c\sum_{1\leq j_{1}<j_{2}\leq N}\langle{\mathrm{Tr}_{x_{j_{1}}=x_{j_{2}}}(\Phi), \mathrm{Tr}_{x_{j_{1}}=x_{j_{2}}}(\Psi)}\rangle_{L^{2}(\mathbb{R}^{N-1})}.$$
If we choose the interaction strength $c=\frac{1}{N-1}$ so that both terms in (1) grow like the number of particles $N$, then (formally), we can rewrite (1) a
$$\frac{1}{(N-1)}\sum_{1\leq j_{1}<j_{2}\leq N}\{-\partial_{x_{j_{1}}}^{2}-\partial_{x_{j_{2}}}^{2}+2\delta(X_{j_{1}}-X_{j_{2}})\}. \tag{2}$$
Each term in the summation in (2) is a permutation the "$2$-body operator"
$$(-\partial_{x_{1}}^{2}-\partial_{x_{2}}^{2}+2\delta(X_{1}-X_{2})) \otimes Id_{x_{3},\ldots,x_{N}} \tag{3}$$
on $L^{2}(\mathbb{R}^{N})$ (I have omitted the $s$ subscript because the operator (3) does not preserve the symmetry). Since each operator
$$-\partial_{x_{j_{1}}}^{2}-\partial_{x_{j_{2}}}^{2}+2\delta(X_{j_{1}}-X_{j_{2}})$$
is a positive, self-adjoint operator on $L^{2}(\mathbb{R}^{N})$, we can take the Friedrich's extension of the sum over $1\leq j_{1}<j_{2}\leq N$ and then restrict to $L_{s}^{2}(\mathbb{R}^{N})$ to obtain $H_{N}$.
Question. Is there some tensor decomposition of $L_{s}^{2}(\mathbb{R}^{N})$ which can be used to reduce the construction of $H_{N}$ to constructing a $2$-body operator?
My dissatisfaction with using the Friedrich's extension argument is that it required positiveness of the operators. I have a situation where I can construct a self-adjoint $2$-body operator $\tilde{H}_{(1,2)}$ with principal part $(-i\partial_{x_{1}})^{3}+(-i\partial_{x_{2}})^{3}$ together with some $\delta$ pair potential involving derivatives, which is not positive. Moreover, I do not know how to generalize my $2$-body construction to the prove that the operator
$$\frac{1}{(N-1)}\sum_{1\leq j_{1}<j_{2}\leq N} \tilde{H}_{(j_{1},j_{2})},$$
where each $\tilde{H}_{(j_{1},j_{2})}$ is just a permutation of the operator ${H}_{(1,2)}$ in the sense previously described.
