Theorem: Quantitative bulk-to-packet bridge

The compact tests (eq:compactpackettest) satisfy

For every $\omega \in {\mathfrak{S}}_C$,

Proof. From (eq:Cauchycoefficient),

Use

and $1+2n_{\mathrm{H}}\le 1+2/(\beta \omega )$. The resulting integral is bounded by

Spectral unitarity gives precisely (eq:packeterror).

For $\lambda \ge 0$, ${\lambda }^{3/2},\lambda \le 1+{\lambda }^2$ and ${\lambda }^{5/2}\le 1+{\lambda }^3$. Also

These facts prove (eq:ordinarytailbound).

Multiplication by $\chi (x/R)$ tends to the identity in every integer Sobolev norm on fixed Sobolev functions. This follows from the $L^2$ derivative tails and the bounds ${{\partial }_x}^j\chi (x/R)=O(R^{-j})$. In particular, $\delta p_R\to 0$ in $H^2$ and $\delta q_R\to 0$ in $H^3$. The potential and its required derivatives are bounded, so $D_p(R)+D_q(R)\to 0$.

Finally, $W_{\varphi }(F_{R,T})=W(k_R)$ in the common regular representation. Apply Lemma lem:Weylcomparison with $l=k_{\ast }$.