Manuscript VII-A@1 · VII-A.tex · section · 11da3fe6

Classical comparison with fixed Cauchy data

Support and comparison of switches › Classical comparison with fixed Cauchy data · lines 1659–1686

VII-A@1#eq:formalGreen

For a fixed metric jet, abbreviate

D_{i} = D_{g_{i}} [g] .

The formal zero-data inverse of $D_{i}$ through second order is

G_{i} = E_{m} - κ E_{m} V_{i} E_{m} + κ^{2} (E_{m} V_{i} E_{m} V_{i} E_{m} - E_{m} W_{i} E_{m}) + O (κ^{3}) .

Multiplication verifies $D_{i} G_{i} = id (mod κ^{3})$ on smooth sources. All positive-order corrections have zero Cauchy data.

Let $Sol_{i}$ denote formal scalar solutions of $D_{i} ψ = 0 (mod κ^{3})$ with arbitrary prescribed formal Cauchy data.

Displayed formulas in this passage

VII-A@1#eq:formalGreen · line 1666
$G_{i} = E_{m} - κ E_{m} V_{i} E_{m} + κ^{2} (E_{m} V_{i} E_{m} V_{i} E_{m} - E_{m} W_{i} E_{m}) + O (κ^{3}) .$
unlabeled · line 1662
$D_{i} = D_{g_{i}} [g] .$