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fac6777 | 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 49 50 51 52 | import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffCoordinateOperator
namespace OAI
/-! # The actual contour coordinates have vanishing torus transfer defect -/
open Set
open scoped SchwartzMap ContDiff NNReal
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "Radius" => {L : ℝ // 1 ≤ L}
theorem sampledCutoffProfile_contour_coordinate_operator (a b k T Q C₀ : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 10 < k) (hT : 0 ≤ T)
(hQ : 0 ≤ Q) (hC₀ : 0 ≤ C₀) (m : ℕ)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(hχzero : ∀ y : E, 1 ≤ ‖y‖ → χ y = 0)
(hχone : ∀ y : E, ‖y‖ ≤ 1 / 2 → χ y = 1)
(Qp : E → ℂ) (hQp : ContDiff ℝ ∞ Qp) (Q₀ : HomogeneousY a k)
(hQ₀ : ∀ y, homogeneousPhysicalCLM a k ha ha1 (by linarith) Q₀ y = Qp y)
(hqb : ∀ L : Radius, ‖cutoffProfileCoefficient a k ha1 (by linarith) χ hχ Qp hQp L‖ ≤ Q)
(hCb : ∀ (L : ℝ) (hL : 1 ≤ L) (f : FourierL2),
‖homogeneousLocalizationCLM a k L ha ha1 (by linarith) hL χ f‖ ≤ C₀ * ‖f‖)
(P : (HomogeneousY a k × HomogeneousY a k) →L[ℂ] (HomogeneousY a k × HomogeneousY a k))
(hfin : FiniteDimensional ℂ P.range)
(hcomm : ∀ t, Commute (homogeneousComplexLinearizedStep a b k ha ha1 (by linarith) m Q₀ t) P) :
let π := homogeneousStableCoordinates a k ha ha1 (by linarith) P
let A := (projectionSemigroupRestriction
(homogeneousComplexLinearizedStep a b k ha ha1 (by linarith) m Q₀)
P hcomm ⟨T, hT⟩).restrictScalars ℝ
∀ ε : ℝ, 0 < ε → ∃ L₀ : ℝ, ∀ L : Radius, L₀ ≤ L.1 →
‖(expandingCoordinates a k ha ha1 (by linarith) χ π
(expandingRadiusCurve L.1 T L.2 ⟨T, hT, le_rfl⟩)).comp
(expandingProfileEndpoint a b k L.1 T ha ha1 (by linarith) L.2 hT m Q hQ
(sampledCutoffProfilePath a k L.1 T ha ha1 (by linarith) L.2 χ hχ Qp hQp)
(fun t => hqb (expandingRadiusCurve L.1 T L.2 t))) -
A.comp (expandingCoordinates a k ha ha1 (by linarith) χ π L)‖ ≤ ε := by
intro π A
let : FiniteDimensional ℂ P.range := hfin
let : FiniteDimensional ℝ P.range := FiniteDimensional.trans ℝ ℂ P.range
apply sampledCutoffProfile_coordinate_operator_norm a b k T Q C₀ ha ha1 hk hT hQ hC₀
m χ hχ hχzero hχone Qp hQp Q₀ hQ₀ hqb hCb π A
intro u
exact homogeneousStableCoordinates_slab a b k T ha ha1 (by linarith) hT m Q₀ P hcomm
⟨T, hT, le_rfl⟩ u
end DefocusingNLS
end OAI
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