openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralAngularParameter.lean
| 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 | |