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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 83 84 85 86 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralResolventAnalytic
namespace OAI
/-! Holomorphic forcing gives the actual bounded spectral correction, with its equation and estimate. -/
open scoped BoundedContinuousFunction
namespace DefocusingNLS
local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ)
local notation "End₄" => CircularTailSpace →L[ℂ] CircularTailSpace
noncomputable def circularResolvedCorrection (κ : ℝ) (hκ : 0 < κ)
(νp νm η : ℂ) (m : ℕ) (hm : 1 ≤ m) (q : ℝ →ᵇ ℂ)
(r : ℂ → CircularTailSpace) (lam : ℂ) : CircularTailSpace :=
circularCorrectionResolvent κ hκ νp νm η m hm q lam (circularTailCLM κ hκ (r lam))
theorem circularResolvedCorrection_analyticAt (κ : ℝ) (hκ : 0 < κ)
(νp νm η : ℂ) (m : ℕ) (hm : 1 ≤ m) (q : ℝ →ᵇ ℂ)
(r : ℂ → CircularTailSpace) (z : ℂ) (hr : AnalyticAt ℂ r z)
(hgap : circularFieldBound (νp-2*z) (νm-2*z) η m ‖q‖ < κ) :
AnalyticAt ℂ (circularResolvedCorrection κ hκ νp νm η m hm q r) z := by
have hi := circularCorrectionResolvent_analyticAt κ hκ νp νm η m hm q z hgap
have ht : AnalyticAt ℂ (fun lam => circularTailCLM κ hκ (r lam)) z := by
convert! ((circularTailCLM κ hκ).analyticAt (r z)).comp hr
have ha : AnalyticAt ℂ (fun p : End₄ × CircularTailSpace => p.1 p.2)
(circularCorrectionResolvent κ hκ νp νm η m hm q z,circularTailCLM κ hκ (r z)) := by
convert! ((ContinuousLinearMap.apply ℂ CircularTailSpace).flip).analyticAt_bilinear
(circularCorrectionResolvent κ hκ νp νm η m hm q z,circularTailCLM κ hκ (r z))
exact ha.comp₂ hi ht
theorem circularResolvedCorrection_norm (κ : ℝ) (hκ : 0 < κ)
(νp νm η : ℂ) (m : ℕ) (hm : 1 ≤ m) (q : ℝ →ᵇ ℂ)
(r : ℂ → CircularTailSpace) (lam : ℂ)
(hgap : circularFieldBound (νp-2*lam) (νm-2*lam) η m ‖q‖ < κ) :
‖circularResolvedCorrection κ hκ νp νm η m hm q r lam‖ ≤
‖r lam‖/(κ-circularFieldBound (νp-2*lam) (νm-2*lam) η m ‖q‖) := by
let v := circularResolvedCorrection κ hκ νp νm η m hm q r lam
let B := circularFieldOperator (νp-2*lam) (νm-2*lam) η m hm q
have hv : v=circularTailCLM κ hκ (B v+r lam) :=
circularCorrectionResolvent_solve κ hκ νp νm η m hm q lam hgap (r lam)
have hBv : ‖B v‖ ≤ circularFieldBound (νp-2*lam) (νm-2*lam) η m ‖q‖*‖v‖ := by
change ‖circularFieldBounded (νp-2*lam) (νm-2*lam) η m hm q v‖ ≤ _
exact circularFieldBounded_norm (νp-2*lam) (νm-2*lam) η m hm q v
have hadd : ‖B v+r lam‖ ≤ ‖B v‖+‖r lam‖ := norm_add_le _ _
have hb : ‖B v+r lam‖ ≤
circularFieldBound (νp-2*lam) (νm-2*lam) η m ‖q‖*‖v‖+‖r lam‖ :=
hadd.trans (add_le_add hBv le_rfl)
have he : ‖v‖ ≤ (circularFieldBound (νp-2*lam) (νm-2*lam) η m ‖q‖*‖v‖+‖r lam‖)/κ := by
calc
_ = ‖circularTailCLM κ hκ (B v+r lam)‖ := congrArg norm hv
_ ≤ ‖B v+r lam‖/κ := circularTail_norm κ hκ _
_ ≤ _ := div_le_div_of_nonneg_right hb hκ.le
apply (le_div_iff₀ (sub_pos.mpr hgap)).mpr
have he' := (le_div_iff₀ hκ).mp he
nlinarith
theorem circularResolvedCorrection_hasDerivAt (κ : ℝ) (hκ : 0 < κ)
(νp νm η : ℂ) (m : ℕ) (hm : 1 ≤ m) (q : ℝ →ᵇ ℂ)
(r : ℂ → CircularTailSpace) (lam : ℂ)
(hgap : circularFieldBound (νp-2*lam) (νm-2*lam) η m ‖q‖ < κ) (t : ℝ) :
let v := circularResolvedCorrection κ hκ νp νm η m hm q r lam
HasDerivAt (circularTailEvaluation v)
(κ • circularTailEvaluation v t+circularLeadingField t (circularTailEvaluation v t)+
circularBoundedField (νp-2*lam) (νm-2*lam) η m (q t) (circularTailEvaluation v t)+
circularTailEvaluation (r lam) t) t := by
intro v
let B := circularFieldOperator (νp-2*lam) (νm-2*lam) η m hm q
have hv : circularTail κ hκ (B v+r lam)=v :=
(circularCorrectionResolvent_solve κ hκ νp νm η m hm q lam hgap (r lam)).symm
have hd := circularTail_hasDerivAt κ hκ (B v+r lam) t
dsimp only at hd
rw [hv] at hd
apply hd.congr_deriv
have he := circularFieldOperator_evaluation (νp-2*lam) (νm-2*lam) η m hm q v t
change circularTailEvaluation (B v) t=_ at he
rw [← he]
apply Prod.ext <;> apply Prod.ext
all_goals
simp only [circularTailEvaluation,circularLeadingField,Prod.fst_add,Prod.snd_add,
Prod.smul_fst,Prod.smul_snd,BoundedContinuousFunction.add_apply]
abel
end DefocusingNLS
end OAI
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