| import Mathlib |
| import OAI.Analysis.CoulombIonization.Variational.SpatialErrorWeight |
| import OAI.Analysis.CoulombIonization.FormDomain.CoreSliceMassIntegrable |
| import OAI.Analysis.CoulombIonization.Variational.SumPairsHalf |
|
|
| namespace OAI |
|
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| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace Convolution |
| open Filter |
| open scoped Topology InnerProductSpace |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace ContDiff |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace Convolution |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace ENNReal |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open Set Filter Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter |
| open scoped Topology |
| open MeasureTheory Filter |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter |
| open scoped Topology BigOperators InnerProductSpace |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter |
| open scoped Topology BigOperators Pointwise |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open scoped Convolution |
| open scoped RealInnerProductSpace |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Set Filter Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Set Filter Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open Filter Set Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open Filter |
| open scoped Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace ContDiff BigOperators |
| open MeasureTheory Filter Set |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators InnerProductSpace |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open scoped BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| namespace CoulombAtom |
|
|
| abbrev cutCoreNumber {L : ℕ} (b : Fin L → Fin 2) : ℕ := Fintype.card {i |
| abbrev cutOutNumber {L : ℕ} (b : Fin L → Fin 2) : ℕ := Fintype.card {i |
|
|
| def cutOrder {L : ℕ} (b : Fin L → Fin 2) : |
| Fin (cutCoreNumber b + cutOutNumber b) ≃ Fin L := |
| finSumFinEquiv.symm.trans ((Equiv.sumCongr |
| (Fintype.equivFin {i |
| (Equiv.sumCompl (fun i => b i = 0))) |
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|
| lemma cutNumbers_sum {L : ℕ} (b : Fin L → Fin 2) : cutCoreNumber b + cutOutNumber b = L := by |
| simpa only [Fintype.card_fin] using Fintype.card_congr (cutOrder b) |
|
|
| lemma cutOrder_labels {L : ℕ} (b : Fin L → Fin 2) : |
| b ∘ cutOrder b = coreCutLabels (cutCoreNumber b) (cutOutNumber b) := by |
| funext i |
| obtain ⟨j,rfl⟩ := finSumFinEquiv.surjective i |
| rcases j with j | j |
| · change b (cutOrder b (finSumFinEquiv (Sum.inl j))) = _ |
| simp only [cutOrder, Equiv.trans_apply, Equiv.symm_apply_apply, Equiv.sumCongr_apply, |
| Sum.map_inl, Equiv.sumCompl_apply_inl, coreCutLabels, joinLists_left] |
| exact ((Fintype.equivFin {i |
| · change b (cutOrder b (finSumFinEquiv (Sum.inr j))) = _ |
| simp only [cutOrder, Equiv.trans_apply, Equiv.symm_apply_apply, Equiv.sumCongr_apply, |
| Sum.map_inr, Equiv.sumCompl_apply_inr, coreCutLabels, joinLists_right] |
| have hh := ((Fintype.equivFin {i |
| omega |
|
|
| lemma spatialProductValue_reindex {M N : ℕ} (e : Fin M ≃ Fin N) |
| (p : Fin 2 → SmoothMultiplier spaceDirections) (b : Fin N → Fin 2) (x : Configuration M) : |
| spatialProductValue p b (x ∘ e.symm) = spatialProductValue p (b ∘ e) x := by |
| unfold spatialProductValue |
| rw [← Equiv.prod_comp e (fun i => (p (b i)).value ((x ∘ e.symm) i))] |
| simp only [Function.comp_apply, Equiv.symm_apply_apply] |
|
|
| def orderedCutForm {L : ℕ} (p : Fin 2 → SmoothMultiplier spaceDirections) |
| (hp : ∀ x, ∑ h : Fin 2, (p h).value x ^ 2 = 1) |
| (ψ : FormVector L) (b : Fin L → Fin 2) : |
| FormVector (cutCoreNumber b + cutOutNumber b) := |
| reindexForm (cutOrder b) (multiplyForm (spatialProduct (N := L) p hp b) ψ) |
|
|
| lemma orderedCutForm_value {L : ℕ} (p : Fin 2 → SmoothMultiplier spaceDirections) |
| (hp : ∀ x, ∑ h : Fin 2, (p h).value x ^ 2 = 1) |
| (ψ : FormVector L) (b : Fin L → Fin 2) (s) (x) : |
| (orderedCutForm p hp ψ b).value s x = |
| (multiplyForm (spatialProduct p hp (coreCutLabels (cutCoreNumber b) (cutOutNumber b))) |
| (reindexForm (cutOrder b) ψ)).value s x := by |
| change (spatialProductValue p b (x ∘ (cutOrder b).symm) : ℂ) * _ = _ |
| rw [spatialProductValue_reindex, cutOrder_labels] |
| rfl |
|
|
| lemma orderedCutForm_sobolev {L : ℕ} (p : Fin 2 → SmoothMultiplier spaceDirections) |
| (hp : ∀ x, ∑ h : Fin 2, (p h).value x ^ 2 = 1) |
| {ψ : FormVector L} (hψ : SobolevVector ψ) (b : Fin L → Fin 2) : |
| SobolevVector (orderedCutForm p hp ψ b) := |
| (hψ.multiply _).reindex _ |
|
|
| lemma orderedCutForm_core_antisymmetric {L : ℕ} (p : Fin 2 → SmoothMultiplier spaceDirections) |
| (hp : ∀ x, ∑ h : Fin 2, (p h).value x ^ 2 = 1) |
| {ψ : FormVector L} (hψ : SobolevFermion ψ) (b : Fin L → Fin 2) : |
| CoreAntisymmetric (orderedCutForm p hp ψ b) := by |
| have ha := (hψ.reindex (cutOrder b)).coreAntisymmetric.multiply |
| (spatialProduct p hp (coreCutLabels (cutCoreNumber b) (cutOutNumber b))) |
| (fun π x => spatialProduct_invariant p _ (corePerm _ π) (coreCutLabels_invariant π) x) |
| intro π s |
| simpa only [orderedCutForm_value] using ha π s |
|
|
| lemma orderedCutForm_mass {L : ℕ} (p : Fin 2 → SmoothMultiplier spaceDirections) |
| (hp : ∀ x, ∑ h : Fin 2, (p h).value x ^ 2 = 1) |
| (ψ : FormVector L) (b : Fin L → Fin 2) : |
| formMass (orderedCutForm p hp ψ b) = formMass (multiplyForm (spatialProduct (N := L) p hp b) ψ) := |
| formMass_reindex _ _ |
|
|
| lemma orderedCutForm_energy {L : ℕ} (p : Fin 2 → SmoothMultiplier spaceDirections) |
| (hp : ∀ x, ∑ h : Fin 2, (p h).value x ^ 2 = 1) |
| (ψ : FormVector L) (b : Fin L → Fin 2) (Z : ℝ) : |
| formEnergy Z (orderedCutForm p hp ψ b) = formEnergy Z (multiplyForm (spatialProduct (N := L) p hp b) ψ) := |
| formEnergy_reindex _ _ _ |
|
|
| def cutCoreEnergy {L : ℕ} (p : Fin 2 → SmoothMultiplier spaceDirections) |
| (hp : ∀ x, ∑ h : Fin 2, (p h).value x ^ 2 = 1) |
| (Z : ℝ) (ψ : FormVector L) (b : Fin L → Fin 2) : ℝ := |
| ∑ t : Spins (cutOutNumber b), ∫ y, |
| formEnergy Z (coreSlice (orderedCutForm p hp ψ b) t y) |
|
|
| theorem cutCoreEnergy_priced {L : ℕ} (p : Fin 2 → SmoothMultiplier spaceDirections) |
| (hp : ∀ x, ∑ h : Fin 2, (p h).value x ^ 2 = 1) |
| {ψ : FormVector L} (hψ : SobolevFermion ψ) (b : Fin L → Fin 2) |
| {Z lam : ℝ} (hZ : 0 ≤ Z) (hlam : 0 < lam) : |
| priceEnergy (energy Z) lam * formMass (multiplyForm (spatialProduct (N := L) p hp b) ψ) ≤ |
| cutCoreEnergy p hp Z ψ b + lam * cutCoreNumber b * |
| formMass (multiplyForm (spatialProduct (N := L) p hp b) ψ) := by |
| have hh := (orderedCutForm_core_antisymmetric p hp hψ b).integrated_priced |
| (orderedCutForm_sobolev p hp hψ.sobolevVector b) hZ hlam |
| simpa only [orderedCutForm_mass, cutCoreEnergy] using hh |
|
|
| theorem fock_cut_deletion {L : ℕ} (p : Fin 2 → SmoothMultiplier spaceDirections) |
| (hp : ∀ x, ∑ h : Fin 2, (p h).value x ^ 2 = 1) |
| {ψ : FormVector L} (hψ : SobolevFermion ψ) {Z lam ε : ℝ} |
| (hZ : 0 ≤ Z) (hlam : 0 < lam) (hm : formMass ψ = 1) |
| (he : formEnergy Z ψ + lam * L ≤ priceEnergy (energy Z) lam + ε) : |
| (∑ b : (Fin L → Fin 2), |
| (formEnergy Z (multiplyForm (spatialProduct (N := L) p hp b) ψ) - cutCoreEnergy p hp Z ψ b + |
| lam * cutOutNumber b * formMass (multiplyForm (spatialProduct (N := L) p hp b) ψ))) ≤ |
| ε + (1/2:ℝ) * weightedParticleCount ψ (spatialErrorWeight p) := by |
| have hb (b : Fin L → Fin 2) := cutCoreEnergy_priced p hp hψ b hZ hlam |
| have hmass := spatial_cut_mass p hp hψ.sobolevVector |
| have henergy := spatial_cut_ims p hp hψ.sobolevVector Z |
| have hsum := Finset.sum_le_sum (s := Finset.univ) (fun b _ => hb b) |
| have hcard (b : Fin L → Fin 2) : (cutCoreNumber b : ℝ) + cutOutNumber b = L := by |
| exact_mod_cast cutNumbers_sum b |
| have hid (b : Fin L → Fin 2) : |
| lam * cutOutNumber b * formMass (multiplyForm (spatialProduct (N := L) p hp b) ψ) = |
| lam * L * formMass (multiplyForm (spatialProduct (N := L) p hp b) ψ) - |
| lam * cutCoreNumber b * formMass (multiplyForm (spatialProduct (N := L) p hp b) ψ) := by |
| rw [← hcard b]; ring |
| simp_rw [hid] |
| simp only [Finset.sum_add_distrib, Finset.sum_sub_distrib, ← Finset.mul_sum] at hsum ⊢ |
| rw [hm] at hmass |
| rw [hmass, mul_one] at hsum ⊢ |
| rw [henergy] |
| change _ ≤ ε + (1/2:ℝ) * weightedParticleCount ψ (spatialErrorWeight p) |
| unfold weightedParticleCount |
| linarith |
|
|
| end CoulombAtom |
|
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| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
|
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| end |
|
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| end OAI |
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