5.1 Far-shell contraction

Proposition 23 Far-shell contraction

✓

Set \(\eta _r:=\frac{1}{3}[1-(1+2r^2)^{-1/4}]\) and \(I_{\max }(\lambda ):=\max _k I_k(\lambda )\). There exist \(q_{\mathrm{sm}},q_{\mathrm{big}}\in (0,1)\) and \(a_r{\gt}0\) such that for all \(n\ge 2\) and \(t\ge 1\) the far shell decomposes as

\[ \begin{aligned} \mathcal F_{\mathrm{sm}} & :=\{ \lambda \in \mathcal{B}_{\pi /4}\setminus \mathcal{D}_r:J(\lambda )\le \eta _r\} ,\\ \mathcal F_{\mathrm{big}} & :=\{ \lambda \in \mathcal{B}_{\pi /4}\setminus \mathcal{D}_r:J(\lambda ){\gt}\eta _r,\; I_{\max }\ge 1\} ,\\ \mathcal F_{\mathrm{mid}} & :=\{ \lambda \in \mathcal{B}_{\pi /4}\setminus \mathcal{D}_r:J(\lambda ){\gt}\eta _r,\; I_{\max }{\lt}1\} , \end{aligned} \]

and

\[ |\psi (\lambda )|^{4t}\le \begin{cases} q_{\mathrm{sm}}^{4t} & \lambda \in \mathcal F_{\mathrm{sm}},\\ q_{\mathrm{big}}^{4t} & \lambda \in \mathcal F_{\mathrm{big}},\\ \exp (-a_r t n^{-2/3}) & \lambda \in \mathcal F_{\mathrm{mid}}. \end{cases} \]

Proof ▶

Small-\(J\): Lemma 10 gives \(|\psi _G(\lambda )|\le (1+2s(\lambda ))^{-1/4}\le q_G\). Corollary 13 gives \(|\psi -\psi _G|\le \frac32 J\le \frac32\eta _r=(1-q_G)/2\), so \(|\psi |\le q_{\mathrm{sm}}=(1+q_G)/2{\lt}1\).

Large-\(I_{\max }\): Lemma 1 gives \(|\psi |^2\le \frac12+\frac12 e^{-8I_{\max }/\pi ^2}\le q_{\mathrm{big}}^2\).

Middle: The same cosine-product bound gives \(|\psi |^2\le e^{-2I_{\max }/\pi ^2}\), and \(J{\gt} \eta _r\) with \(J\le nI_{\max }^{3/2}\) implies \(I_{\max }\ge (\eta _r/n)^{2/3}\), so \(|\psi |^{4t}\le \exp (-a_r tn^{-2/3})\).