| import Mathlib |
| import OAI.Analysis.CoulombIonization.RadialBounds.Radial |
| import OAI.Analysis.CoulombIonization.Variational.SpinFinite |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace CoulombAtom |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
|
|
| open MeasureTheory Filter |
| open scoped BigOperators ContDiff |
|
|
| lemma compact_potential_bounded {d : Space → ℝ} (hd : Continuous d) (hcd : HasCompactSupport d) : |
| ∃ C : ℝ, ∀ x : Space, |CoulombAnalysis.tfPotential d x| ≤ C := by |
| have h1 : Integrable d := hd.integrable_of_hasCompactSupport hcd |
| have hp : MemLp d (5/3 : ENNReal) := hd.memLp_of_hasCompactSupport hcd |
| obtain ⟨R,_,hR⟩ := hcd.isBounded.exists_pos_norm_le |
| have hs (y : Space) (hy : d y ≠ 0) : ‖y‖ ≤ R := hR y (subset_tsupport d hy) |
| obtain ⟨S,_,hS⟩ := CoulombAnalysis.tfPotential_compact_decay h1 hp hs 1 zero_lt_one |
| obtain ⟨C,hC⟩ := (isCompact_closedBall (0 : Space) S).exists_bound_of_continuousOn |
| (CoulombAnalysis.tfPotential_continuous h1 hp).continuousOn |
| refine ⟨max C 1,fun x => ?_⟩ |
| by_cases hx : ‖x‖ ≤ S |
| · exact (hC x (by simpa using hx)).trans (le_max_left _ _) |
| · exact (hS x (le_of_not_ge hx)).trans (le_max_right _ _) |
|
|
| lemma core_bounded_potential_integrable {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) |
| {V : Space → ℝ} (hV : Continuous V) {C : ℝ} (hC : ∀ x, |V x| ≤ C) |
| (s : Spins N) (j : Fin N) : |
| Integrable (fun x : Configuration N => ‖ψ.value s x‖^2 * V (x j)) := by |
| apply ((hψ.1 s).norm.integrable_sq.mul_const C).mono' |
| (((hψ.1 s).aestronglyMeasurable.norm.pow 2).mul |
| (hV.comp (continuous_apply j)).aestronglyMeasurable) |
| filter_upwards [] with x |
| change ‖‖ψ.value s x‖^2 * V (x j)‖ ≤ _ |
| rw [norm_mul,Real.norm_of_nonneg (sq_nonneg _),Real.norm_eq_abs] |
| exact mul_le_mul_of_nonneg_left (hC (x j)) (sq_nonneg _) |
|
|
| lemma core_compact_potential_integrable {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) |
| {d : Space → ℝ} (hd : Continuous d) (hcd : HasCompactSupport d) (s : Spins N) (j : Fin N) : |
| Integrable (fun x : Configuration N => ‖ψ.value s x‖^2 * CoulombAnalysis.tfPotential d (x j)) := by |
| obtain ⟨C,hC⟩ := compact_potential_bounded hd hcd |
| exact core_bounded_potential_integrable hψ |
| (CoulombAnalysis.tfPotential_continuous (hd.integrable_of_hasCompactSupport hcd) |
| (hd.memLp_of_hasCompactSupport hcd)) hC s j |
|
|
| def coreDensityInteraction {N : ℕ} (ψ : FormVector N) (d : Space → ℝ) : ℝ := |
| ∑ s : Spins N, ∑ j : Fin N, ∫ x : Configuration N, |
| ‖ψ.value s x‖^2 * CoulombAnalysis.tfPotential d (x j) |
|
|
| lemma spinLift_pole (u : Space → ℂ) (σ : Fin 2) (a : Space) : |
| orbitalPole (spinLift u σ) a = CoulombAnalysis.tfPotential (fun y => ‖u y‖^2) a := by |
| unfold orbitalPole CoulombAnalysis.tfPotential |
| simpa only [div_eq_mul_inv,mul_comm] using spinLift_weight u σ (fun x => ‖a-x‖⁻¹) |
|
|
| lemma spinLift_core_interaction {N : ℕ} (ψ : FormVector N) (u : Space → ℂ) (σ : Fin 2) : |
| orbitalCoreInteraction ψ (spinLift u σ) = coreDensityInteraction ψ (fun y => ‖u y‖^2) := by |
| simp only [orbitalCoreInteraction,coreDensityInteraction,spinLift_pole] |
|
|
| section Work_CompactDirect_scope |
|
|
| open MeasureTheory Filter |
| open scoped BigOperators ContDiff |
|
|
| lemma continuous_compact_coulomb_integrable {f g : Space → ℝ} |
| (hf : Continuous f) (hg : Continuous g) (hcf : HasCompactSupport f) (hcg : HasCompactSupport g) : |
| Integrable (fun r : Space × Space => f r.1 * g r.2 / ‖r.1-r.2‖) := by |
| have hm : Measurable (fun r : Space × Space => f r.1 * g r.2 / ‖r.1-r.2‖) := |
| ((hf.measurable.comp measurable_fst).mul (hg.measurable.comp measurable_snd)).div |
| ((measurable_fst.sub measurable_snd).norm) |
| have hg1 : Integrable g := hg.integrable_of_hasCompactSupport hcg |
| have hgp : MemLp g (5/3 : ENNReal) := hg.memLp_of_hasCompactSupport hcg (p := (5/3 : ENNReal)) |
| have hgn1 : Integrable (fun x => ‖g x‖) := hg.norm.integrable_of_hasCompactSupport hcg.norm |
| have hgnp : MemLp (fun x => ‖g x‖) (5/3 : ENNReal) := hg.norm.memLp_of_hasCompactSupport hcg.norm (p := (5/3 : ENNReal)) |
| apply (integrable_prod_iff hm.aestronglyMeasurable).2 |
| refine ⟨Eventually.of_forall (fun x => ?_),?_⟩ |
| · simpa only [mul_div_assoc] using (CoulombAnalysis.tfPotential_integrable hg1 hgp x).const_mul (f x) |
| · have hi : Integrable (fun x => ‖f x‖ * CoulombAnalysis.tfPotential (fun y => ‖g y‖) x) := (hf.norm.mul (CoulombAnalysis.tfPotential_continuous hgn1 hgnp)).integrable_of_hasCompactSupport |
| hcf.norm.mul_right |
| apply hi.congr |
| filter_upwards [] with x |
| simp only [norm_div,norm_mul,norm_norm,CoulombAnalysis.tfPotential,← integral_const_mul] |
| congr 1 |
| funext y |
| ring |
|
|
| lemma spatialFiniteDensity_continuous {ι : Type*} [Fintype ι] (p : ι → ℝ) {u : ι → Space → ℂ} |
| (hu : ∀ i, Continuous (u i)) : Continuous (spatialFiniteDensity p u) := by |
| unfold spatialFiniteDensity |
| fun_prop |
|
|
| lemma spatialFiniteDensity_compact {ι : Type*} [Fintype ι] (p : ι → ℝ) {u : ι → Space → ℂ} |
| (hu : ∀ i, HasCompactSupport (u i)) : HasCompactSupport (spatialFiniteDensity p u) := by |
| unfold spatialFiniteDensity |
| classical |
| induction (Finset.univ : Finset ι) using Finset.induction_on with |
| | empty => |
| simp only [Finset.sum_empty] |
| exact HasCompactSupport.zero |
| | @insert a s ha ih => |
| simp only [Finset.sum_insert ha] |
| exact ((compact_norm_sq (hu a)).mul_left).add ih |
|
|
| lemma spatialFiniteDensity_nonneg {ι : Type*} [Fintype ι] {p : ι → ℝ} {u : ι → Space → ℂ} |
| (hp : ∀ i, 0 ≤ p i) (x : Space) : 0 ≤ spatialFiniteDensity p u x := |
| Finset.sum_nonneg (fun i _ => mul_nonneg (hp i) (sq_nonneg _)) |
|
|
| lemma compact_direct_mono {f g : Space → ℝ} (hf : Continuous f) (hg : Continuous g) |
| (hcf : HasCompactSupport f) (hcg : HasCompactSupport g) |
| (hn : ∀ x, 0 ≤ f x) (hb : ∀ x, f x ≤ 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 |
| apply integral_mono (continuous_compact_coulomb_integrable hf hf hcf hcf) |
| (continuous_compact_coulomb_integrable hg hg hcg hcg) |
| intro r |
| exact div_le_div_of_nonneg_right (mul_le_mul (hb r.1) (hb r.2) (hn r.2) |
| ((hn r.1).trans (hb r.1))) (norm_nonneg _) |
|
|
| end Work_CompactDirect_scope |
|
|
| open MeasureTheory Filter Set Metric |
| open scoped BigOperators ContDiff |
|
|
| lemma bounded_compact_memLp {f : Space → ℝ} (hf : Measurable f) |
| {J : Set Space} (hJ : IsCompact J) (hs : Function.support f ⊆ J) |
| {B : ℝ} (hb : ∀ x, |f x| ≤ B) (p : ENNReal) : MemLp f p := by |
| have hi := memLp_indicator_const (μ := volume) p hJ.measurableSet B (Or.inr hJ.measure_lt_top.ne) |
| apply hi.mono' hf.aestronglyMeasurable |
| filter_upwards [] with x |
| by_cases hx : x ∈ J |
| · simpa only [Set.indicator_of_mem hx,Real.norm_eq_abs] using hb x |
| · have hz : f x = 0 := by by_contra hn; exact hx (hs hn) |
| simp only [Set.indicator_of_notMem hx,hz,norm_zero,le_refl] |
|
|
| lemma bounded_compact_integrable {f : Space → ℝ} (hf : Measurable f) |
| {J : Set Space} (hJ : IsCompact J) (hs : Function.support f ⊆ J) |
| {B : ℝ} (hb : ∀ x, |f x| ≤ B) : Integrable f := |
| memLp_one_iff_integrable.mp (bounded_compact_memLp hf hJ hs hb 1) |
|
|
| lemma compact_support_potential_bounded {f : Space → ℝ} (h1 : Integrable f) |
| (hp : MemLp f (5/3 : ENNReal)) {J : Set Space} (hJ : IsCompact J) |
| (hs : Function.support f ⊆ J) : |
| ∃ C : ℝ, ∀ x : Space, |CoulombAnalysis.tfPotential f x| ≤ C := by |
| obtain ⟨R,_,hR⟩ := hJ.isBounded.exists_pos_norm_le |
| have hr (y : Space) (hy : f y ≠ 0) : ‖y‖ ≤ R := hR y (hs hy) |
| obtain ⟨S,_,hS⟩ := CoulombAnalysis.tfPotential_compact_decay h1 hp hr 1 zero_lt_one |
| obtain ⟨C,hC⟩ := (isCompact_closedBall (0 : Space) S).exists_bound_of_continuousOn |
| (CoulombAnalysis.tfPotential_continuous h1 hp).continuousOn |
| refine ⟨max C 1,fun x => ?_⟩ |
| by_cases hx : ‖x‖ ≤ S |
| · exact (hC x (by simpa using hx)).trans (le_max_left _ _) |
| · exact (hS x (le_of_not_ge hx)).trans (le_max_right _ _) |
|
|
| lemma compact_support_coulomb_integrable {f g : Space → ℝ} |
| (hfm : Measurable f) (hgm : Measurable g) (hf : Integrable f) (hg : Integrable g) |
| (hgp : MemLp g (5/3 : ENNReal)) {J : Set Space} (hJ : IsCompact J) |
| (hs : Function.support g ⊆ J) : |
| Integrable (fun r : Space × Space => f r.1*g r.2/‖r.1-r.2‖) := by |
| have hm : Measurable (fun r : Space × Space => f r.1*g r.2/‖r.1-r.2‖) := by fun_prop |
| apply (integrable_prod_iff hm.aestronglyMeasurable).2 |
| refine ⟨Eventually.of_forall (fun x => ?_),?_⟩ |
| · simpa only [mul_div_assoc] using (CoulombAnalysis.tfPotential_integrable hg hgp x).const_mul (f x) |
| · obtain ⟨C,hC⟩ := compact_support_potential_bounded hg.norm hgp.norm hJ |
| (fun x hx => hs (by intro hz; exact hx (by simp [hz]))) |
| have hi : Integrable (fun x => ‖f x‖*CoulombAnalysis.tfPotential (fun y => ‖g y‖) x) := by |
| apply (hf.norm.mul_const C).mono' |
| ((hfm.norm.mul (CoulombAnalysis.tfPotential_continuous hg.norm hgp.norm).measurable).aestronglyMeasurable) |
| filter_upwards [] with x |
| change ‖‖f x‖*CoulombAnalysis.tfPotential (fun y => ‖g y‖) x‖ ≤ ‖f x‖*C |
| rw [norm_mul,norm_norm,Real.norm_eq_abs] |
| exact mul_le_mul_of_nonneg_left (hC x) (abs_nonneg _) |
| apply hi.congr |
| filter_upwards [] with x |
| simp only [norm_div,norm_mul,norm_norm,CoulombAnalysis.tfPotential,← integral_const_mul] |
| congr 1 |
| funext y |
| ring |
|
|
| lemma bounded_compact_coulomb_integrable {f g : Space → ℝ} |
| (hfm : Measurable f) (hgm : Measurable g) |
| {J K : Set Space} (hJ : IsCompact J) (hK : IsCompact K) |
| (hsf : Function.support f ⊆ J) (hsg : Function.support g ⊆ K) |
| {B C : ℝ} (hb : ∀ x, |f x| ≤ B) (hc : ∀ x, |g x| ≤ C) : |
| Integrable (fun r : Space × Space => f r.1*g r.2/‖r.1-r.2‖) := |
| compact_support_coulomb_integrable hfm hgm (bounded_compact_integrable hfm hJ hsf hb) |
| (bounded_compact_integrable hgm hK hsg hc) (bounded_compact_memLp hgm hK hsg hc _) hK hsg |
|
|
| end CoulombAtom |
|
|
| end |
|
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| end OAI |
|
|
