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ef96466 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralCanonicalFreeFirstLimit
import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralCircularTailLimit
namespace OAI
/-! Convergence of the canonical first column at a fixed exterior endpoint. -/
open Filter Topology Set Polynomial
namespace DefocusingNLS
local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ)
theorem canonical_circular_first_H_fixed_limit
(ν m νp νm : ℕ → ℂ) (ν₀ m₀ νm₀ q : ℂ) (ell : ℕ)
(hν : Tendsto ν atTop (𝓝 ν₀)) (hm : Tendsto m atTop (𝓝 m₀))
(hp : Tendsto νp atTop (𝓝 ((ell : ℂ)-2*q)))
(hn : Tendsto νm atTop (𝓝 νm₀)) (hq : -1 < q.re)
(δ L : ℝ) (hδ : 0 < δ) (hsmall : ‖m₀‖+2*δ < 1)
(hX : ∀ᶠ n in atTop, HasRadialExterior (ν n) n (m n) L)
(W : ℕ → ℝ → E₄)
(hW : ∀ᶠ n in atTop, ∀ t, 0 ≤ t → HasDerivAt (W n)
(circularLeadingField t (W n t)+circularBoundedField (νp n) (νm n)
((ell*(ell+10) : ℕ) : ℂ) n (radialExteriorCanonical (ν n) n (m n) L t).1
(W n t)) t)
(heW : ∀ᶠ n in atTop, ∀ J : ℕ, ∃ k : ℕ, J ≤ k ∧ ∃ u : CircularTailSpace,
∀ t, 0 ≤ t → W n t=circularPolynomialJet
(spectralOutgoingPolynomial (νp n) (νm n) ((ell*(ell+10) : ℕ) : ℂ) n
(radialExteriorExpansion (ν n) n (m n) k) (1,0) k) t+
circularUnweight (2*(k : ℝ)) u t)
(A : ℝ) (hA : 0 ≤ A)
(ε : ℕ → ℝ) (hε : ∀ n, 0 ≤ ε n) (hεlim : Tendsto ε atTop (𝓝 0))
(hc : ∀ᶠ n in atTop, ∀ t ∈ Ici A,
‖spectralDiagonalCoefficient n (radialExteriorCanonical (ν n) n (m n) L t).1‖+
‖spectralCrossCoefficient n (radialExteriorCanonical (ν n) n (m n) L t).1‖ ≤ ε n) :
TendstoUniformlyOn W (spectralFreeFirstColumn ell q) atTop (Ici A) := by
obtain ⟨T,_hT,_hLT,hlim⟩ := canonical_circular_first_H_limit
ν m νp νm ν₀ m₀ νm₀ q ell hν hm hp hn hq δ L hδ hsmall hX W hW heW
apply circularSolutions_tail_uniform_limit νp νm
((ell : ℂ)-2*q) νm₀ ((ell*(ell+10) : ℕ) : ℂ)
hp hn id (eventually_ge_atTop 1)
(fun n t => (radialExteriorCanonical (ν n) n (m n) L t).1)
ε hε hεlim A T W (spectralFreeFirstColumn ell q) _ _ hc hlim
· exact hW.mono (fun n h t ht => h t (hA.trans ht))
· exact fun t _ => spectralFreeFirstColumn_hasDerivAt ell q νm₀ hq t
end DefocusingNLS
end OAI
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