Manuscript IV@1 · IV.tex · section · 351cb2d7

Cutoff kernel estimates

Cutoff ledger: Model 3 › Cutoff kernel estimates · lines 3668–3694

At fixed scalar momentum cutoff $Λ$ ,

∣ K_{J}^{pair} (p, q) ∣ \leq C_{J, Λ} (1 + ∣ p + q ∣^{- 1}),

∣ K_{J}^{mixed} (p, q) ∣ \leq C_{J, Λ} (1 + ∣ p - q ∣^{- 2}) .

The first is square integrable on the cutoff support. The second has finite Schur integrals there. The cancellation

C_{00}^{++} = O (∣ p + q ∣^{2}), C_{0 i}^{++} = O (∣ p + q ∣)

is essential for the pair estimate. No Newton infrared mask is used.

The resulting symmetric operator obeys

∥ B_{Λ} (J) Ψ ∥ \leq C_{Λ} ∥ (N + 2) Ψ ∥,

and $B_{Λ, M}$ is bounded self-adjoint for each finite $M$ . This is not a consequence of number compression alone.

Displayed formulas in this passage

unlabeled · line 3671
$∣ K_{J}^{pair} (p, q) ∣ \leq C_{J, Λ} (1 + ∣ p + q ∣^{- 1}),$
unlabeled · line 3675
$∣ K_{J}^{mixed} (p, q) ∣ \leq C_{J, Λ} (1 + ∣ p - q ∣^{- 2}) .$
unlabeled · line 3681
$C_{00}^{++} = O (∣ p + q ∣^{2}), C_{0 i}^{++} = O (∣ p + q ∣)$
unlabeled · line 3688
$∥ B_{Λ} (J) Ψ ∥ \leq C_{Λ} ∥ (N + 2) Ψ ∥,$