Two Body Schrodinger Equations

Question

I have a question involving the eigenvalues of a two-body Schrodinger equation. Let $$H=-\frac{1}{2m}\Delta_{x_1}-\frac{1}{2m}\Delta_{x_2}+\frac{e^2}{|{{x_1}-{x_2}}|}$$ over the Hilbert space $L^2(\mathbb{R}^6)$ be self adjoint on the domain $D(H)$ ($D(H)$ would be $H^2(\mathbb{R}^6)$ would it not?).
Then, let $H_f$ be the same operator restricted to the domain $D(H)\cap\mathcal{H}_f$ where $\mathcal{H_f}=\{\varphi\in L^2(\mathbb{R}^6)|\varphi(x_1,x_2)=-\varphi(x_2,x_1)\}$, i.e., $\mathcal{H_f}$ is the fermionic subspace.

I want to show that $inf \sigma(H) \leq inf \sigma(H_f)$.

Ideas:
1. I have shown before that $H$ has eigenvalues below zero using Birman-Schwinger. If I could somehow show that $H_f$ only has non-negative eigenvalues, this would work.
2. Use the Rayliegh-Ritz principle, i.e. $$inf\sigma(H)\leq \frac{<\psi,H\psi>}{||\psi||}$$ for all $\psi\in D(H)$ (since D(H) is self-adjoint). I'm not sure exactly how to use this though, because $H_f$ is not self adjoint, and I need it as an upper bound, not a lower.

Any help is appreciated!