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

Theorem: Supported fixed-profile expansion

Frozen family and principal statements · lines 305–339 · proof included or linked

Em01-G@5#thm:supported, Em01-G@5#eq:supported-expansion, Em01-G@5#eq:supported-remainder

There are distributions

A_{- 3}^{supp}, A_{- 2}^{supp}, A_{- 1}^{supp}, A_{l o g}^{supp}, T_{supp} \in D^{'} (U; S)

such that

t_{δ, supp}^{app} = λ^{3} ℏ^{2} [δ^{- 3} A_{- 3}^{supp} + δ^{- 2} A_{- 2}^{supp} + δ^{- 1} A_{- 1}^{supp} + (lo g δ) A_{l o g}^{supp} + T_{supp}] + R_{δ}^{supp},

and

∣ R_{δ}^{supp} (Φ) ∣ \leq C_{L} ∣ λ ∣^{3} ℏ^{2} δ (1 + ∣ lo g δ ∣) ∥Φ ∥_{40, L}, supp Φ \subset L .

The four singular coefficients are supported on $N_{mark}$ . They are constructed by (eq:scalar-coefficients) and (eq:supported-coefficients). The inverse-cubic coefficient is (eq:supported-leading).

Displayed formulas in this passage

Em01-G@5#eq:supported-expansion · line 315
$t_{δ, supp}^{app} = λ^{3} ℏ^{2} [δ^{- 3} A_{- 3}^{supp} + δ^{- 2} A_{- 2}^{supp} + δ^{- 1} A_{- 1}^{supp} + (lo g δ) A_{l o g}^{supp} + T_{supp}] + R_{δ}^{supp},$
Em01-G@5#eq:supported-remainder · line 328
$∣ R_{δ}^{supp} (Φ) ∣ \leq C_{L} ∣ λ ∣^{3} ℏ^{2} δ (1 + ∣ lo g δ ∣) ∥Φ ∥_{40, L}, supp Φ \subset L .$
unlabeled · line 308
$A_{- 3}^{supp}, A_{- 2}^{supp}, A_{- 1}^{supp}, A_{l o g}^{supp}, T_{supp} \in D^{'} (U; S)$

Theorem: Supported fixed-profile expansion

Displayed formulas in this passage

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