openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralCanonicalPhysicalParameter.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralPhysicalParameterEquation | |
| namespace OAI | |
| /-! Canonical physical outgoing columns satisfy the differentiated equation | |
| used in matching generalized eigenvectors. -/ | |
| open Filter Topology | |
| namespace DefocusingNLS | |
| local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ) | |
| theorem canonical_physical_parameter_equation (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) (hr : 1 < r) | |
| (hscale : (Complex.exp (ν * (Real.log r : ℂ)) * | |
| star (Complex.exp (ν * (Real.log r : ℂ)))) ^ n = 1 / (r : ℂ) ^ 2) : | |
| let V := fun lam => spectralPhysicalPair (ν - 2 * lam) (star ν - 2 * lam) (Y lam) | |
| HasDerivAt (fun s => deriv (fun lam => V lam s) z) | |
| (spectralPhysicalCircularField (ν - 2 * z) (star ν - 2 * z) η n | |
| (Complex.exp (ν * (Real.log r : ℂ)) * (radialExteriorCanonical ν n b L (Real.log r)).1) r | |
| (deriv (fun lam => V lam r) z) + | |
| ((0, -Complex.I * (V z r).1.1), (0, Complex.I * (V z r).2.1))) r := by | |
| let V := fun lam => spectralPhysicalPair (ν - 2 * lam) (star ν - 2 * lam) (Y lam) | |
| let D := fun t => deriv (fun lam => Y lam t) z | |
| let W := spectralPhysicalPair (ν - 2 * z) (star ν - 2 * z) | |
| (fun t => circularParameterShear t (Y z t) (D t)) | |
| have he (s : ℝ) (hs : 1 < s) : deriv (fun lam => V lam s) z = W s := by | |
| have hp := spectralPhysicalPair_hasParameterDerivAt ν (star ν) z Y D s | |
| (hY.2.2.1 z (Real.log s) (Real.log_pos hs).le).differentiableAt.hasDerivAt | |
| exact hp.deriv | |
| have hd := spectralPhysicalShear_hasDerivAt ν z η n hn | |
| (fun t => (radialExteriorCanonical ν n b L t).1) (Y z) D r | |
| (lt_trans zero_lt_one hr) hscale (hY.1 z (Real.log r) (Real.log_pos hr).le) | |
| (canonical_circular_parameter_equation ν η b n hn L hX hb c Y hY z | |
| (Real.log r) (Real.log_pos hr)) | |
| have htime : (fun s => deriv (fun lam => V lam s) z) =ᶠ[𝓝 r] W := by | |
| filter_upwards [eventually_gt_nhds hr] with s hs | |
| exact he s hs | |
| have hnew := hd.congr_of_eventuallyEq htime | |
| change HasDerivAt (fun s => deriv (fun lam => V lam s) z) | |
| (spectralPhysicalCircularField (ν - 2 * z) (star ν - 2 * z) η n | |
| (Complex.exp (ν * (Real.log r : ℂ)) * (radialExteriorCanonical ν n b L (Real.log r)).1) r | |
| (deriv (fun lam => V lam r) z) + | |
| ((0, -Complex.I * (V z r).1.1), (0, Complex.I * (V z r).2.1))) r | |
| rw [he r hr] | |
| exact hnew | |
| end DefocusingNLS | |
| end OAI | |