Is the free Hamiltonian to the three body problem self-adjoint?

Question

I have three particles: $x_0$ with mass $m$ and $x_1, x_2$ fermions with unitary mass. If I consider their free Hamiltonian I have $$H_0=-\frac{1}{2m}\Delta_{x_0}-\frac{1}{2}\Delta_{x_1}-\frac{1}{2}\Delta_{x_2}$$ acting on the space $$ \{f\in L^2(\mathbb{R}^9), f(x_0,x_1,x_2)=-f(x_0,x_2,x_1)\}$$ Is this operator selfadjoint in the domain $$D=\{f\in H^2(\mathbb{R}^9)|f(x_0,x_1,x_2)=-f(x_0,x_2,x_1)\}$$ ?