| import Mathlib |
| import OAI.Analysis.CoulombIonization.FormDomain.FormRawPotential |
|
|
| 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 Set |
| open scoped BigOperators |
|
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| namespace CoulombAtom |
|
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| lemma rawPotential_weighted_integrable {N : ℕ} {ψ : FormVector N} |
| (hψ : SobolevVector ψ) (s : Spins N) (y : Space) : |
| Integrable (fun x => rawPotential y x*‖ψ.value s x‖^2) := by |
| simp only [rawPotential,Finset.sum_mul] |
| apply integrable_finsetSum |
| intro i _ |
| simpa only [one_div,mul_comm,←div_eq_mul_inv] using hψ.shifted_nuclear_integrable s i y |
|
|
| lemma integral_rawPotential_formRawLaw {N : ℕ} {ψ : FormVector N} |
| (hψ : SobolevVector ψ) (y : Space) : |
| (∫ x, rawPotential y x ∂formRawLaw ψ) = coreCoulombAt ψ y := by |
| rw [formRawLaw_integral hψ] |
| simp only [formRawDensity,Finset.sum_mul] |
| rw [integral_finsetSum _ (fun s _ => by |
| simpa only [mul_comm] using rawPotential_weighted_integrable hψ s y)] |
| unfold coreCoulombAt |
| congr 1 |
| ext s |
| simp only [rawPotential,Finset.mul_sum,mul_one_div] |
| exact integral_finsetSum _ (fun i _ => hψ.shifted_nuclear_integrable s i y) |
|
|
| lemma coreSlice_coreCoulombAt_integrable {N M : ℕ} {ψ : FormVector (N+M)} |
| (hψ : SobolevVector ψ) (t : Spins M) (y : Space) : |
| Integrable (fun u : Configuration M => coreCoulombAt (coreSlice ψ t u) y) := by |
| unfold coreCoulombAt |
| apply integrable_finsetSum |
| intro s _ |
| apply integrable_finsetSum |
| intro i _ |
| have hh := (integrable_join (N := N) (M := M) |
| (hψ.shifted_nuclear_integrable (joinLists s t) (finSumFinEquiv (Sum.inl i)) y)).integral_prod_right |
| simpa only [coreSlice,joinLists_left] using hh |
|
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| lemma rawPotential_coreSlice_integrable {N M : ℕ} {ψ : FormVector (N+M)} |
| (hψ : SobolevVector ψ) (t : Spins M) (y : Space) : |
| Integrable (fun u : Configuration M => rawPotential y u*formMass (coreSlice ψ t u)) := by |
| simp only [rawPotential,Finset.sum_mul] |
| apply integrable_finsetSum |
| intro i _ |
| simpa only [one_div,mul_comm,←div_eq_mul_inv] using coreSlice_mass_shifted_integrable hψ t i y |
|
|
| lemma integral_coreSlice_coreCoulombAt {N M : ℕ} {ψ : FormVector (N+M)} |
| (hψ : SobolevVector ψ) (y : Space) : |
| (∑ t : Spins M, ∫ u : Configuration M, coreCoulombAt (coreSlice ψ t u) y) = |
| ∑ s : Spins (N+M), ∑ i : Fin N, ∫ x, ‖ψ.value s x‖^2/‖x (finSumFinEquiv (Sum.inl i))-y‖ := by |
| have hi (s : Spins N) (t : Spins M) (i : Fin N) : Integrable (fun u : Configuration M => |
| ∫ v : Configuration N, ‖ψ.value (joinLists s t) (joinLists v u)‖^2/‖v i-y‖) := by |
| simpa only [joinLists_left] using (integrable_join (N := N) (M := M) |
| (hψ.shifted_nuclear_integrable (joinLists s t) (finSumFinEquiv (Sum.inl i)) y)).integral_prod_right |
| simp only [coreCoulombAt,coreSlice] |
| simp_rw [integral_finsetSum _ (fun s _ => integrable_finsetSum _ (fun i _ => hi s _ i)), |
| integral_finsetSum _ (fun i _ => hi _ _ i)] |
| have he (s : Spins N) (t : Spins M) (i : Fin N) : |
| (∫ u : Configuration M, ∫ v : Configuration N, |
| ‖ψ.value (joinLists s t) (joinLists v u)‖^2/‖v i-y‖) = |
| ∫ x, ‖ψ.value (joinLists s t) x‖^2/‖x (finSumFinEquiv (Sum.inl i))-y‖ := by |
| simpa only [joinLists_left] using integral_join (N := N) (M := M) |
| (hψ.shifted_nuclear_integrable (joinLists s t) (finSumFinEquiv (Sum.inl i)) y) |
| simp_rw [he] |
| exact sum_spin_join (fun s : Spins (N+M) => ∑ i : Fin N, |
| ∫ x, ‖ψ.value s x‖^2/‖x (finSumFinEquiv (Sum.inl i))-y‖) |
|
|
| lemma integral_rawPotential_coreSlice {N M : ℕ} {ψ : FormVector (N+M)} |
| (hψ : SobolevVector ψ) (y : Space) : |
| (∑ t : Spins M, ∫ u : Configuration M, rawPotential y u*formMass (coreSlice ψ t u)) = |
| ∑ s : Spins (N+M), ∑ i : Fin M, ∫ x, ‖ψ.value s x‖^2/‖x (finSumFinEquiv (Sum.inr i))-y‖ := by |
| have hi (s : Spins (N+M)) : Integrable (fun x => |
| rawPotential y (outProjection x)*‖ψ.value s x‖^2) := by |
| simp only [rawPotential,Finset.sum_mul,outProjection] |
| apply integrable_finsetSum |
| intro i _ |
| simpa only [one_div,mul_comm,←div_eq_mul_inv] using |
| hψ.shifted_nuclear_integrable s (finSumFinEquiv (Sum.inr i)) y |
| rw [weighted_coreSlice_eq_full_integrable _ _ hi] |
| apply Finset.sum_congr rfl |
| intro s _ |
| simp only [rawPotential,outProjection] |
| simp only [Finset.sum_mul,div_mul_eq_mul_div,one_mul] |
| exact integral_finsetSum _ (fun i _ => hψ.shifted_nuclear_integrable s (finSumFinEquiv (Sum.inr i)) y) |
|
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| lemma fresh_core_outer_potential_tower {N M : ℕ} {ψ : FormVector (N+M)} |
| (hψ : SobolevVector ψ) (y : Space) : |
| (∑ t : Spins M, ∫ u : Configuration M, |
| coreCoulombAt (coreSlice ψ t u) y+rawPotential y u*formMass (coreSlice ψ t u)) = |
| coreCoulombAt ψ y := by |
| simp_rw [integral_add (coreSlice_coreCoulombAt_integrable hψ _ y) |
| (rawPotential_coreSlice_integrable hψ _ y)] |
| rw [Finset.sum_add_distrib,integral_coreSlice_coreCoulombAt hψ y, |
| integral_rawPotential_coreSlice hψ y,←Finset.sum_add_distrib] |
| apply Finset.sum_congr rfl |
| intro s _ |
| change (∑ i : Fin N, ∫ x, ‖ψ.value s x‖^2/‖x (finSumFinEquiv (Sum.inl i))-y‖) + |
| (∑ i : Fin M, ∫ x, ‖ψ.value s x‖^2/‖x (finSumFinEquiv (Sum.inr i))-y‖) = |
| ∑ i : Fin (N+M), ∫ x, ‖ψ.value s x‖^2/‖x i-y‖ |
| simpa only [Fintype.sum_sum_type] using |
| (Equiv.sum_comp finSumFinEquiv (fun i : Fin (N+M) => ∫ x, ‖ψ.value s x‖^2/‖x i-y‖)) |
|
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| end CoulombAtom |
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| end |
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| end OAI |
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