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import OAI.Analysis.CoulombIonization.RadialBounds.PatchMass
import OAI.Analysis.CoulombIonization.ThomasFermi.PatchRandomVariables
import OAI.Analysis.CoulombIonization.FieldAnalysis.ConditionalObservables
import OAI.Analysis.CoulombIonization.Ionization.ConditionalPatchMinimizer
import OAI.Analysis.CoulombIonization.FieldAnalysis.RetainedFieldIdentity
namespace OAI
noncomputable section
namespace CoulombAtom
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff
section Work_ConditionalRetainedGap_scope
open MeasureTheory Filter Set Metric
open scoped BigOperators ContDiff ENNReal
open CoulombAnalysis CoulombNeumann
theorem conditional_retained_patch_gap {M N : ℕ} {ψ : FormVector M}
(hψ : SobolevFermion ψ) (y : Space) (R : ℝ) (Φ : TFField R)
{Z lam : ℝ} (hZ : 0 ≤ Z) (hlam : 0 < lam)
(hΦ : Φ =ᵐ[ballMeasure R] fun z => normalizedCoreField Z lam ψ (y+z))
{F : ℝ} (hF : ∀ᵐ z ∂ballMeasure R, Φ z ≤ F)
{g : Space → ℝ} (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g)
(hgn : ∫ z : Space, (g z)^2 = 1) (hr : IsRadial g)
(hgs : tsupport g ⊆ ball 0 1) {b : ℝ} (hb : 0 < b)
(A : Set Space) (hcore : ∀ u i, u i ∉ A → FormZeroAt ψ u)
(hsep : Disjoint A (packetRegion (scaledRealPacket b g) (closedBall y R)))
(hnuc : ∀ z ∈ closedBall y R, b ≤ ‖z‖)
(hcs : ∀ a ∈ A, ∀ z ∈ closedBall y R, b ≤ ‖a-z‖)
(S : Finset (Fin N)) (x : Configuration N)
(hmargin : ∀ i ∈ S, ‖x i-y‖+Real.sqrt 3*b ≤ R)
(hret_nuc : ∀ i ∈ S, Real.sqrt 3*b ≤ ‖x i‖)
(hret_core : ∀ i ∈ S, ∀ a ∈ A, Real.sqrt 3*b ≤ ‖a-x i‖) :
let σ := retainedPatchLp hb S x y R
let ρ := tfPatchMinimizer R tfKinetic tfKinetic_pos Φ
formMass ψ*((1/2:ℝ)*tfCoulombL R (σ-ρ) (σ-ρ)+
(∫ z, max (tfBallPotential R ρ z-Φ z) 0*σ z ∂ballMeasure R)) ≤
formEnergy Z ψ+lam*M*formMass ψ-priceEnergy (energy Z) lam*formMass ψ+
formMass ψ*(tfKinetic*retainedPressure b S x-
(∑ i ∈ S, normalizedCoreField Z lam ψ (x i))+
(1/2:ℝ)*(∫ p : Space × Space,
retainedSmear b S x p.1*retainedSmear b S x p.2/‖p.1-p.2‖)+
(packetDirichlet g/2)*b⁻¹^2*(∫ z, ρ z ∂ballMeasure R)) := by
dsimp only
have hu := conditional_priced_patch_minimizer_upper hψ y R Φ hZ hlam hΦ hF
hg hcg hgn hr hgs hb A hcore hsep hnuc hcs
have hg := mul_le_mul_of_nonneg_left
(tfPatchFunctional_clipping_gap R Φ tfKinetic_pos (retainedPatchLp_nonneg hb S x y R))
(formMass_nonneg ψ)
rw [retainedPatchLp_functional hb S x y hmargin Φ (normalizedCoreField Z lam ψ) hΦ tfKinetic, retainedSmear_core_field hψ.sobolevVector hb S x A hcore hret_nuc hret_core Z lam] at hg
dsimp only at hu
nlinarith
theorem actual_retained_patch_gap {M N : ℕ} {ψ : FormVector M}
(hψ : SobolevFermion ψ) (y : Space) (R : ℝ)
{Z lam : ℝ} (hZ : 0 ≤ Z) (hlam : 0 < lam)
{g : Space → ℝ} (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g)
(hgn : ∫ z : Space, (g z)^2 = 1) (hr : IsRadial g)
(hgs : tsupport g ⊆ ball 0 1) {b : ℝ} (hb : 0 < b)
(A : Set Space) (hcore : ∀ u i, u i ∉ A → FormZeroAt ψ u)
(hsep : Disjoint A (packetRegion (scaledRealPacket b g) (closedBall y R)))
(hnuc : ∀ z ∈ closedBall y R, b ≤ ‖z‖)
(hcs : ∀ a ∈ A, ∀ z ∈ closedBall y R, b ≤ ‖a-z‖)
(S : Finset (Fin N)) (x : Configuration N)
(hmargin : ∀ i ∈ S, ‖x i-y‖+Real.sqrt 3*b ≤ R)
(hret_nuc : ∀ i ∈ S, Real.sqrt 3*b ≤ ‖x i‖)
(hret_core : ∀ i ∈ S, ∀ a ∈ A, Real.sqrt 3*b ≤ ‖a-x i‖) :
let hfield := normalizedCoreField_memLp_patch hψ.sobolevVector A hcore y R hb hb hnuc hcs Z lam
let Φ := hfield.toLp (fun z => normalizedCoreField Z lam ψ (y+z))
let σ := retainedPatchLp hb S x y R
let ρ := tfPatchMinimizer R tfKinetic tfKinetic_pos Φ
formMass ψ*((1/2:ℝ)*tfCoulombL R (σ-ρ) (σ-ρ)+
(∫ z, max (tfBallPotential R ρ z-Φ z) 0*σ z ∂ballMeasure R)) ≤
formEnergy Z ψ+lam*M*formMass ψ-priceEnergy (energy Z) lam*formMass ψ+
formMass ψ*(tfKinetic*retainedPressure b S x-
(∑ i ∈ S, normalizedCoreField Z lam ψ (x i))+
(1/2:ℝ)*(∫ p : Space × Space,
retainedSmear b S x p.1*retainedSmear b S x p.2/‖p.1-p.2‖)+
(packetDirichlet g/2)*b⁻¹^2*(∫ z, ρ z ∂ballMeasure R)) := by
dsimp only
let hfield := normalizedCoreField_memLp_patch hψ.sobolevVector A hcore y R hb hb hnuc hcs Z lam
have hbound : ∀ᵐ z ∂ballMeasure R,
hfield.toLp (fun z => normalizedCoreField Z lam ψ (y+z)) z ≤ |Z|/b+|lam|+M/b := by
filter_upwards [hfield.coeFn_toLp,ae_restrict_mem measurableSet_ball] with z hz hmem
rw [hz]
have hz' : y+z ∈ closedBall y R := by
simpa only [mem_closedBall,dist_eq_norm,add_sub_cancel_left] using
(show ‖z‖ ≤ R from (by simpa only [mem_ball,dist_eq_norm,sub_zero] using hmem : ‖z‖ < R).le)
exact (le_abs_self _).trans (normalizedCoreField_abs_le hψ.sobolevVector A hcore hb
(y+z) (fun a ha => hcs a ha _ hz') hb (hnuc _ hz') Z lam)
exact conditional_retained_patch_gap hψ y R _ hZ hlam hfield.coeFn_toLp hbound
hg hcg hgn hr hgs hb A hcore hsep hnuc hcs S x hmargin hret_nuc hret_core
end Work_ConditionalRetainedGap_scope
open MeasureTheory Filter Set Metric
open scoped BigOperators ENNReal ContDiff
open CoulombAnalysis CoulombNeumann
def finiteCorePatchBound (N : ℕ) (Z lam b : ℝ) : ℝ := |Z|/b+|lam|+N/b
lemma finiteCorePatchBound_nonneg (N : ℕ) (Z lam : ℝ) {b : ℝ} (hb : 0 < b) :
0 ≤ finiteCorePatchBound N Z lam b := by unfold finiteCorePatchBound; positivity
lemma conditionalPatchField_upper {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) (t : Spins M) (y : Space) (R : ℝ) (Z lam : ℝ)
{b : ℝ} (hb : 0 < b) (A : Set Space)
(hcore : ∀ u x i, x i ∉ A → FormZeroAt (coreSlice ψ t u) x)
(hnuc : ∀ z ∈ closedBall y R, b ≤ ‖z‖)
(hcs : ∀ a ∈ A, ∀ z ∈ closedBall y R, b ≤ ‖a-z‖) :
∀ᵐ u : Configuration M, ∀ᵐ z ∂ballMeasure R,
conditionalPatchField ψ t Z lam y R u z ≤ finiteCorePatchBound N Z lam b := by
apply (hψ.ae_coreSlice t).mono
intro u hu
have hf := normalizedCoreField_memLp_patch hu A (hcore u) y R hb hb hnuc hcs Z lam
filter_upwards [conditionalPatchField_ae ψ t Z lam y R u hf,
ae_restrict_mem measurableSet_ball] with z hz hmem
have hpos : y+z ∈ closedBall y R := by
simpa only [mem_closedBall,dist_eq_norm,add_sub_cancel_left] using
((mem_ball_zero_iff.mp hmem).le)
rw [hz]
exact (le_abs_self _).trans (normalizedCoreField_abs_le hu A (hcore u) hb (y+z)
(fun a ha => hcs a ha _ hpos) hb (hnuc _ hpos) Z lam)
def weightedPatchMass {N M : ℕ} (ψ : FormVector (N+M)) (t : Spins M)
(Z lam : ℝ) (y : Space) (R : ℝ) (u : Configuration M) : ℝ :=
formMass (coreSlice ψ t u)*(∫ z,
tfPatchMinimizer R tfKinetic tfKinetic_pos (conditionalPatchField ψ t Z lam y R u) z ∂ballMeasure R)
lemma weightedPatchMass_nonneg {N M : ℕ} (ψ : FormVector (N+M)) (t : Spins M)
(Z lam : ℝ) (y : Space) (R : ℝ) (u : Configuration M) :
0 ≤ weightedPatchMass ψ t Z lam y R u :=
mul_nonneg (formMass_nonneg _) (integral_nonneg_of_ae (tfPatchMinimizer_nonneg R tfKinetic tfKinetic_pos _))
lemma weightedPatchMass_integrable {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) (t : Spins M) (y : Space) {R : ℝ} (hR : 0 < R) (Z lam : ℝ)
{b : ℝ} (hb : 0 < b) (A : Set Space)
(hcore : ∀ u x i, x i ∉ A → FormZeroAt (coreSlice ψ t u) x)
(hnuc : ∀ z ∈ closedBall y R, b ≤ ‖z‖)
(hcs : ∀ a ∈ A, ∀ z ∈ closedBall y R, b ≤ ‖a-z‖) :
Integrable (weightedPatchMass ψ t Z lam y R) := by
have hm := (conditionalPatchMass_aemeasurable hψ t Z lam y R).aestronglyMeasurable
have hbnd : ∀ᵐ u : Configuration M, ‖(∫ z,
tfPatchMinimizer R tfKinetic tfKinetic_pos (conditionalPatchField ψ t Z lam y R u) z ∂ballMeasure R)‖ ≤
4*R*finiteCorePatchBound N Z lam b := by
apply (conditionalPatchField_upper hψ t y R Z lam hb A hcore hnuc hcs).mono
intro u hu
rw [Real.norm_of_nonneg (integral_nonneg_of_ae (tfPatchMinimizer_nonneg R tfKinetic tfKinetic_pos _))]
exact tfPatchMinimizer_mass_le hR tfKinetic_pos (finiteCorePatchBound_nonneg N Z lam hb) _ hu
exact ((hψ.coreSlice_mass_integrable t).bdd_mul hm hbnd).congr
(Eventually.of_forall fun _ => mul_comm _ _)
lemma coreSlice_priceExcess_integrable {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) (t : Spins M) (Z lam : ℝ) :
Integrable (fun u => corePriceExcess Z lam (coreSlice ψ t u)) := by
exact ((hψ.coreSlice_energy_integrable Z t).add ((hψ.coreSlice_mass_integrable t).const_mul (lam*N))).sub
((hψ.coreSlice_mass_integrable t).const_mul (priceEnergy (energy Z) lam))
def weightedPatchGap {N M : ℕ} (ψ : FormVector (N+M)) (t : Spins M)
(Z lam : ℝ) (y : Space) (R : ℝ) {b : ℝ} (hb : 0 < b)
(S : Configuration M → Finset (Fin M)) (u : Configuration M) : ℝ :=
formMass (coreSlice ψ t u)*tfPatchGap R tfKinetic tfKinetic_pos
(conditionalPatchField ψ t Z lam y R u) (retainedPatchLp hb (S u) u y R)
lemma weightedPatchGap_nonneg {N M : ℕ} (ψ : FormVector (N+M)) (t : Spins M)
(Z lam : ℝ) (y : Space) (R : ℝ) {b : ℝ} (hb : 0 < b)
(S : Configuration M → Finset (Fin M)) (u : Configuration M) :
0 ≤ weightedPatchGap ψ t Z lam y R hb S u :=
mul_nonneg (formMass_nonneg _) (tfPatchGap_nonneg _ _ _ _ (retainedPatchLp_nonneg hb (S u) u y R))
theorem conditionalPatchGap_upper {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) (ha : CoreAntisymmetric ψ) (t : Spins M)
(y : Space) (R : ℝ) {Z lam : ℝ} (hZ : 0 ≤ Z) (hlam : 0 < lam)
{g : Space → ℝ} (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g)
(hgn : ∫ z : Space, (g z)^2 = 1) (hr : IsRadial g)
(hgs : tsupport g ⊆ ball 0 1) {b : ℝ} (hb : 0 < b)
(A : Set Space) (hcore : ∀ u x i, x i ∉ A → FormZeroAt (coreSlice ψ t u) x)
(hsep : Disjoint A (packetRegion (scaledRealPacket b g) (closedBall y R)))
(hnuc : ∀ z ∈ closedBall y R, b ≤ ‖z‖)
(hcs : ∀ a ∈ A, ∀ z ∈ closedBall y R, b ≤ ‖a-z‖)
(S : Configuration M → Finset (Fin M))
(hmargin : ∀ u, ∀ i ∈ S u, ‖u i-y‖+Real.sqrt 3*b ≤ R)
(hret_nuc : ∀ u, ∀ i ∈ S u, Real.sqrt 3*b ≤ ‖u i‖)
(hret_core : ∀ u, ∀ i ∈ S u, ∀ a ∈ A, Real.sqrt 3*b ≤ ‖a-u i‖) :
∀ᵐ u : Configuration M,
weightedPatchGap ψ t Z lam y R hb S u ≤
corePriceExcess Z lam (coreSlice ψ t u)+
tfKinetic*(retainedPressure b (S u) u*formMass (coreSlice ψ t u))-
conditionalRetainedFieldSum Z lam ψ t S u+
retainedDirect b (S u) u*formMass (coreSlice ψ t u)+
((packetDirichlet g/2)*b⁻¹^2)*weightedPatchMass ψ t Z lam y R u := by
filter_upwards [ha.ae_coreSlice hψ t,
conditionalPatchField_upper hψ t y R Z lam hb A hcore hnuc hcs] with u hu hfield
have hf := normalizedCoreField_memLp_patch hu.sobolevVector A (hcore u) y R hb hb hnuc hcs Z lam
have hh := conditional_retained_patch_gap hu y R (conditionalPatchField ψ t Z lam y R u)
hZ hlam (conditionalPatchField_ae ψ t Z lam y R u hf) hfield
hg hcg hgn hr hgs hb A (hcore u) hsep hnuc hcs (S u) u
(hmargin u) (hret_nuc u) (hret_core u)
dsimp only at hh
change formMass (coreSlice ψ t u)*_ ≤ _
unfold tfPatchGap
unfold corePriceExcess conditionalRetainedFieldSum retainedDirect weightedPatchMass
nlinarith
theorem weightedPatchGap_integrable {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) (ha : CoreAntisymmetric ψ) (t : Spins M)
(y : Space) {R : ℝ} (hR : 0 < R) {Z lam : ℝ} (hZ : 0 ≤ Z) (hlam : 0 < lam)
{g : Space → ℝ} (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g)
(hgn : ∫ z : Space, (g z)^2 = 1) (hr : IsRadial g)
(hgs : tsupport g ⊆ ball 0 1) {b : ℝ} (hb : 0 < b)
(A : Set Space) (hcore : ∀ u x i, x i ∉ A → FormZeroAt (coreSlice ψ t u) x)
(hsep : Disjoint A (packetRegion (scaledRealPacket b g) (closedBall y R)))
(hnuc : ∀ z ∈ closedBall y R, b ≤ ‖z‖)
(hcs : ∀ a ∈ A, ∀ z ∈ closedBall y R, b ≤ ‖a-z‖)
(S : Configuration M → Finset (Fin M)) (hS : ∀ i, MeasurableSet {u | i ∈ S u})
(hmargin : ∀ u, ∀ i ∈ S u, ‖u i-y‖+Real.sqrt 3*b ≤ R)
(hret_nuc : ∀ u, ∀ i ∈ S u, Real.sqrt 3*b ≤ ‖u i‖)
(hret_core : ∀ u, ∀ i ∈ S u, ∀ a ∈ A, Real.sqrt 3*b ≤ ‖a-u i‖) :
Integrable (weightedPatchGap ψ t Z lam y R hb S) := by
have hm := (hψ.coreSlice_mass_integrable t).aemeasurable.mul
(conditionalPatchGap_aemeasurable hψ t Z lam y R hb S hS)
have hi := ((((coreSlice_priceExcess_integrable hψ t Z lam).add
((retained_pressure_weighted_integrable hψ hb t S hS).const_mul tfKinetic)).sub
(conditionalRetainedFieldSum_integrable hψ Z lam t S hS)).add
(retained_direct_weighted_integrable hψ hb t S hS)).add
((weightedPatchMass_integrable hψ t y hR Z lam hb A hcore hnuc hcs).const_mul
((packetDirichlet g/2)*b⁻¹^2))
apply hi.mono' hm.aestronglyMeasurable
filter_upwards [conditionalPatchGap_upper hψ ha t y R hZ hlam hg hcg hgn hr hgs hb A hcore hsep hnuc hcs S hmargin hret_nuc hret_core] with u hu
simp only [Pi.mul_apply,Pi.add_apply,Pi.sub_apply]
change ‖weightedPatchGap ψ t Z lam y R hb S u‖ ≤ _
rw [Real.norm_of_nonneg (weightedPatchGap_nonneg ψ t Z lam y R hb S u)]
exact hu
end CoulombAtom
end
end OAI
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