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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 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralCanonicalParameterPower
namespace OAI
/-! Smoothness of the physical parameter values on a sufficiently large exterior interval. -/
open Set Filter Topology
open scoped ContDiff
namespace DefocusingNLS
local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ)
theorem canonicalPhysicalParameterValues_smooth_tail (ν η b : ℂ) (n : ℕ) (hn : 1≤n)
(L : ℝ) (hX : HasRadialExterior ν n b L) (hb : b ≠ 0)
(c : ℂ × ℂ) (Y : ℂ → ℝ → E₄)
(hY : IsCanonicalHolomorphicColumn ν η b n L c Y) (z : ℂ) :
∃ R : ℝ, 1<R ∧ ContDiffOn ℝ ∞ (canonicalPhysicalParameterValues ν Y z) (Ioi R) := by
let f := fun t => (circularParameterShear t (Y z t) (deriv (fun lam => Y lam t) z)).1.1
let g := fun t => (circularParameterShear t (Y z t) (deriv (fun lam => Y lam t) z)).2.1
have hj := canonical_parameterShear_value_logJets ν η b n hn L hX hb c Y hY z 1 zero_lt_one
obtain ⟨Lf,hfs⟩ := hj.1.smooth
obtain ⟨Lg,hgs⟩ := hj.2.smooth
let T := max Lf Lg
let R := max (Real.exp T) 1+1
have hR : 1<R := by dsimp only [R]; linarith [le_max_right (Real.exp T) 1]
have hER : Real.exp T≤R := by dsimp only [R]; linarith [le_max_left (Real.exp T) 1]
have hp := spectralPowerValues_smooth (ν-2*z) (star ν-2*z) f g T
(hfs.mono (Ioi_subset_Ioi (le_max_left _ _)))
(hgs.mono (Ioi_subset_Ioi (le_max_right _ _)))
refine ⟨R,hR,(hp.mono (Ioi_subset_Ioi hER)).congr ?_⟩
intro r hr
exact canonicalPhysicalParameterValues_eq ν η b n L c Y hY z r (hR.trans hr)
end DefocusingNLS
end OAI
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