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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 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralRegularPhysical
namespace OAI
/-! Restoring the angular power in a forced regular equation. -/
namespace DefocusingNLS
local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ)
theorem spectralRegularPhysical_source_hasDerivAt (ell m : ℕ) (νp νm Q F G : ℂ)
(Y : ℝ → E₄) (r : ℝ) (hr : 0 < r)
(hY : HasDerivAt Y
(spectralRegularField (2 * ell + 11)
(spectralDiagonalCoefficient m Q) (spectralCrossCoefficient m Q)
(-Complex.I * νp / 2 + Complex.I * (ell : ℂ) / 2)
(Complex.I * νm / 2 - Complex.I * (ell : ℂ) / 2) r (Y r) +
((0, F), (0, G))) r) :
HasDerivAt (spectralAngularPair ell Y)
(spectralPhysicalCircularField νp νm ((ell * (ell + 10) : ℕ) : ℂ) m Q r
(spectralAngularPair ell Y r) +
((0, (r : ℂ) ^ ell * F), (0, (r : ℂ) ^ ell * G))) r := by
let cp := -Complex.I * νp / 2 + Complex.I * (ell : ℂ) / 2
let cm := Complex.I * νm / 2 - Complex.I * (ell : ℂ) / 2
let Fp := spectralDiagonalCoefficient m Q * (Y r).1.1 +
spectralCrossCoefficient m Q * (Y r).2.1 + F
let Fm := star (spectralCrossCoefficient m Q) * (Y r).1.1 +
star (spectralDiagonalCoefficient m Q) * (Y r).2.1 + G
have hp : HasDerivAt (fun t => (Y t).1) ((Y r).1.2,
-(((2 * ell + 11 : ℕ) : ℂ) / (r : ℂ) + (1 : ℂ) * Complex.I * (r : ℂ) / 2) *
(Y r).1.2 - cp * (Y r).1.1 + Fp) r := by
apply ((ContinuousLinearMap.fst ℝ (ℂ × ℂ) (ℂ × ℂ)).hasFDerivAt.comp_hasDerivAt r hY).congr_deriv
simp only [ContinuousLinearMap.coe_fst']
apply Prod.ext
· simp [spectralRegularField]
· dsimp only [spectralRegularField, cp, Fp, Prod.fst_add, Prod.snd_add]
ring
have hm : HasDerivAt (fun t => (Y t).2) ((Y r).2.2,
-(((2 * ell + 11 : ℕ) : ℂ) / (r : ℂ) + (-1 : ℂ) * Complex.I * (r : ℂ) / 2) *
(Y r).2.2 - cm * (Y r).2.1 + Fm) r := by
apply ((ContinuousLinearMap.snd ℝ (ℂ × ℂ) (ℂ × ℂ)).hasFDerivAt.comp_hasDerivAt r hY).congr_deriv
simp only [ContinuousLinearMap.coe_snd']
apply Prod.ext
· simp [spectralRegularField]
· dsimp only [spectralRegularField, cm, Fm, Prod.fst_add, Prod.snd_add]
ring
have hdp := spectralAngularJet_hasDerivAt ell 1 cp Fp (fun t => (Y t).1) r hr.ne' hp
have hdm := spectralAngularJet_hasDerivAt ell (-1) cm Fm (fun t => (Y t).2) r hr.ne' hm
apply (hdp.prodMk hdm).congr_deriv
apply Prod.ext <;> apply Prod.ext
· simp only [Prod.fst_add, add_zero]
rfl
· dsimp only [spectralAngularPair, spectralAngularJet, spectralPhysicalCircularField,
cp, Fp, Prod.fst_add, Prod.snd_add]
ring
· simp only [Prod.snd_add, Prod.fst_add, add_zero]
rfl
· dsimp only [spectralAngularPair, spectralAngularJet, spectralPhysicalCircularField,
cm, Fm, Prod.fst_add, Prod.snd_add]
ring
end DefocusingNLS
end OAI
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