openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Nonlinear /CutoffContourTransfer.lean
| 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 | |