| import Mathlib |
| import OAI.Analysis.CoulombIonization.Fermionic.Slater |
|
|
| namespace OAI |
|
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| noncomputable section |
|
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| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
|
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| open MeasureTheory Filter |
| open scoped BigOperators ComplexConjugate |
|
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| namespace CoulombAtom |
|
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| lemma complex_integral_norm_sq {α : Type*} [MeasurableSpace α] |
| {μ : Measure α} (f : α → ℂ) : |
| (∫ x, conj (f x) * f x ∂μ) = (∫ x, ‖f x‖^2 ∂μ : ℝ) := by |
| calc |
| _ = ∫ x, (Complex.ofReal (‖f x‖^2)) ∂μ := by |
| apply integral_congr_ae |
| exact Filter.Eventually.of_forall (fun x => by |
| simpa only [Complex.sq_norm] using |
| (Complex.normSq_eq_conj_mul_self (z := f x)).symm) |
| _ = _ := integral_ofReal (𝕜 := ℂ) |
|
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| lemma integral_norm_sq_weighted_sum {α ι : Type*} [MeasurableSpace α] |
| {μ : Measure α} [Fintype ι] (a : ι → ℂ) (f : ι → α → ℂ) |
| (hf : ∀ i, MemLp (f i) 2 μ) : |
| (Complex.ofReal (∫ x, ‖∑ i, a i * f i x‖^2 ∂μ)) = |
| ∑ i, ∑ j, conj (a i) * a j * (∫ x, conj (f i x) * f j x ∂μ) := by |
| rw [← complex_integral_norm_sq] |
| have hi (i j : ι) : Integrable (fun x => conj (a i) * a j * |
| (conj (f i x) * f j x)) μ := |
| ((hf i).star.integrable_mul (hf j)).const_mul _ |
| have he (x : α) : conj (∑ i, a i * f i x) * (∑ i, a i * f i x) = |
| ∑ i, ∑ j, conj (a i) * a j * (conj (f i x) * f j x) := by |
| simp only [map_sum, map_mul] |
| rw [Finset.sum_mul] |
| simp_rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring |
| simp_rw [he] |
| rw [integral_finsetSum _ (fun i _ => integrable_finsetSum _ (fun j _ => hi i j))] |
| simp_rw [integral_finsetSum _ (fun j _ => hi _ j), integral_const_mul] |
|
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| variable {α : Type*} [MeasurableSpace α] {μ : Measure α} [SigmaFinite μ] {n : ℕ} |
|
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| def slaterCoordinateTensor (φ ψ : Fin n → α → ℂ) (j : Fin n) |
| (σ : Equiv.Perm (Fin n)) (x : Fin n → α) : ℂ := |
| ∏ i, (if i=j then ψ else φ) (σ i) (x i) |
|
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| def slaterCoordinateRaw (φ ψ : Fin n → α → ℂ) (j : Fin n) (x : Fin n → α) : ℂ := |
| ∑ σ : Equiv.Perm (Fin n), ((σ.sign : ℤ) : ℂ) * slaterCoordinateTensor φ ψ j σ x |
|
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| lemma memLp_slaterCoordinateTensor {φ ψ : Fin n → α → ℂ} |
| (hφ : ∀ i, MemLp (φ i) 2 μ) (hψ : ∀ i, MemLp (ψ i) 2 μ) |
| (j : Fin n) (σ : Equiv.Perm (Fin n)) : |
| MemLp (slaterCoordinateTensor φ ψ j σ) 2 (Measure.pi fun _ : Fin n => μ) := by |
| have h (i : Fin n) : MemLp ((if i=j then ψ else φ) (σ i)) 2 μ := by |
| split_ifs <;> first | exact hψ _ | exact hφ _ |
| have hm : AEStronglyMeasurable (slaterCoordinateTensor φ ψ j σ) |
| (Measure.pi fun _ : Fin n => μ) := by |
| unfold slaterCoordinateTensor |
| apply Finset.aestronglyMeasurable_fun_prod |
| intro i _ |
| exact (h i).aestronglyMeasurable.comp_quasiMeasurePreserving |
| (Measure.quasiMeasurePreserving_eval (fun _ : Fin n => μ) i) |
| apply (memLp_two_iff_integrable_sq_norm hm).mpr |
| simpa only [slaterCoordinateTensor, norm_prod, Finset.prod_pow] using |
| Integrable.fintype_prod (fun i : Fin n => (h i).norm.integrable_sq) |
|
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| lemma memLp_slaterCoordinateRaw {φ ψ : Fin n → α → ℂ} |
| (hφ : ∀ i, MemLp (φ i) 2 μ) (hψ : ∀ i, MemLp (ψ i) 2 μ) (j : Fin n) : |
| MemLp (slaterCoordinateRaw φ ψ j) 2 (Measure.pi fun _ : Fin n => μ) := |
| memLp_finsetSum _ (fun σ _ => (memLp_slaterCoordinateTensor hφ hψ j σ).const_mul _) |
|
|
| lemma slaterCoordinateTensor_inner (φ ψ : Fin n → α → ℂ) |
| (ho : ∀ i k, (∫ t, conj (φ i t) * φ k t ∂μ) = if i=k then 1 else 0) |
| (j : Fin n) (σ τ : Equiv.Perm (Fin n)) : |
| (∫ x, conj (slaterCoordinateTensor φ ψ j σ x) * slaterCoordinateTensor φ ψ j τ x |
| ∂Measure.pi (fun _ : Fin n => μ)) = |
| if σ=τ then (∫ t, ‖ψ (σ j) t‖^2 ∂μ : ℝ) else 0 := by |
| simp only [slaterCoordinateTensor, map_prod, ← Finset.prod_mul_distrib] |
| rw [integral_fintype_prod_eq_prod (μ := fun _ : Fin n => μ) |
| (fun i t => conj ((if i=j then ψ else φ) (σ i) t) * |
| (if i=j then ψ else φ) (τ i) t)] |
| by_cases he : σ=τ |
| · subst τ |
| rw [ite_eq_left rfl, Finset.prod_eq_single j] |
| · simpa only [ite_eq_left (show j=j from rfl), ite_true] using |
| complex_integral_norm_sq (μ := μ) (ψ (σ j)) |
| · intro i _ hij |
| rw [ite_eq_right hij, ho, ite_eq_left rfl] |
| · simp |
| · rw [ite_eq_right he] |
| have hh : ∃ i, i ≠ j ∧ σ i ≠ τ i := by |
| by_contra! hh |
| exact he (permutations_eq_of_agree_except σ τ j hh) |
| obtain ⟨i,hij,hi⟩ := hh |
| apply Finset.prod_eq_zero (Finset.mem_univ i) |
| rw [ite_eq_right hij, ho, ite_eq_right hi] |
|
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| lemma slaterCoordinateRaw_norm_sq_integral {φ ψ : Fin n → α → ℂ} |
| (hφ : ∀ i, MemLp (φ i) 2 μ) (hψ : ∀ i, MemLp (ψ i) 2 μ) |
| (ho : ∀ i k, (∫ t, conj (φ i t) * φ k t ∂μ) = if i=k then 1 else 0) |
| (j : Fin n) : |
| (∫ x, ‖slaterCoordinateRaw φ ψ j x‖^2 ∂Measure.pi (fun _ : Fin n => μ)) = |
| ∑ σ : Equiv.Perm (Fin n), ∫ t, ‖ψ (σ j) t‖^2 ∂μ := by |
| apply Complex.ofReal_injective |
| rw [Complex.ofReal_sum] |
| change (Complex.ofReal (∫ x, ‖∑ σ : Equiv.Perm (Fin n), |
| ((σ.sign : ℤ) : ℂ) * slaterCoordinateTensor φ ψ j σ x‖^2 |
| ∂Measure.pi (fun _ : Fin n => μ))) = _ |
| rw [integral_norm_sq_weighted_sum _ _ (memLp_slaterCoordinateTensor hφ hψ j)] |
| apply Finset.sum_congr rfl |
| intro σ _ |
| simp_rw [slaterCoordinateTensor_inner φ ψ ho] |
| rw [Finset.sum_eq_single σ] |
| · simp only [ite_true, map_intCast] |
| rw [← pow_two, complex_sign_sq, one_mul] |
| · intro τ _ ht |
| simp only [ite_eq_right (Ne.symm ht), Complex.ofReal_zero, mul_zero] |
| · simp |
|
|
| lemma slaterCoordinateRaw_sum_norm_sq_integral {φ ψ : Fin n → α → ℂ} |
| (hφ : ∀ i, MemLp (φ i) 2 μ) (hψ : ∀ i, MemLp (ψ i) 2 μ) |
| (ho : ∀ i k, (∫ t, conj (φ i t) * φ k t ∂μ) = if i=k then 1 else 0) : |
| (∑ j : Fin n, ∫ x, ‖slaterCoordinateRaw φ ψ j x‖^2 |
| ∂Measure.pi (fun _ : Fin n => μ)) = |
| (n.factorial : ℝ) * ∑ i, ∫ t, ‖ψ i t‖^2 ∂μ := by |
| simp_rw [slaterCoordinateRaw_norm_sq_integral hφ hψ ho] |
| rw [Finset.sum_comm] |
| have he (σ : Equiv.Perm (Fin n)) : |
| (∑ j, ∫ t, ‖ψ (σ j) t‖^2 ∂μ) = ∑ i, ∫ t, ‖ψ i t‖^2 ∂μ := |
| Equiv.sum_comp σ (fun i => ∫ t, ‖ψ i t‖^2 ∂μ) |
| simp_rw [he] |
| simp only [Finset.sum_const, Finset.card_univ, Fintype.card_perm, Fintype.card_fin, |
| nsmul_eq_mul] |
|
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| end CoulombAtom |
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| end |
|
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| end OAI |
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