| import Mathlib |
| import OAI.Analysis.CoulombIonization.FormDomain.Form |
| import OAI.Analysis.CoulombIonization.Fermionic.SlaterSymmetry |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace CoulombAtom |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
|
|
| open MeasureTheory Filter |
| open scoped BigOperators ComplexConjugate |
|
|
| lemma pi_count_eq_count {ι β : Type*} [Fintype ι] [Fintype β] |
| [MeasurableSpace β] [MeasurableSingletonClass β] : |
| (Measure.pi fun _ : ι => (Measure.count : Measure β)) = Measure.count := by |
| apply Measure.ext_of_singleton |
| intro x |
| rw [Measure.pi_singleton, Measure.count_singleton] |
| simp only [Measure.count_singleton, Finset.prod_const_one] |
|
|
| lemma spinSpace_measurePreserving (n : ℕ) : |
| MeasurePreserving (MeasurableEquiv.arrowProdEquivProdArrow (Fin 2) Space (Fin n)) |
| (Measure.pi fun _ : Fin n => (Measure.count : Measure (Fin 2)).prod volume) |
| ((Measure.count : Measure (Spins n)).prod volume) := by |
| simpa only [pi_count_eq_count, ← volume_pi] using |
| measurePreserving_arrowProdEquivProdArrow (Fin 2) Space (Fin n) |
| (fun _ => (Measure.count : Measure (Fin 2))) (fun _ => (volume : Measure Space)) |
|
|
| lemma spinSpace_integral {n : ℕ} {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
| (F : (Fin n → Fin 2 × Space) → E) |
| (hF : Integrable F (Measure.pi fun _ : Fin n => |
| (Measure.count : Measure (Fin 2)).prod volume)) : |
| (∫ z, F z ∂Measure.pi (fun _ : Fin n => (Measure.count : Measure (Fin 2)).prod volume)) = |
| ∑ s : Spins n, ∫ x : Configuration n, F (fun i => (s i,x i)) := by |
| have hp := (spinSpace_measurePreserving n).symm |
| have hi := (hp.integrable_comp_emb |
| (MeasurableEquiv.arrowProdEquivProdArrow (Fin 2) Space (Fin n)).symm.measurableEmbedding).2 hF |
| rw [← hp.integral_comp' F] |
| change Integrable (fun z : Spins n × Configuration n => |
| F (fun i => (z.1 i,z.2 i))) (Measure.count.prod volume) at hi |
| change (∫ z : Spins n × Configuration n, F (fun i => (z.1 i,z.2 i)) |
| ∂Measure.count.prod volume) = _ |
| rw [integral_prod _ hi, integral_count] |
|
|
| open MeasureTheory Filter |
| open scoped BigOperators ComplexConjugate ContDiff Topology |
|
|
| abbrev slaterParticleMeasure : Measure SlaterParticle := |
| (Measure.count : Measure (Fin 2)).prod volume |
|
|
| lemma compact_spinSpace {f : SlaterParticle → ℂ} |
| (hc : ∀ s, HasCompactSupport (fun x : Space => f (s,x))) : HasCompactSupport f := by |
| apply HasCompactSupport.of_support_subset_isCompact |
| (isCompact_univ.prod (isCompact_iUnion hc)) |
| intro z hz |
| exact ⟨Set.mem_univ _, Set.mem_iUnion.mpr ⟨z.1, subset_tsupport _ hz⟩⟩ |
|
|
| lemma memLp_spinSpace {f : SlaterParticle → ℂ} |
| (hf : ∀ s, Continuous (fun x : Space => f (s,x))) |
| (hc : ∀ s, HasCompactSupport (fun x : Space => f (s,x))) : |
| MemLp f 2 slaterParticleMeasure := |
| (continuous_prod_of_discrete_left.mpr hf).memLp_of_hasCompactSupport (compact_spinSpace hc) |
|
|
| lemma memLp_spatialOrbitalDerivative {n : ℕ} {φ : Fin n → SlaterParticle → ℂ} |
| (hφ : ∀ i s, ContDiff ℝ ∞ (fun x : Space => φ i (s,x))) |
| (hc : ∀ i s, HasCompactSupport (fun x : Space => φ i (s,x))) |
| (a : Fin 3) (i : Fin n) : |
| MemLp (spatialOrbitalDerivative φ a i) 2 slaterParticleMeasure := |
| memLp_spinSpace (fun s => smooth_derivative_continuous (hφ i s) _) |
| (fun s => compact_derivative (hc i s) _) |
|
|
| lemma slaterForm_mass {n : ℕ} {φ : Fin n → SlaterParticle → ℂ} |
| (hφ : ∀ i, MemLp (φ i) 2 slaterParticleMeasure) |
| (ho : ∀ i k, (∫ z, conj (φ i z) * φ k z ∂slaterParticleMeasure) = if i=k then 1 else 0) : |
| formMass (slaterForm φ) = 1 := by |
| have he := spinSpace_integral (fun z => ‖slater φ z‖^2) |
| (memLp_slater hφ).norm.integrable_sq |
| have hn := slater_norm_sq_integral hφ ho |
| rw [he] at hn |
| exact hn |
|
|
| theorem slaterForm_admissible {n : ℕ} {φ : Fin n → SlaterParticle → ℂ} |
| (hφ : ∀ i s, ContDiff ℝ ∞ (fun x : Space => φ i (s,x))) |
| (hc : ∀ i s, HasCompactSupport (fun x : Space => φ i (s,x))) |
| (ho : ∀ i k, (∫ z, conj (φ i z) * φ k z ∂slaterParticleMeasure) = if i=k then 1 else 0) : |
| FormAdmissible (slaterForm φ) := |
| (slaterForm_sobolevFermion hφ hc).admissible |
| (slaterForm_mass (fun i => memLp_spinSpace (fun s => (hφ i s).continuous) (hc i)) ho) |
|
|
| theorem slaterForm_kinetic {n : ℕ} {φ : Fin n → SlaterParticle → ℂ} |
| (hφ : ∀ i s, ContDiff ℝ ∞ (fun x : Space => φ i (s,x))) |
| (hc : ∀ i s, HasCompactSupport (fun x : Space => φ i (s,x))) |
| (ho : ∀ i k, (∫ z, conj (φ i z) * φ k z ∂slaterParticleMeasure) = if i=k then 1 else 0) : |
| formKinetic (slaterForm φ) = (1/2:ℝ) * ∑ a : Fin 3, ∑ i : Fin n, |
| ∫ z, ‖spatialOrbitalDerivative φ a i z‖^2 ∂slaterParticleMeasure := by |
| have hl (i : Fin n) : MemLp (φ i) 2 slaterParticleMeasure := |
| memLp_spinSpace (fun s => (hφ i s).continuous) (hc i) |
| have hd := memLp_spatialOrbitalDerivative hφ hc |
| have he (a : Fin 3) (j : Fin n) := spinSpace_integral |
| (fun z => ‖slaterCoordinate φ (spatialOrbitalDerivative φ a) j z‖^2) |
| (memLp_slaterCoordinate hl (hd a) j).norm.integrable_sq |
| unfold formKinetic |
| congr 1 |
| simp only [slaterForm] |
| rw [Finset.sum_comm] |
| simp_rw [Finset.sum_comm (s := (Finset.univ : Finset (Spins n))) (t := (Finset.univ : Finset (Fin 3)))] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro a _ |
| calc |
| _ = ∑ j : Fin n, ∫ z, ‖slaterCoordinate φ (spatialOrbitalDerivative φ a) j z‖^2 |
| ∂Measure.pi (fun _ : Fin n => slaterParticleMeasure) := by |
| apply Finset.sum_congr rfl |
| intro j _ |
| exact (he a j).symm |
| _ = _ := slaterCoordinate_sum_norm_sq_integral hl (hd a) ho |
|
|
| end CoulombAtom |
|
|
| end |
|
|
| end OAI |
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|
