Manuscript Em03-F@1 · Em03-F-v1.tex · theorem · 0d4d148b

Theorem: Nonzero compact-source packet coupling

A compact exterior profile with a proved outgoing signal · lines 1300–1341 · proof included or linked

Em03-F@1#thm:nonzero, Em03-F@1#eq:workedgap

The explicitly specified compact profile and packet satisfy $c_{h} \neq = 0$ . For $ϑ > 0$ , choose

l = \frac{ϑ}{2} e^{i (a r g c_{h} + π /2)} v_{h} .

Then

σ (k_{F}, l) = 2 ϑ ∣ c_{h} ∣ > 0, ∥ l ∥_{U}^{2} = ϑ^{2} (ν_{h} + 1/2) .

Consequently

Δ_{s i n} = e^{- ϑ^{2} (2 ν_{h} + 1) /4} ∣ sin (2 g ϑ ∣ c_{h} ∣) ∣, δ ν_{h} = g^{2} ∣ c_{h} ∣^{2} .

For $0 < ∣ g ∣ 2 ϑ ∣ c_{h} ∣ \leq π /2$ , the gap is strictly positive.

Proof. The preceding bump estimates prove $d_{0} \neq = 0$ , and the band estimate proves $c_{h} \neq = 0$ . Since

⟨ k_{F}, l ⟩ = \frac{ϑ}{2} e^{i (a r g c_{h} + π /2)} c_{h},

its imaginary part is $ϑ ∣ c_{h} ∣/ 2$ . Scattering unitarity and the baseline covariance give the stated norm. Apply Theorems thm:witness and thm:counts.

Displayed formulas in this passage

Em03-F@1#eq:workedgap · line 1315
$Δ_{s i n} = e^{- ϑ^{2} (2 ν_{h} + 1) /4} ∣ sin (2 g ϑ ∣ c_{h} ∣) ∣, δ ν_{h} = g^{2} ∣ c_{h} ∣^{2} .$
unlabeled · line 1305
$l = \frac{ϑ}{2} e^{i (a r g c_{h} + π /2)} v_{h} .$
unlabeled · line 1310
$σ (k_{F}, l) = 2 ϑ ∣ c_{h} ∣ > 0, ∥ l ∥_{U}^{2} = ϑ^{2} (ν_{h} + 1/2) .$
unlabeled · line 1333
$⟨ k_{F}, l ⟩ = \frac{ϑ}{2} e^{i (a r g c_{h} + π /2)} c_{h},$

Theorem: Nonzero compact-source packet coupling

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