| import Mathlib |
| import OAI.Analysis.CoulombIonization.Variational.CoherentDensity |
| import OAI.Analysis.CoulombIonization.Localization.RadialPacket |
| import OAI.Analysis.CoulombIonization.RadialBounds.FermiMeasure |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace CoulombAtom |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
|
|
| open MeasureTheory Filter Set Metric |
| open scoped BigOperators ContDiff |
|
|
| def smearedPacketDensity (g ρ : Space → ℝ) (x : Space) : ℝ := |
| ∫ z : Space, ρ z * (g (x-z))^2 |
|
|
| lemma coherentDensity_fermi {ρ g : Space → ℝ} (hm : Measurable ρ) (hn : ∀ z, 0 ≤ ρ z) |
| {J : Set Space} (hJ : IsCompact J) (hs : Function.support ρ ⊆ J) {B : ℝ} |
| (hb : ∀ z, ρ z ≤ B) (hg : Continuous g) (x : Space) : |
| 2*coherentDensity (fun x => (g x : ℂ)) (fermiMeasure ρ) x = smearedPacketDensity g ρ x := by |
| have hi := fermiMeasure_integrable_continuous hn hJ hs hb |
| (show Continuous (fun q : Space × Space => (g (x-q.1))^2) by fun_prop) |
| simp only [coherentDensity,coherentPacket_norm,Complex.norm_real,Real.norm_eq_abs,sq_abs] |
| rw [← mul_assoc,fermiMeasure_integral hm hi,← integral_const_mul] |
| apply integral_congr_ae |
| filter_upwards [] with z |
| rw [integral_const] |
| simp only [smul_eq_mul,Measure.real,Measure.restrict_apply_univ] |
| rw [← mul_assoc] |
| exact congrArg (fun a : ℝ => a*(g (x-z))^2) (fermiRadius_ball_mass hn z) |
|
|
| lemma smearedPacketDensity_nonneg {ρ : Space → ℝ} (hn : ∀ z, 0 ≤ ρ z) (g : Space → ℝ) (x : Space) : |
| 0 ≤ smearedPacketDensity g ρ x := integral_nonneg (fun z => mul_nonneg (hn z) (sq_nonneg _)) |
|
|
| lemma smearedPacketDensity_continuous {ρ g : Space → ℝ} (hm : Measurable ρ) (hn : ∀ z, 0 ≤ ρ z) |
| {J : Set Space} (hJ : IsCompact J) (hs : Function.support ρ ⊆ J) {B : ℝ} |
| (hb : ∀ z, ρ z ≤ B) (hg : Continuous g) : Continuous (smearedPacketDensity g ρ) := by |
| let := fermiMeasure_finite hn hJ hs hb |
| have he : smearedPacketDensity g ρ = fun x => 2*coherentDensity (fun x => (g x : ℂ)) (fermiMeasure ρ) x := |
| funext (fun x => (coherentDensity_fermi hm hn hJ hs hb hg x).symm) |
| rw [he] |
| exact continuous_const.mul (coherentDensity_continuous (Complex.continuous_ofReal.comp hg) |
| (fermiMeasure ρ) (hJ.prod (isCompact_closedBall _ _)) (fermiMeasure_mem hm hn hs hb)) |
|
|
| lemma smearedPacketDensity_compact {ρ g : Space → ℝ} (hm : Measurable ρ) (hn : ∀ z, 0 ≤ ρ z) |
| {J : Set Space} (hJ : IsCompact J) (hs : Function.support ρ ⊆ J) {B : ℝ} |
| (hb : ∀ z, ρ z ≤ B) (hg : Continuous g) (hcg : HasCompactSupport g) : |
| HasCompactSupport (smearedPacketDensity g ρ) := by |
| let := fermiMeasure_finite hn hJ hs hb |
| have he : smearedPacketDensity g ρ = fun x => 2*coherentDensity (fun x => (g x : ℂ)) (fermiMeasure ρ) x := |
| funext (fun x => (coherentDensity_fermi hm hn hJ hs hb hg x).symm) |
| rw [he] |
| have hcc : HasCompactSupport (fun x => (g x : ℂ)) := hcg.comp_left (g := Complex.ofReal) Complex.ofReal_zero |
| have hd : HasCompactSupport (coherentDensity (fun x => (g x : ℂ)) (fermiMeasure ρ)) := |
| coherentDensity_compact hcc (fermiMeasure ρ) (hJ.prod (isCompact_closedBall _ _)) |
| (fermiMeasure_mem hm hn hs hb) |
| exact hd.comp_left (g := fun t : ℝ => 2*t) (mul_zero 2) |
|
|
| lemma smearedPacketDensity_mass {ρ g : Space → ℝ} (hm : Measurable ρ) (hn : ∀ z, 0 ≤ ρ z) |
| {J : Set Space} (hJ : IsCompact J) (hs : Function.support ρ ⊆ J) {B : ℝ} |
| (hb : ∀ z, ρ z ≤ B) (hg : Continuous g) (hcg : HasCompactSupport g) |
| (hgn : ∫ x : Space, (g x)^2 = 1) : (∫ x : Space, smearedPacketDensity g ρ x) = ∫ z, ρ z := by |
| let := fermiMeasure_finite hn hJ hs hb |
| have he : smearedPacketDensity g ρ = fun x => 2*coherentDensity (fun x => (g x : ℂ)) (fermiMeasure ρ) x := |
| funext (fun x => (coherentDensity_fermi hm hn hJ hs hb hg x).symm) |
| have hgc : Continuous (fun x => (g x : ℂ)) := Complex.continuous_ofReal.comp hg |
| have hcc : HasCompactSupport (fun x => (g x : ℂ)) := hcg.comp_left (g := Complex.ofReal) Complex.ofReal_zero |
| rw [he,integral_const_mul,coherentDensity_mass hgc hcc (fermiMeasure ρ) |
| (hJ.prod (isCompact_closedBall _ _)) (fermiMeasure_mem hm hn hs hb)] |
| simp only [Complex.norm_real,Real.norm_eq_abs,sq_abs,hgn,mul_one,← mul_assoc] |
| exact fermiMeasure_mass hm hn hJ hs hb |
|
|
| open MeasureTheory Filter Set Metric |
| open scoped BigOperators ContDiff |
|
|
| lemma coulomb_double_eq_potential {f g : Space → ℝ} |
| (hi : Integrable (fun r : Space × Space => f r.1*g r.2/‖r.1-r.2‖)) : |
| (∫ r : Space × Space, f r.1*g r.2/‖r.1-r.2‖) = |
| ∫ x : Space, f x*CoulombAnalysis.tfPotential g x := by |
| rw [Measure.volume_eq_prod] at hi ⊢ |
| rw [integral_prod _ hi] |
| simp only [CoulombAnalysis.tfPotential,mul_div_assoc,integral_const_mul] |
|
|
| lemma coulomb_potential_pair_integrable {f g : Space → ℝ} |
| (hi : Integrable (fun r : Space × Space => f r.1*g r.2/‖r.1-r.2‖)) : |
| Integrable (fun x : Space => f x*CoulombAnalysis.tfPotential g x) := by |
| simpa only [CoulombAnalysis.tfPotential,mul_div_assoc,integral_const_mul] using hi.integral_prod_left |
|
|
| lemma coulomb_potential_pair_comm {f g : Space → ℝ} |
| (hi : Integrable (fun r : Space × Space => f r.1*g r.2/‖r.1-r.2‖)) : |
| (∫ x : Space, f x*CoulombAnalysis.tfPotential g x) = |
| ∫ x : Space, g x*CoulombAnalysis.tfPotential f x := by |
| simp only [CoulombAnalysis.tfPotential,← integral_const_mul] |
| have hi' : Integrable (fun r : Space × Space => f r.1*(g r.2/‖r.1-r.2‖)) (volume.prod volume) := by |
| simpa only [mul_div_assoc,Measure.volume_eq_prod] using hi |
| rw [integral_integral_swap hi'] |
| apply integral_congr_ae |
| filter_upwards [] with x |
| apply integral_congr_ae |
| filter_upwards [] with y |
| rw [norm_sub_rev x y] |
| ring |
|
|
| lemma coulomb_direct_le_of_potential_le {f g : Space → ℝ} |
| (hff : Integrable (fun r : Space × Space => f r.1*f r.2/‖r.1-r.2‖)) |
| (hfg : Integrable (fun r : Space × Space => f r.1*g r.2/‖r.1-r.2‖)) |
| (hgg : Integrable (fun r : Space × Space => g r.1*g r.2/‖r.1-r.2‖)) |
| (hfn : ∀ x, 0 ≤ f x) (hgn : ∀ x, 0 ≤ g x) |
| (hp : ∀ x, CoulombAnalysis.tfPotential f x ≤ CoulombAnalysis.tfPotential g x) : |
| (∫ r : Space × Space, f r.1*f r.2/‖r.1-r.2‖) ≤ |
| ∫ r : Space × Space, g r.1*g r.2/‖r.1-r.2‖ := by |
| have hgf : Integrable (fun r : Space × Space => g r.1*f r.2/‖r.1-r.2‖) := by |
| rw [Measure.volume_eq_prod] at hfg ⊢ |
| apply hfg.swap.congr |
| filter_upwards [] with r |
| change f r.2*g r.1/‖r.2-r.1‖ = g r.1*f r.2/‖r.1-r.2‖ |
| rw [norm_sub_rev r.1 r.2] |
| ring |
| rw [coulomb_double_eq_potential hff,coulomb_double_eq_potential hgg] |
| calc |
| _ ≤ ∫ x : Space, f x*CoulombAnalysis.tfPotential g x := |
| integral_mono (coulomb_potential_pair_integrable hff) (coulomb_potential_pair_integrable hfg) |
| (fun x => mul_le_mul_of_nonneg_left (hp x) (hfn x)) |
| _ = ∫ x : Space, g x*CoulombAnalysis.tfPotential f x := coulomb_potential_pair_comm hfg |
| _ ≤ _ := integral_mono (coulomb_potential_pair_integrable hgf) (coulomb_potential_pair_integrable hgg) |
| (fun x => mul_le_mul_of_nonneg_left (hp x) (hgn x)) |
|
|
| lemma square_compact {g : Space → ℝ} (hg : HasCompactSupport g) : |
| HasCompactSupport (fun x => (g x)^2) := hg.comp_left (g := fun t : ℝ => t^2) (by simp) |
|
|
| lemma smearedPacketDensity_direct_le {ρ g : Space → ℝ} (hm : Measurable ρ) (hn : ∀ z, 0 ≤ ρ z) |
| {J : Set Space} (hJ : IsCompact J) (hs : Function.support ρ ⊆ J) {B : ℝ} |
| (hb : ∀ z, ρ z ≤ B) (hg : Continuous g) (hcg : HasCompactSupport g) |
| (hgn : ∫ x : Space, (g x)^2 = 1) (hr : CoulombAnalysis.IsRadial (fun x => (g x)^2)) : |
| (∫ r : Space × Space, smearedPacketDensity g ρ r.1*smearedPacketDensity g ρ r.2/‖r.1-r.2‖) ≤ |
| ∫ r : Space × Space, ρ r.1*ρ r.2/‖r.1-r.2‖ := by |
| have hab (z : Space) : |ρ z| ≤ B := by rw [abs_of_nonneg (hn z)]; exact hb z |
| have h1 : Integrable ρ := bounded_compact_integrable hm hJ hs hab |
| have hp : MemLp ρ (5/3 : ENNReal) := bounded_compact_memLp hm hJ hs hab _ |
| have hd := smearedPacketDensity_continuous hm hn hJ hs hb hg |
| have hc := smearedPacketDensity_compact hm hn hJ hs hb hg hcg |
| apply coulomb_direct_le_of_potential_le |
| (continuous_compact_coulomb_integrable hd hd hc hc) |
| (compact_support_coulomb_integrable hd.measurable hm (hd.integrable_of_hasCompactSupport hc) h1 hp hJ hs) |
| (compact_support_coulomb_integrable hm hm h1 h1 hp hJ hs) |
| (smearedPacketDensity_nonneg hn g) hn |
| intro x |
| exact spatialSmearing_potential_le (hg.pow 2) (square_compact hcg) (fun y => sq_nonneg (g y)) |
| hr hgn hm h1 hp hn x |
|
|
| end CoulombAtom |
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| end |
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| end OAI |
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