| import Mathlib |
| import OAI.Analysis.CoulombIonization.Fermionic.Density |
| import OAI.Analysis.CoulombIonization.Variational.Space |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| open MeasureTheory Filter Complex TopologicalSpace |
| open scoped Topology InnerProductSpace ENNReal |
| open MeasureTheory Filter Complex |
| open scoped Topology BigOperators ComplexConjugate FourierTransform SchwartzMap ENNReal |
| open MeasureTheory Filter |
| open scoped Topology ContDiff SchwartzMap FourierTransform ENNReal |
| open MeasureTheory Filter |
| open scoped ContDiff InnerProductSpace Topology |
| open MeasureTheory Filter |
| open scoped ENNReal |
| open MeasureTheory Set Metric |
| open scoped ENNReal |
| namespace CoulombAtom |
|
|
| namespace Pauli |
| open CoulombDensity |
|
|
| def globalDensityConstant : ℝ := |
| 192 + (8192/3:ℝ)*(8*Real.pi*(3/8192:ℝ)^(-3/2:ℝ)+3) + 8192/3 |
|
|
| lemma globalDensityConstant_pos : 0 < globalDensityConstant := by |
| unfold globalDensityConstant |
| positivity |
|
|
| lemma density_nuclear_energy {N : ℕ} (ψ : FormVector (N+1)) (hψ : FormAdmissible ψ) (Z : ℝ) : |
| (∫ x, density ψ hψ.1 x*(Z/‖x‖)) = |
| Z*(∑ s : Spins (N+1), ∑ i : Fin (N+1), |
| ∫ x : Configuration (N+1), ‖ψ.value s x‖^2/‖x i‖) := by |
| calc |
| _ = ∫ x, Z*(density ψ hψ.1 x/‖x‖) := by |
| apply integral_congr_ae |
| exact Filter.Eventually.of_forall (fun x => by ring) |
| _ = _ := by rw [integral_const_mul,(density_coulomb_integral ψ hψ).2] |
|
|
| theorem global_density_of_energy_le {N : ℕ} {Z δ : ℝ} |
| (hZ : 1 ≤ Z) (hδ : 0 ≤ δ) (hN : (N+1:ℝ) ≤ 3*Z) |
| (ψ : FormVector (N+1)) (hψ : FormAdmissible ψ) (he : formEnergy Z ψ ≤ δ) : |
| (∫ x, density ψ hψ.1 x^(5/3:ℝ)) ≤ globalDensityConstant * (Z^(7/3:ℝ)+δ) := by |
| let D := ∫ x, density ψ hψ.1 x^(5/3:ℝ) |
| let G := ∑ s : Spins (N+1), ∑ i : Fin (N+1), ∑ a : Fin 3, |
| ∫ x : Configuration (N+1), ‖ψ.gradient s i a x‖^2 |
| let A := ∑ s : Spins (N+1), ∑ i : Fin (N+1), |
| ∫ x : Configuration (N+1), ‖ψ.value s x‖^2/‖x i‖ |
| let B := 8*Real.pi*(3/8192:ℝ)^(-3/2:ℝ)+3 |
| have hDp : 0 ≤ D := integral_nonneg (fun _ => Real.rpow_nonneg (density_nonneg _ _ _) _) |
| have hB : 0 ≤ B := by dsimp [B]; positivity |
| have hZp : 0 < Z := lt_of_lt_of_le zero_lt_one hZ |
| have hpow : Z ≤ Z^(7/3:ℝ) := by |
| calc |
| _ = Z^(1:ℝ) := (Real.rpow_one _).symm |
| _ ≤ _ := Real.rpow_le_rpow_of_exponent_le hZ (by norm_num) |
| have hp : 0 ≤ Z^(7/3:ℝ) := Real.rpow_nonneg hZp.le _ |
| have hlt : D ≤ 32*(N+1:ℝ)+(2048/3:ℝ)*G := density_power_bound ψ hψ |
| have ha : Z*A ≤ (3/8192:ℝ)*D + B*Z^(7/3:ℝ) := by |
| have h := nuclear_density_scaled (density ψ hψ.1) (density_nonneg ψ hψ.1) |
| (density_integrable ψ hψ.1) (density_power_integrable ψ hψ) hZp |
| (by norm_num : (0:ℝ) < 3/8192) (by rw [density_mass ψ hψ]; exact hN) |
| rwa [density_nuclear_energy ψ hψ Z] at h |
| have hr : 0 ≤ (∑ s : Spins (N+1), ∑ i : Fin (N+1), ∑ j : Fin (N+1), |
| if i < j then (∫ x : Configuration (N+1), ‖ψ.value s x‖^2/‖x i-x j‖) else 0) := by |
| apply Finset.sum_nonneg |
| intro s _ |
| apply Finset.sum_nonneg |
| intro i _ |
| apply Finset.sum_nonneg |
| intro j _ |
| split_ifs <;> positivity |
| have hkin : (1/2:ℝ)*G - Z*A ≤ δ := by |
| dsimp [formEnergy] at he |
| dsimp [G,A] |
| linarith |
| have hbase : D ≤ (192+(8192/3:ℝ)*B)*Z^(7/3:ℝ)+(8192/3:ℝ)*δ := by |
| nlinarith |
| have hc : globalDensityConstant = 192+(8192/3:ℝ)*B+8192/3 := rfl |
| rw [hc] |
| nlinarith |
| end Pauli |
| end CoulombAtom |
|
|
| end |
|
|
| end OAI |
|
|
