| import Mathlib |
| import OAI.Analysis.CoulombIonization.RadialBounds.Radial |
| import OAI.Analysis.CoulombIonization.Ionization.InsertionTermsDisjointSet |
| import OAI.Analysis.CoulombIonization.Localization.RadialPacketBase |
|
|
| namespace OAI |
|
|
| 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 |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| namespace CoulombAtom |
|
|
| lemma integral_one_position (f : Space → ℝ) : |
| (∫ x : Configuration 1, f (x 0)) = ∫ y, f y := by |
| exact (volume_preserving_funUnique (Fin 1) Space).integral_comp |
| (MeasurableEquiv.funUnique (Fin 1) Space).measurableEmbedding f |
|
|
| lemma integral_one_configuration (f : Configuration 1 → ℝ) : |
| (∫ y : Space, f (fun _ => y)) = ∫ x, f x := by |
| exact ((volume_preserving_funUnique (Fin 1) Space).symm |
| (MeasurableEquiv.funUnique (Fin 1) Space)).integral_comp |
| (MeasurableEquiv.funUnique (Fin 1) Space).symm.measurableEmbedding f |
|
|
| def packetDensity (y : Space) (R : ℝ) (x : Space) : ℝ := |
| ∑ s : Spins 1, ‖(radialLocalOrbital y R).value s (fun _ => x)‖^2 |
|
|
| lemma packetDensity_formula (y : Space) (R : ℝ) (x : Space) : |
| packetDensity y R x = (Fintype.card (Spins 1) : ℝ) * |
| (Real.sqrt (formMass (radialScaledRaw R)))⁻¹ ^ 2 * |
| ‖radialPacketBase (R⁻¹ • (x-y))‖^2 := by |
| simp only [packetDensity,radialLocalOrbital,scaleForm,norm_real_mul_sq,radialLocalRaw_mass] |
| simp only [radialLocalRaw,smoothForm,radialLocalSeed,radialScaledSeed, |
| radialPacketSeed,radialCenterShift,Pi.smul_apply,Pi.add_apply,Finset.sum_const, |
| Finset.card_univ,nsmul_eq_mul,sub_eq_add_neg,mul_assoc] |
|
|
| lemma packetDensity_continuous (y : Space) (R : ℝ) : Continuous (packetDensity y R) := by |
| change Continuous (fun x => packetDensity y R x) |
| simp_rw [packetDensity_formula] |
| have hc : Continuous (fun x : Space => R⁻¹ • (x-y)) := |
| (continuous_id.sub continuous_const).const_smul R⁻¹ |
| have hn : Continuous (fun x : Space => ‖radialPacketBase (R⁻¹ • (x-y))‖^2) := |
| (radialPacketBase_smooth.continuous.comp hc).norm.pow 2 |
| exact continuous_const.mul hn |
|
|
| lemma packetDensity_support (y : Space) {R : ℝ} (hR : 0 < R) {x : Space} |
| (hx : packetDensity y R x ≠ 0) : ‖x-y‖ ≤ R := by |
| by_contra hh |
| have hz := radialLocalOrbital_zero y hR (x := fun _ => x) (not_le.mp hh) |
| apply hx |
| unfold packetDensity |
| apply Finset.sum_eq_zero |
| intro s _ |
| rw [(hz s).1,norm_zero,zero_pow (by decide : (2:ℕ) ≠ 0)] |
|
|
| lemma packetDensity_compact (y : Space) {R : ℝ} (hR : 0 < R) : |
| HasCompactSupport (packetDensity y R) := by |
| apply HasCompactSupport.of_support_subset_isCompact (isCompact_closedBall y R) |
| intro x hx |
| exact (Metric.mem_closedBall).mpr (packetDensity_support y hR hx) |
|
|
| lemma packetDensity_integrable (y : Space) {R : ℝ} (hR : 0 < R) : |
| Integrable (packetDensity y R) := |
| (packetDensity_continuous y R).integrable_of_hasCompactSupport (packetDensity_compact y hR) |
|
|
| lemma packetDensity_memLp (y : Space) {R : ℝ} (hR : 0 < R) : |
| MemLp (packetDensity y R) (5/3) := |
| (packetDensity_continuous y R).memLp_of_hasCompactSupport (packetDensity_compact y hR) |
|
|
| lemma packetDensity_mass (y : Space) {R : ℝ} (hR : 0 < R) : |
| (∫ x, packetDensity y R x) = 1 := by |
| have hm := (radialLocalOrbital_admissible y hR).2.2.2.2.1 |
| change formMass (radialLocalOrbital y R) = 1 at hm |
| rw [← hm] |
| unfold packetDensity formMass |
| rw [integral_finsetSum] |
| · apply Finset.sum_congr rfl |
| intro s _ |
| exact integral_one_configuration (fun x => ‖(radialLocalOrbital y R).value s x‖^2) |
| · intro s _ |
| have hh := ((radialLocalOrbital_admissible y hR).1 s).integrable_norm_pow (by norm_num) |
| exact ((volume_preserving_funUnique (Fin 1) Space).symm |
| (MeasurableEquiv.funUnique (Fin 1) Space)).integrable_comp_of_integrable hh |
|
|
| lemma packetDensity_translate (y : Space) (R : ℝ) (x : Space) : |
| packetDensity y R x = packetDensity 0 R (x-y) := by |
| simp only [packetDensity_formula,sub_zero] |
|
|
| lemma packetDensity_radial (R : ℝ) : CoulombAnalysis.IsRadial (packetDensity 0 R) := by |
| intro x y hxy |
| simp only [packetDensity_formula,sub_zero,radialPacketBase,norm_smul,hxy] |
|
|
| theorem packetDensity_potential (y : Space) {R : ℝ} (hR : 0 < R) |
| {z : Space} (hz : R < ‖z-y‖) : |
| (∫ x, packetDensity y R x / ‖z-x‖) = 1 / ‖z-y‖ := by |
| have ht := integral_add_right_eq_self (μ := volume) |
| (fun x : Space => packetDensity y R x / ‖z-x‖) y |
| rw [← ht] |
| have he : (fun x => packetDensity y R (x+y) / ‖z-(x+y)‖) = |
| (fun x => packetDensity 0 R x / ‖(z-y)-x‖) := by |
| funext x |
| rw [packetDensity_translate,add_sub_cancel_right] |
| congr 2 |
| abel |
| rw [he] |
| have hn := CoulombAnalysis.tfPotential_newton_exterior |
| (packetDensity_integrable 0 hR) (packetDensity_memLp 0 hR) (packetDensity_radial R) |
| hR.le (fun x hx => by simpa only [sub_zero] using packetDensity_support 0 hR hx) hz |
| rw [packetDensity_mass 0 hR] at hn |
| exact hn |
|
|
| lemma packetDensity_eq_two (y : Space) (R : ℝ) (x : Space) (s : Spins 1) : |
| packetDensity y R x = 2 * ‖(radialLocalOrbital y R).value s (fun _ => x)‖^2 := by |
| simp only [packetDensity,radialLocalOrbital,scaleForm,radialLocalRaw,smoothForm, |
| Finset.sum_const,Finset.card_univ,nsmul_eq_mul] |
| congr 1 |
|
|
| lemma radialLocalOrbital_value_config (y : Space) (R : ℝ) (s : Spins 1) |
| (x : Configuration 1) : |
| (radialLocalOrbital y R).value s x = |
| (radialLocalOrbital y R).value s (fun _ => x 0) := by |
| congr 1 |
| funext i |
| exact congrArg x (Subsingleton.elim i 0) |
|
|
| lemma radialLocalOrbital_potential (y : Space) {R : ℝ} (hR : 0 < R) |
| {z : Space} (hz : R < ‖z-y‖) (s : Spins 1) : |
| (∫ x : Configuration 1, ‖(radialLocalOrbital y R).value s x‖^2 / ‖z-x 0‖) = |
| (1/2 : ℝ) / ‖z-y‖ := by |
| have ht := packetDensity_potential y hR hz |
| simp_rw [packetDensity_eq_two y R _ s,mul_div_assoc] at ht |
| rw [integral_const_mul] at ht |
| have he : (∫ x : Configuration 1, ‖(radialLocalOrbital y R).value s x‖^2 / ‖z-x 0‖) = |
| ∫ v : Space, ‖(radialLocalOrbital y R).value s (fun _ => v)‖^2 / ‖z-v‖ := by |
| rw [← integral_one_configuration] |
| rw [he] |
| calc |
| _ = (1 / ‖z-y‖) / 2 := by linarith |
| _ = _ := by ring |
|
|
| lemma radialLocalOrbital_nuclear (y : Space) {R : ℝ} (hR : 0 < R) |
| (hy : R < ‖y‖) : formNuclear (radialLocalOrbital y R) = 1 / ‖y‖ := by |
| have hh (s : Spins 1) := radialLocalOrbital_potential y hR |
| (z := 0) (by simpa only [zero_sub,norm_neg] using hy) s |
| simp only [zero_sub,norm_neg] at hh |
| unfold formNuclear |
| simp only [Fin.sum_univ_one,hh,Finset.sum_const,Finset.card_univ,nsmul_eq_mul] |
| norm_num [Spins] |
| ring |
|
|
| lemma integral_join_left {N M : ℕ} {f : Configuration (N+M) → ℝ} |
| (hf : Integrable f) : |
| (∫ x : Configuration N, ∫ y : Configuration M, f (joinLists x y)) = ∫ z, f z := by |
| have hj := integrable_join (N := N) (M := M) hf |
| have he := (configurationJoin_preserving N M).integral_comp |
| (configurationJoin N M).toHomeomorph.measurableEmbedding f |
| simp only [configurationJoin_apply_prod] at he |
| rw [Measure.volume_eq_prod] at hj he |
| exact (integral_prod _ hj).symm.trans he |
|
|
| def coreCoulombAt {N : ℕ} (ψ : FormVector N) (y : Space) : ℝ := |
| ∑ s : Spins N, ∑ i : Fin N, ∫ x, ‖ψ.value s x‖^2 / ‖x i-y‖ |
|
|
| lemma tensor_packet_pair {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) |
| (y : Space) {R B : ℝ} (hR : 0 < R) (hRB : R < B) |
| (hc : ∀ x i, ‖x i-y‖ < B → FormZeroAt ψ x) |
| (s : Spins N) (t : Spins 1) (i : Fin N) : |
| (∫ z, ‖(tensorForm ψ (radialLocalOrbital y R)).value (joinLists s t) z‖^2 / |
| ‖z i.castSucc-z (Fin.last N)‖) = |
| (1/2 : ℝ) * ∫ x, ‖ψ.value s x‖^2 / ‖x i-y‖ := by |
| have hφ := (radialLocalOrbital_admissible y hR).sobolevFermion.sobolevVector |
| have hT := hψ.tensor hφ |
| have hi := pair_integrable i.castSucc (Fin.last N) (Fin.castSucc_ne_last i) |
| (hT.1 (joinLists s t)) (hT.2.1 (joinLists s t) _) (hT.2.2 (joinLists s t) _) |
| rw [← integral_join_left hi] |
| change (∫ x : Configuration N, ∫ v : Configuration 1, |
| ‖(tensorForm ψ (radialLocalOrbital y R)).value (joinLists s t) (joinLists x v)‖^2 / |
| ‖joinLists x v (finSumFinEquiv (Sum.inl i)) - |
| joinLists x v (finSumFinEquiv (Sum.inr (0 : Fin 1)))‖) = _ |
| simp only [tensorForm_value_join,joinLists_left,joinLists_right,norm_mul,mul_pow, |
| mul_div_assoc,integral_const_mul] |
| rw [← integral_const_mul] |
| apply integral_congr_ae |
| apply Eventually.of_forall |
| intro x |
| by_cases hz : ψ.value s x = 0 |
| · simp only [hz,norm_zero,zero_pow (by decide : (2:ℕ) ≠ 0),zero_mul,zero_div,mul_zero] |
| · have hd : R < ‖x i-y‖ := hRB.trans_le (le_of_not_gt fun hh => hz ((hc x i hh s).1)) |
| dsimp only |
| rw [radialLocalOrbital_potential y hR hd] |
| ring |
|
|
| lemma tensorCross_packet {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) |
| (y : Space) {R B : ℝ} (hR : 0 < R) (hRB : R < B) |
| (hc : ∀ x i, ‖x i-y‖ < B → FormZeroAt ψ x) : |
| tensorCross ψ (radialLocalOrbital y R) = coreCoulombAt ψ y := by |
| unfold tensorCross coreCoulombAt |
| simp_rw [tensor_packet_pair hψ y hR hRB hc,← Finset.mul_sum] |
| simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul] |
| norm_num [Spins] |
| simp only [← Finset.mul_sum] |
| ring |
|
|
| theorem local_point_field {N : ℕ} {ψ : FormVector N} (hψ : SobolevFermion ψ) |
| (y : Space) {R B : ℝ} (hR : 0 < R) (hRB : R < B) (hy : R < ‖y‖) |
| (hc : ∀ x i, ‖x i-y‖ < B → FormZeroAt ψ x) |
| {Z lam : ℝ} (hZ : 0 ≤ Z) (hlam : 0 < lam) : |
| Z * formMass ψ / ‖y‖ - coreCoulombAt ψ y ≤ |
| (formEnergy Z ψ + lam*N*formMass ψ - priceEnergy (energy Z) lam * formMass ψ) + |
| formMass ψ * (radialPacketKinetic / R^2 + lam) := by |
| have hh := local_screened_insertion hψ (radialLocalOrbital_admissible y hR) |
| (Metric.ball y B) (fun x i hi => hc x i (Metric.mem_ball.mp hi)) |
| (fun x hx => radialLocalOrbital_zero y hR (hRB.trans_le (le_of_not_gt |
| (fun hh => hx (Metric.mem_ball.mpr hh))))) hZ hlam |
| rw [radialLocalOrbital_nuclear y hR hy,tensorCross_packet hψ.sobolevVector y hR hRB hc, |
| radialLocalOrbital_kinetic y hR] at hh |
| simpa only [mul_one_div] using hh |
|
|
| end CoulombAtom |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
|
|
| end |
|
|
| end OAI |
|
|
