Manuscript Em03-D@1 · Em03-D-v1.tex · section · 450d878a

Explicit packet error

Bridge A: compact bulk tests approaching the packet › Explicit packet error · lines 847–886

Em03-D@1#eq:packeterror, Em03-D@1#eq:ordinarytailbound

Write

δ q_{R} = q_{R} - q, δ p_{R} = p_{R} - p .

Define

Δ_{R}^{2} = ∥ L^{3/4} δ p_{R} ∥_{2}^{2} + ∥ L^{5/4} δ q_{R} ∥_{2}^{2} + \frac{2}{β} (∥ L^{1/2} δ p_{R} ∥_{2}^{2} + ∥ L δ q_{R} ∥_{2}^{2}) .

All fractional powers are those of the positive self-adjoint radial operator.

For a bound using only ordinary spatial derivatives, put

D_{p} (R) = ∥ δ p_{R} ∥_{2}^{2} + ∥ L δ p_{R} ∥_{2}^{2},

D_{q} (R) = ∥ δ q_{R} ∥_{2}^{2} + ∥ L δ q_{R} ∥_{2}^{2} + ∥ \partial_{x} (L δ q_{R}) ∥_{2}^{2} + \int_{R} V ∣ L δ q_{R} ∣^{2} d x .

Then

Δ_{R}^{2} \leq (1 + 2/ β) (D_{p} (R) + D_{q} (R)) .

Displayed formulas in this passage

Em03-D@1#eq:packeterror · line 854
$Δ_{R}^{2} = ∥ L^{3/4} δ p_{R} ∥_{2}^{2} + ∥ L^{5/4} δ q_{R} ∥_{2}^{2} + \frac{2}{β} (∥ L^{1/2} δ p_{R} ∥_{2}^{2} + ∥ L δ q_{R} ∥_{2}^{2}) .$
Em03-D@1#eq:ordinarytailbound · line 881
$Δ_{R}^{2} \leq (1 + 2/ β) (D_{p} (R) + D_{q} (R)) .$
unlabeled · line 850
$δ q_{R} = q_{R} - q, δ p_{R} = p_{R} - p .$
unlabeled · line 869
$D_{p} (R) = ∥ δ p_{R} ∥_{2}^{2} + ∥ L δ p_{R} ∥_{2}^{2},$
unlabeled · line 872
$D_{q} (R) = ∥ δ q_{R} ∥_{2}^{2} + ∥ L δ q_{R} ∥_{2}^{2} + ∥ \partial_{x} (L δ q_{R}) ∥_{2}^{2} + \int_{R} V ∣ L δ q_{R} ∣^{2} d x .$