Manuscript Hawk02-D@1 · Hawk02-D.tex · lemma · e6e4b72f

Lemma: Survival and collection certificates

Exact amplitudes and quantitative collection bounds · lines 722–762 · proof included or linked

Hawk02-D@1#lem:tails

For $t > 0$ ,

q (t) \leq min {1, (c t)^{- 2}}, ℓ_{M} (t) \leq η_{M} (2 c t)^{2} .

Consequently,

s_{M} (t) \leq min {1, (c t)^{- 2} + η_{M} (2 c t)^{2}} .

Proof. The integer-order Bessel Fourier integral gives $∣ J_{1} (z) ∣ \leq 1$ for real $z$ . Unitarity gives $q \leq 1$ .

For the collection bound, remove the scalar phase $e^{- i ω_{0} t}$ . Since $A^{r} e_{0}$ is supported on sites $0, \dots, r$ ,

P_{> M} e^{- i c t A} e_{0} = P_{> M} r = M + 1 \sum^{\infty} \frac{( - i c t A ) ^{r}}{r !} e_{0} .

Using $∥ A ∥ \leq 2$ , its norm is at most

r = M + 1 \sum^{\infty} \frac{( 2 c t ) ^{r}}{r !} \leq e^{2 c t} \frac{( 2 c t ) ^{M + 1}}{( M + 1 )!} .

The last inequality follows from $(m + r)! \geq m! r!$ . Squaring gives the result.

Displayed formulas in this passage

unlabeled · line 725
$q (t) \leq min {1, (c t)^{- 2}}, ℓ_{M} (t) \leq η_{M} (2 c t)^{2} .$
unlabeled · line 731
$s_{M} (t) \leq min {1, (c t)^{- 2} + η_{M} (2 c t)^{2}} .$
unlabeled · line 747
$P_{> M} e^{- i c t A} e_{0} = P_{> M} r = M + 1 \sum^{\infty} \frac{( - i c t A ) ^{r}}{r !} e_{0} .$
unlabeled · line 754
$r = M + 1 \sum^{\infty} \frac{( 2 c t ) ^{r}}{r !} \leq e^{2 c t} \frac{( 2 c t ) ^{M + 1}}{( M + 1 )!} .$