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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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralAngularSource
import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralRegularParameterEquation
namespace OAI
/-! The differentiated regular physical columns, with the angular factor
restored, satisfy the same source system as a radial Jordan chain. -/
open Set
open scoped BoundedContinuousFunction
namespace DefocusingNLS
local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ)
theorem spectralAngularPair_hasParameterDerivAt (ell : ℕ)
(Y : ℂ → ℝ → E₄) (D : ℝ → E₄) (z : ℂ) (r : ℝ)
(hY : HasDerivAt (fun lam => Y lam r) (D r) z) :
HasDerivAt (fun lam => spectralAngularPair ell (Y lam) r)
(spectralAngularPair ell D r) z := by
have hp := (ContinuousLinearMap.fst ℂ (ℂ × ℂ) (ℂ × ℂ)).hasFDerivAt.comp_hasDerivAt z hY
have hm := (ContinuousLinearMap.snd ℂ (ℂ × ℂ) (ℂ × ℂ)).hasFDerivAt.comp_hasDerivAt z hY
have hp0 := (ContinuousLinearMap.fst ℂ ℂ ℂ).hasFDerivAt.comp_hasDerivAt z hp
have hp1 := (ContinuousLinearMap.snd ℂ ℂ ℂ).hasFDerivAt.comp_hasDerivAt z hp
have hm0 := (ContinuousLinearMap.fst ℂ ℂ ℂ).hasFDerivAt.comp_hasDerivAt z hm
have hm1 := (ContinuousLinearMap.snd ℂ ℂ ℂ).hasFDerivAt.comp_hasDerivAt z hm
have h0 := hp0.const_mul ((r : ℂ) ^ ell)
have h1 := ((hp0.const_mul ((ell : ℂ) / (r : ℂ))).add hp1).const_mul ((r : ℂ) ^ ell)
have h2 := hm0.const_mul ((r : ℂ) ^ ell)
have h3 := ((hm0.const_mul ((ell : ℂ) / (r : ℂ))).add hm1).const_mul ((r : ℂ) ^ ell)
convert! (h0.prodMk h1).prodMk (h2.prodMk h3) using 1
theorem spectralRegularParameterPhysical_hasDerivAt (ell m : ℕ) (νp νm Q : ℂ)
(R α : ℝ) (hR : 0 ≤ R) (hα : 0 < α) (A B : ℝ →ᵇ ℂ)
(c : ℂ × ℂ) (z : ℂ)
(hgap : spectralRegularSourceBound A B
(-Complex.I * νp / 2 + Complex.I * (ell : ℂ) / 2 + Complex.I * z)
(Complex.I * νm / 2 - Complex.I * (ell : ℂ) / 2 - Complex.I * z) < 2 * α)
(r : ℝ) (hr : r ∈ Ioc 0 R)
(hA : A r = spectralDiagonalCoefficient m Q)
(hB : B r = spectralCrossCoefficient m Q) :
let s := spectralRegularSolutionSource (2 * ell + 11) R α hR hα A B
(-Complex.I * νp / 2 + Complex.I * (ell : ℂ) / 2)
(Complex.I * νm / 2 - Complex.I * (ell : ℂ) / 2) c
let W := spectralRegularState (2 * ell + 11) α c (s z)
let V := spectralRegularState (2 * ell + 11) α 0 (deriv s z)
HasDerivAt (spectralAngularPair ell V)
(spectralPhysicalCircularField (νp - 2 * z) (νm - 2 * z)
((ell * (ell + 10) : ℕ) : ℂ) m Q r (spectralAngularPair ell V r) +
((0, -Complex.I * (spectralAngularPair ell W r).1.1),
(0, Complex.I * (spectralAngularPair ell W r).2.1))) 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 s := spectralRegularSolutionSource (2 * ell + 11) R α hR hα A B cp cm c
let W := spectralRegularState (2 * ell + 11) α c (s z)
let V := spectralRegularState (2 * ell + 11) α 0 (deriv s z)
change HasDerivAt (spectralAngularPair ell V)
(spectralPhysicalCircularField (νp - 2 * z) (νm - 2 * z)
((ell * (ell + 10) : ℕ) : ℂ) m Q r (spectralAngularPair ell V r) +
((0, -Complex.I * (spectralAngularPair ell W r).1.1),
(0, Complex.I * (spectralAngularPair ell W r).2.1))) r
have hv := spectralRegularParameterState_hasDerivAt (2 * ell + 11) R α hR hα A B
cp cm c z hgap r hr
have hp : cp + Complex.I * z = -Complex.I * (νp - 2 * z) / 2 +
Complex.I * (ell : ℂ) / 2 := by dsimp only [cp]; ring
have hm : cm - Complex.I * z = Complex.I * (νm - 2 * z) / 2 -
Complex.I * (ell : ℂ) / 2 := by dsimp only [cm]; ring
dsimp only at hv
rw [hp, hm, hA, hB] at hv
have hh := spectralRegularPhysical_source_hasDerivAt ell m (νp - 2 * z) (νm - 2 * z)
Q (-Complex.I * (W r).1.1) (Complex.I * (W r).2.1) V r hr.1 hv
apply hh.congr_deriv
apply Prod.ext <;> apply Prod.ext
· rfl
· dsimp only [spectralAngularPair, spectralAngularJet, Prod.fst_add, Prod.snd_add]
ring
· rfl
· dsimp only [spectralAngularPair, spectralAngularJet, Prod.fst_add, Prod.snd_add]
ring
end DefocusingNLS
end OAI
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