Manuscript V-B@1 · V-B-v1.tex · section · 3fadc1ad

One metric oscillator and the switched identity

One metric oscillator and the switched identity · lines 1281–1308

This conclusion concerns the displayed degree-at-most-two polynomial, not the mixed cubic interaction vertex or an all-orders interacting Hamiltonian.

For comparison with an unswitched response starting at $0$ , set

v_{a}^{0} = K (- 1_{[0, b]} (t) \partial_{i} (a^{ij} \partial_{j} u_{f})) .

These particular discontinuous tests still give Schwartz mass-shell vectors: their spatial Fourier transforms decrease rapidly, and their time support is bounded. Then

R_{g, f} - B_{f}^{0} = Q \otimes Φ (v_{e} - v_{e}^{0}) + Π \otimes Φ (v_{r} - v_{r}^{0}) .

On total-number sectors at most $M$ , its norm is bounded by

2 (M + 2) (v_{e} - v_{e}^{0} + v_{r} - v_{r}^{0}) .

It is a switching and initial-data remainder, not an independent renormalization ambiguity.

Displayed formulas in this passage

unlabeled · line 1288
$v_{a}^{0} = K (- 1_{[0, b]} (t) \partial_{i} (a^{ij} \partial_{j} u_{f})) .$
unlabeled · line 1296
$R_{g, f} - B_{f}^{0} = Q \otimes Φ (v_{e} - v_{e}^{0}) + Π \otimes Φ (v_{r} - v_{r}^{0}) .$
unlabeled · line 1301
$2 (M + 2) (v_{e} - v_{e}^{0} + v_{r} - v_{r}^{0}) .$

One metric oscillator and the switched identity

Displayed formulas in this passage

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