| import Mathlib |
| import OAI.Analysis.CoulombIonization.Variational.IntegrableCutTower |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
|
|
| open MeasureTheory Filter Set Metric |
| open scoped BigOperators ENNReal |
|
|
| namespace CoulombAtom |
|
|
| def rawPotential {N : ℕ} (y : Space) (x : Configuration N) : ℝ := ∑ i, 1/‖x i-y‖ |
|
|
| lemma rawPotential_nonneg {N : ℕ} (y : Space) (x : Configuration N) : 0 ≤ rawPotential y x := |
| Finset.sum_nonneg (fun _ _ => by positivity) |
|
|
| lemma rawPotential_measurable {N : ℕ} (y : Space) : Measurable (rawPotential (N := N) y) := |
| Finset.measurable_sum _ (fun i _ => measurable_const.div (configuration_distance_measurable y i)) |
|
|
| lemma formRawLaw_integrable_iff {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) |
| {f : Configuration N → ℝ} : |
| Integrable f (formRawLaw ψ) ↔ Integrable (fun x => formRawDensity ψ x*f x) := by |
| rw [formRawLaw,integrable_withDensity_iff_integrable_smul₀' |
| (formRawDensity_integrable hψ).aemeasurable.ennreal_ofReal |
| (ae_of_all _ (fun _ => ENNReal.ofReal_lt_top))] |
| simp only [ENNReal.toReal_ofReal (formRawDensity_nonneg ψ _),smul_eq_mul] |
|
|
| lemma rawPotential_form_integrable {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) (y : Space) : |
| Integrable (rawPotential y) (formRawLaw ψ) := by |
| rw [formRawLaw_integrable_iff hψ] |
| simp only [formRawDensity,rawPotential,Finset.sum_mul,Finset.mul_sum] |
| apply integrable_finsetSum |
| intro s _ |
| apply integrable_finsetSum |
| intro i _ |
| simpa only [mul_one_div] using hψ.shifted_nuclear_integrable i s y |
|
|
| lemma rawLocalPotential_le_raw {N : ℕ} (y : Space) (q : ℝ) (x : Configuration N) : |
| rawLocalPotential y q x ≤ rawPotential y x := by |
| apply Finset.sum_le_sum |
| intro i _ |
| split_ifs |
| · exact le_rfl |
| · positivity |
|
|
| lemma rawLocalPotential_form_integrable {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) |
| (y : Space) (q : ℝ) : Integrable (rawLocalPotential y q) (formRawLaw ψ) := by |
| apply (rawPotential_form_integrable hψ y).mono' (rawLocalPotential_measurable N y q).aestronglyMeasurable |
| exact ae_of_all _ (fun x => by simpa only [Real.norm_of_nonneg (rawLocalPotential_nonneg y q x)] using |
| (rawLocalPotential_le_raw y q x)) |
|
|
| lemma formRawLaw_ae_no_poles {N : ℕ} (ψ : FormVector N) (y : Space) : |
| ∀ᵐ x ∂formRawLaw ψ, ∀ i, x i ≠ y := by |
| apply (withDensity_absolutelyContinuous volume _).ae_le |
| exact ae_all_iff.mpr (fun i => Measure.ae_eval_ne (fun _ : Fin N => (volume : Measure Space)) i y) |
|
|
| end CoulombAtom |
|
|
| end |
|
|
| end OAI |
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|
