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d5b562f | 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 53 54 55 56 57 58 59 60 | import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.ContinuousTorusAssembly
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.DiagonalRealCoordinates
namespace OAI
/-! # Continuous normalized linear data in the real diagonal symmetry coordinates -/
open scoped SchwartzMap NNReal
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "Radius" => {L : ℝ // 1 ≤ L}
def HasContinuousFiniteDiagonalTorusLinearSteps {H V : Type*}
[NormedAddCommGroup H] [NormedSpace ℝ H]
[NormedAddCommGroup V] [NormedSpace ℂ V] [FiniteDimensional ℂ V]
(π : H →L[ℝ] V) (A : ℝ≥0 → Radius → FourierL2 →L[ℝ] FourierL2) : Prop :=
∃ G : V →L[ℂ] V,
∃ hspan : (⨆ lam : ℂ, Module.End.eigenspace G.toLinearMap lam) = ⊤,
∃ hspec : ∀ (lam : ℂ) (v : V), v ≠ 0 → G v = lam • v →
lam = 0 ∨ lam = 1 ∨ lam = 1 / 2,
∃ T₀ : ℝ, 0 ≤ T₀ ∧ ∀ T : ℝ≥0, T₀ ≤ T →
HasContinuousTorusLinearStep (F := (diagonalRealCoordinates π G hspan hspec).range) T (A T)
theorem continuous_finiteDiagonalTorusLinearSteps_of_transfer {V : Type*}
[NormedAddCommGroup V] [NormedSpace ℂ V] [FiniteDimensional ℂ V]
(a k : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (χ : 𝓢(E, ℂ))
(ρ : ℝ) (hρ : 0 < ρ) (hχ : ∀ y : E, ‖y‖ ≤ ρ → χ y = 1)
(π : HomogeneousY a k →L[ℝ] V) (G : V →L[ℂ] V)
(hspan : (⨆ lam : ℂ, Module.End.eigenspace G.toLinearMap lam) = ⊤)
(hspec : ∀ (lam : ℂ) (v : V), v ≠ 0 → G v = lam • v →
lam = 0 ∨ lam = 1 ∨ lam = 1 / 2)
(A : ℝ≥0 → Radius → FourierL2 →L[ℝ] FourierL2)
(hkernel : ∀ β : ℝ, 0 < β → ∃ T₀ : ℝ, 0 ≤ T₀ ∧
∀ T : ℝ≥0, T₀ ≤ T → ∃ L₀ : ℝ, ∀ L : Radius, L₀ ≤ L.1 →
∀ f : FourierL2, ‖f‖ ≤ 1 →
π (homogeneousLocalizationCLM a k L.1 ha ha1 hk L.2 χ f) = 0 → ‖A T L f‖ < β)
(hcoord : ∀ T, HasContractingTorusCoordinateTransfer a k ha ha1 hk χ
(diagonalRealCoordinates π G hspan hspec) T (A T))
(hA : ∀ T, ∃ C : ℝ, 0 ≤ C ∧ ∀ L, ‖A T L‖ ≤ C) :
HasContinuousFiniteDiagonalTorusLinearSteps π A := by
let : FiniteDimensional ℝ (SymmetryCoordinates G) :=
FiniteDimensional.trans ℝ ℂ (SymmetryCoordinates G)
refine ⟨G, hspan, hspec, ?_⟩
apply continuous_torusLinearSteps_of_transfer a k ha ha1 hk χ ρ hρ hχ
(diagonalRealCoordinates π G hspan hspec) A
· intro β hβ
obtain ⟨T₀, hT₀, hT⟩ := hkernel β hβ
refine ⟨T₀, hT₀, fun T hTT => ?_⟩
obtain ⟨L₀, hL₀⟩ := hT T hTT
refine ⟨L₀, fun L hL f hf hz => ?_⟩
exact hL₀ L hL f hf ((diagonalRealCoordinates_zero_iff π G hspan hspec _).mp hz)
· exact hcoord
· exact hA
end DefocusingNLS
end OAI
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