| import Mathlib |
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| namespace OAI |
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| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
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| noncomputable section |
| namespace CoulombAtom |
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| abbrev Space := EuclideanSpace ℝ (Fin 3) |
| abbrev Configuration (N : ℕ) := Fin N → Space |
| abbrev Spins (N : ℕ) := Fin N → Fin 2 |
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| structure FormVector (N : ℕ) where |
| value : Spins N → Configuration N → ℂ |
| gradient : Spins N → Fin N → Fin 3 → Configuration N → ℂ |
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| def direction {N : ℕ} (i : Fin N) (a : Fin 3) : Configuration N := |
| Pi.single i (EuclideanSpace.single a 1) |
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| def FormAdmissible {N : ℕ} (ψ : FormVector N) : Prop := |
| (∀ s, MemLp (ψ.value s) 2) ∧ |
| (∀ s i a, MemLp (ψ.gradient s i a) 2) ∧ |
| (∀ s i a (φ : Configuration N → ℝ), |
| ContDiff ℝ ∞ φ → HasCompactSupport φ → |
| (∫ x, ψ.value s x * Complex.ofReal (lineDeriv ℝ φ x (direction i a))) = |
| -(∫ x, ψ.gradient s i a x * (φ x : ℂ))) ∧ |
| (∀ (π : Equiv.Perm (Fin N)) s, ∀ᵐ x, |
| ψ.value (s ∘ π) (x ∘ π) = |
| (((Equiv.Perm.sign π : ℤ) : ℂ) * ψ.value s x)) ∧ |
| (∑ s : Spins N, ∫ x, ‖ψ.value s x‖ ^ 2) = 1 ∧ |
| (∀ s i, Integrable (fun x => ‖ψ.value s x‖ ^ 2 / ‖x i‖)) ∧ |
| (∀ s i j, i ≠ j → |
| Integrable (fun x => ‖ψ.value s x‖ ^ 2 / ‖x i - x j‖)) |
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| def formEnergy {N : ℕ} (Z : ℝ) (ψ : FormVector N) : ℝ := |
| (1 / 2 : ℝ) * |
| (∑ s : Spins N, ∑ i : Fin N, ∑ a : Fin 3, |
| (∫ x, ‖ψ.gradient s i a x‖ ^ 2)) - |
| Z * (∑ s : Spins N, ∑ i : Fin N, |
| (∫ x, ‖ψ.value s x‖ ^ 2 / ‖x i‖)) + |
| (∑ s : Spins N, ∑ i : Fin N, ∑ j : Fin N, |
| if i < j then (∫ x, ‖ψ.value s x‖ ^ 2 / ‖x i - x j‖) else 0) |
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| def energy (Z : ℝ) (N : ℕ) : ℝ := |
| if N = 0 then 0 else sInf {e | ∃ ψ : FormVector N, |
| FormAdmissible ψ ∧ formEnergy Z ψ = e} |
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| def ionization (m Z : ℕ) : ℝ := |
| energy Z (Z - m) - energy Z Z |
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| def tfKinetic : ℝ := (3 / 10 : ℝ) * (3 * Real.pi ^ 2) ^ (2 / 3 : ℝ) |
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| def directIntegrand (ρ : Space → ℝ) (p : Space × Space) : ℝ := |
| ρ p.1 * ρ p.2 / ‖p.1 - p.2‖ |
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| def TFAdmissible (M : ℝ) (ρ : Space → ℝ) : Prop := |
| (∀ᵐ x, 0 ≤ ρ x) ∧ Integrable ρ ∧ (∫ x, ρ x) = M ∧ |
| Integrable (fun x => ρ x ^ (5 / 3 : ℝ)) ∧ |
| Integrable (fun x => ρ x / ‖x‖) ∧ Integrable (directIntegrand ρ) |
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| def tfFunctional (Z : ℝ) (ρ : Space → ℝ) : ℝ := |
| tfKinetic * (∫ x, ρ x ^ (5 / 3 : ℝ)) - Z * (∫ x, ρ x / ‖x‖) + |
| (1 / 2 : ℝ) * ∫ p, directIntegrand ρ p |
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| def tfEnergy (Z M : ℝ) : ℝ := |
| sInf {e | ∃ ρ : Space → ℝ, TFAdmissible M ρ ∧ tfFunctional Z ρ = e} |
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| def tfIonization (m Z : ℝ) : ℝ := tfEnergy Z (Z - m) - tfEnergy Z Z |
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| def TFCharacterization (a : ℝ) : Prop := |
| ∀ m : ℝ, 0 < m → |
| Tendsto (tfIonization m) atTop (𝓝 (a * m ^ (7 / 3 : ℝ))) |
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| def JointLimit (a : ℝ) : Prop := |
| ∀ m Z : ℕ → ℕ, (∀ j, 1 ≤ m j ∧ m j < Z j) → |
| Tendsto m atTop atTop → |
| Tendsto (fun j => (Z j : ℝ) / (m j : ℝ)) atTop atTop → |
| Tendsto (fun j => ionization (m j) (Z j) / (m j : ℝ) ^ (7 / 3 : ℝ)) |
| atTop (𝓝 a) |
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| def IteratedLimits (a : ℝ) : Prop := |
| Tendsto (fun m : ℕ => (limsup (ionization m) atTop) / (m : ℝ) ^ (7 / 3 : ℝ)) |
| atTop (𝓝 a) ∧ |
| Tendsto (fun m : ℕ => (liminf (ionization m) atTop) / (m : ℝ) ^ (7 / 3 : ℝ)) |
| atTop (𝓝 a) |
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| def MainStatement : Prop := |
| ∃ a : ℝ, 0 < a ∧ TFCharacterization a ∧ JointLimit a ∧ IteratedLimits a |
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| end CoulombAtom |
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| end |
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| end OAI |
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