Manuscript Em01-G@5 · Em01-G-v5.tex · section · 3fbfe951

Supported output-plateau inverse-quadratic inventory

Supported output-plateau inverse-quadratic inventory · lines 2474–2517

Em01-G@5#eq:contact-operator

This appendix preserves the precise unfinished supported calculation. It is not needed for the complete-family certificate.

Put

S = B_{2} - B_{1}, E = A_{2} - A_{1}, κ = \frac{4 α d}{β}, Z_{0} = \frac{d Z _{1} + α Z _{2}}{β},

and write $⟨ P ⟩_{ρ} = \int ρ_{n} P d H$ . In addition to $M_{2}$ , define

B_{3} = ⟨ B_{2} (S - i 0)^{- 3} ⟩_{ρ}, Z_{0, 3}^{A} = ⟨ Z_{0}^{A} (S - i 0)^{- 3} ⟩_{ρ}, E_{3} = ⟨ E (S - i 0)^{- 3} ⟩_{ρ},

D_{3}^{A} = ⟨(Z_{1}^{A} - Z_{2}^{A}) (S - i 0)^{- 3} ⟩_{ρ}, Q_{4} = ⟨ ∣ Z_{1} - Z_{2} ∣^{2} (S - i 0)^{- 4} ⟩_{ρ} .

For an amplitude $A$ , evaluate its displayed jets at zero and define

L_{μ} [A] = ℓ_{μ} [(\frac{3}{2} M_{2} - B_{3}) \partial_{v} A - Z_{0, 3}^{C} \partial_{ξ^{C}} A + \frac{κ M _{2}}{2} (\frac{1}{4} Δ_{ξ} A - \frac{1}{4 d} \partial_{v} A - \frac{m ^{2} β}{4 α} A (0)) + (\frac{E _{3}}{β} - \frac{3 Q _{4}}{4 β}) A (0)] - \frac{α M _{2}}{β} (E_{C})_{μ} \partial_{ξ^{C}} A + \frac{D _{3} ^{C}}{β} (E_{C})_{μ} A (0) - \frac{M _{2}}{2 β} k_{μ} A (0) .

Here $C = 1, 2$ is summed.

Displayed formulas in this passage

Em01-G@5#eq:contact-operator · line 2498
$L_{μ} [A] = ℓ_{μ} [(\frac{3}{2} M_{2} - B_{3}) \partial_{v} A - Z_{0, 3}^{C} \partial_{ξ^{C}} A + \frac{κ M _{2}}{2} (\frac{1}{4} Δ_{ξ} A - \frac{1}{4 d} \partial_{v} A - \frac{m ^{2} β}{4 α} A (0)) + (\frac{E _{3}}{β} - \frac{3 Q _{4}}{4 β}) A (0)] - \frac{α M _{2}}{β} (E_{C})_{μ} \partial_{ξ^{C}} A + \frac{D _{3} ^{C}}{β} (E_{C})_{μ} A (0) - \frac{M _{2}}{2 β} k_{μ} A (0) .$
unlabeled · line 2481
$S = B_{2} - B_{1}, E = A_{2} - A_{1}, κ = \frac{4 α d}{β}, Z_{0} = \frac{d Z _{1} + α Z _{2}}{β},$
unlabeled · line 2488
$B_{3} = ⟨ B_{2} (S - i 0)^{- 3} ⟩_{ρ}, Z_{0, 3}^{A} = ⟨ Z_{0}^{A} (S - i 0)^{- 3} ⟩_{ρ}, E_{3} = ⟨ E (S - i 0)^{- 3} ⟩_{ρ},$
unlabeled · line 2493
$D_{3}^{A} = ⟨(Z_{1}^{A} - Z_{2}^{A}) (S - i 0)^{- 3} ⟩_{ρ}, Q_{4} = ⟨ ∣ Z_{1} - Z_{2} ∣^{2} (S - i 0)^{- 4} ⟩_{ρ} .$