File size: 2,722 Bytes
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 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralRegularPhysical
import Mathlib.LinearAlgebra.LinearIndependent.Lemmas
namespace OAI
/-! Restoring an angular factor preserves the rank of the regular columns away from zero. -/
namespace DefocusingNLS
local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ)
noncomputable def spectralAngularState (ell : ℕ) (r : ℝ) (Z : E₄) : E₄ :=
(((r : ℂ)^ell*Z.1.1,(r : ℂ)^ell*((ell : ℂ)/(r : ℂ)*Z.1.1+Z.1.2)),
((r : ℂ)^ell*Z.2.1,(r : ℂ)^ell*((ell : ℂ)/(r : ℂ)*Z.2.1+Z.2.2)))
noncomputable def spectralAngularStateLinear (ell : ℕ) (r : ℝ) : E₄ →ₗ[ℂ] E₄ where
toFun := spectralAngularState ell r
map_add' := by
intro Z W
apply Prod.ext <;> apply Prod.ext <;>
simp only [spectralAngularState,Prod.fst_add,Prod.snd_add]
all_goals ring
map_smul' := by
intro a Z
apply Prod.ext <;> apply Prod.ext <;>
simp only [spectralAngularState,Prod.smul_fst,Prod.smul_snd,smul_eq_mul,RingHom.id_apply]
all_goals ring
theorem spectralAngularState_injective (ell : ℕ) (r : ℝ) (hr : r ≠ 0) :
Function.Injective (spectralAngularStateLinear ell r) := by
intro Z W he
have hp : (r : ℂ)^ell ≠ 0 := pow_ne_zero _ (Complex.ofReal_ne_zero.mpr hr)
have h11 := congrArg (fun z : E₄ => z.1.1) he
have h12 := congrArg (fun z : E₄ => z.1.2) he
have h21 := congrArg (fun z : E₄ => z.2.1) he
have h22 := congrArg (fun z : E₄ => z.2.2) he
change (r : ℂ)^ell*Z.1.1=(r : ℂ)^ell*W.1.1 at h11
change (r : ℂ)^ell*Z.2.1=(r : ℂ)^ell*W.2.1 at h21
have hZ11 := mul_left_cancel₀ hp h11
have hZ21 := mul_left_cancel₀ hp h21
change (r : ℂ)^ell*((ell : ℂ)/(r : ℂ)*Z.1.1+Z.1.2)=
(r : ℂ)^ell*((ell : ℂ)/(r : ℂ)*W.1.1+W.1.2) at h12
change (r : ℂ)^ell*((ell : ℂ)/(r : ℂ)*Z.2.1+Z.2.2)=
(r : ℂ)^ell*((ell : ℂ)/(r : ℂ)*W.2.1+W.2.2) at h22
have hZ12 := mul_left_cancel₀ hp h12
have hZ22 := mul_left_cancel₀ hp h22
rw [hZ11] at hZ12
rw [hZ21] at hZ22
exact Prod.ext (Prod.ext hZ11 (add_left_cancel hZ12))
(Prod.ext hZ21 (add_left_cancel hZ22))
theorem spectralAngularPair_rank (ell : ℕ) (Yp Ym : ℝ → E₄)
(r : ℝ) (hr : r ≠ 0) (hY : LinearIndependent ℂ ![Yp r,Ym r]) :
LinearIndependent ℂ ![spectralAngularPair ell Yp r,spectralAngularPair ell Ym r] := by
have h := hY.map' (spectralAngularStateLinear ell r)
(LinearMap.ker_eq_bot.mpr (spectralAngularState_injective ell r hr))
have he : spectralAngularStateLinear ell r ∘ ![Yp r,Ym r]=
![spectralAngularPair ell Yp r,spectralAngularPair ell Ym r] := by
funext i
fin_cases i <;> rfl
rw [he] at h
exact h
end DefocusingNLS
end OAI
|