\nonumber  \\
&-& \Lambda^4\int w(y) {\rm d} y \int \langle \Psi(x), \Psi(x) 
\rangle {\rm d} x \ .
\label{eq:2.35}\end{aligned}$$ By Schwarz’s inequality the second and third term together are bounded below by $$\sqrt{\frac{\alpha}{2}} \Lambda^3\int \langle \Psi(x), \Psi(y) 
\rangle \frac{w(x-y)}{\sqrt{\int w(z)
{\rm d}z}} {\rm d} x {\rm d} y \ .
\label{eq:2.36}$$ If we restate our bound in terms of Fourier space variables we get $$\int \left[\frac{|p|}{2}^2 + \sqrt{\frac{\alpha}{2}} ~ \Lambda^3
\frac{\widehat{w}(p)}{\sqrt{ \widehat{w}(0)}} 
\right] \langle \widehat{\Psi}(p), \widehat{\Psi}(p)\rangle {\rm d} p
 -\Lambda^4 \widehat{w}(0) \int \langle \widehat{\Psi}(p),
\widehat{\Psi}(p) \rangle {\rm d}p \ .
\label{eq:2.37}$$ Choosing the function $\widehat{w}(p)$ to be $(2 \pi)^3 
\Lambda^{-18/7}$ times the characteristic function of the ball of radius $\Lambda^{6/7}$, we have that $w(x)
\leq 1$ and it remains to optimize (\[eq:2.37\]) over all normalized states $\widehat{\Psi}(p)$. This is easily achieved by noting that the function $$\frac{1}{2} |p|^2 + \sqrt{\frac{\alpha}{ 2}} \Lambda^3 \frac{\widehat{w}(p)}{\sqrt{ \widehat{w}(0)}} 
\label{eq:2.38}$$ is everywhere larger than $\Lambda ^{12/7}$. Strictly speaking, the function $w(x)$ should be positive in order for the argument that led to (\[eq:2.36\]) to be valid. This can be achieved with a different choice of $w(x)$ that is more complicated but does not change the argument in an essential way.

NON-RELATIVISTIC MANY-BODY ENERGIES {#non-relativistic}
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