| import Mathlib |
| import OAI.Analysis.CoulombIonization.FormDomain.SobolevVector |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace Convolution |
| open Filter |
| open scoped Topology InnerProductSpace |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace ContDiff |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace Convolution |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace ENNReal |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open Set Filter Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter |
| open scoped Topology |
| open MeasureTheory Filter |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter |
| open scoped Topology BigOperators InnerProductSpace |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter |
| open scoped Topology BigOperators Pointwise |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open scoped Convolution |
| open scoped RealInnerProductSpace |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Set Filter Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Set Filter Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open Filter Set Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open Filter |
| open scoped Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace ContDiff BigOperators |
| open MeasureTheory Filter Set |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators InnerProductSpace |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open scoped BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| namespace CoulombAtom |
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| def sumForm {ι : Type*} [Fintype ι] {N : ℕ} (ψ : ι → FormVector N) : FormVector N where |
| value s x := ∑ k, (ψ k).value s x |
| gradient s i a x := ∑ k, (ψ k).gradient s i a x |
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| lemma SobolevVector.sum {ι : Type*} [Fintype ι] {N : ℕ} {ψ : ι → FormVector N} |
| (hψ : ∀ k, SobolevVector (ψ k)) : SobolevVector (sumForm ψ) := by |
| classical |
| refine ⟨fun s => memLp_finsetSum _ (fun k _ => (hψ k).1 s), |
| fun s i a => memLp_finsetSum _ (fun k _ => (hψ k).2.1 s i a), ?_⟩ |
| intro s i a φ hφ hcφ |
| change (∫ x, (∑ k, (ψ k).value s x) * Complex.ofReal (lineDeriv ℝ φ x (direction i a))) = |
| -(∫ x, (∑ k, (ψ k).gradient s i a x) * (φ x : ℂ)) |
| simp only [Finset.sum_mul] |
| rw [integral_finsetSum Finset.univ (fun k _ => show Integrable (fun x => |
| (ψ k).value s x * Complex.ofReal (lineDeriv ℝ φ x (direction i a))) from |
| ((hψ k).1 s).integrable_mul (test_deriv_memLp hφ hcφ (direction i a))), |
| integral_finsetSum Finset.univ (fun k _ => show Integrable (fun x => |
| (ψ k).gradient s i a x * (φ x : ℂ)) from |
| ((hψ k).2.1 s i a).integrable_mul (test_memLp hφ hcφ))] |
| simp only [(hψ _).2.2 s i a φ hφ hcφ, Finset.sum_neg_distrib] |
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| lemma SobolevVector.scale {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) (c : ℝ) : |
| SobolevVector (scaleForm c ψ) := by |
| refine ⟨fun s => (hψ.1 s).const_mul _, fun s i a => (hψ.2.1 s i a).const_mul _, ?_⟩ |
| intro s i a φ hφ hcφ |
| change (∫ x, ((c : ℂ) * ψ.value s x) * Complex.ofReal (lineDeriv ℝ φ x (direction i a))) = |
| -(∫ x, ((c : ℂ) * ψ.gradient s i a x) * (φ x : ℂ)) |
| simp only [mul_assoc, integral_const_mul, hψ.2.2 s i a φ hφ hcφ, mul_neg] |
|
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| lemma norm_sum_sq_disjoint {ι : Type*} [Fintype ι] (v : ι → ℂ) |
| (h : ∀ i j, i ≠ j → v i = 0 ∨ v j = 0) : |
| ‖∑ i, v i‖^2 = ∑ i, ‖v i‖^2 := by |
| classical |
| by_cases hz : ∀ i, v i = 0 |
| · simp [hz] |
| · push Not at hz |
| obtain ⟨i,hi⟩ := hz |
| have hh (j : ι) (hji : j ≠ i) : v j = 0 := (h j i hji).resolve_right hi |
| rw [Finset.sum_eq_single i (fun j _ hj => hh j hj) (by simp), |
| Finset.sum_eq_single i (fun j _ hj => by simp [hh j hj]) (by simp)] |
|
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| def FormsDisjoint {ι : Type*} {N : ℕ} (ψ : ι → FormVector N) : Prop := |
| (∀ k l, k ≠ l → ∀ s x, (ψ k).value s x = 0 ∨ (ψ l).value s x = 0) ∧ |
| (∀ k l, k ≠ l → ∀ s i a x, |
| (ψ k).gradient s i a x = 0 ∨ (ψ l).gradient s i a x = 0) |
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| lemma formMass_sum_disjoint {ι : Type*} [Fintype ι] {N : ℕ} {ψ : ι → FormVector N} |
| (hψ : ∀ k, SobolevVector (ψ k)) (hd : FormsDisjoint ψ) : |
| formMass (sumForm ψ) = ∑ k, formMass (ψ k) := by |
| unfold formMass |
| have hh (s : Spins N) : (∫ x, ‖(sumForm ψ).value s x‖^2) = |
| ∑ k, ∫ x, ‖(ψ k).value s x‖^2 := by |
| simp only [sumForm, norm_sum_sq_disjoint _ (fun k l hkl => hd.1 k l hkl s _)] |
| exact integral_finsetSum _ (fun k _ => ((hψ k).1 s).norm.integrable_sq) |
| simp_rw [hh] |
| exact Finset.sum_comm |
|
|
| lemma formEnergy_sum_disjoint {ι : Type*} [Fintype ι] {N : ℕ} {ψ : ι → FormVector N} |
| (hψ : ∀ k, SobolevVector (ψ k)) (hd : FormsDisjoint ψ) (Z : ℝ) : |
| formEnergy Z (sumForm ψ) = ∑ k, formEnergy Z (ψ k) := by |
| classical |
| have hk (s : Spins N) (i : Fin N) (a : Fin 3) : |
| (∫ x, ‖(sumForm ψ).gradient s i a x‖^2) = ∑ k, ∫ x, ‖(ψ k).gradient s i a x‖^2 := by |
| simp only [sumForm, norm_sum_sq_disjoint _ (fun k l hkl => hd.2 k l hkl s i a _)] |
| exact integral_finsetSum _ (fun k _ => ((hψ k).2.1 s i a).norm.integrable_sq) |
| have hn (s : Spins N) (i : Fin N) : |
| (∫ x, ‖(sumForm ψ).value s x‖^2 / ‖x i‖) = |
| ∑ k, ∫ x, ‖(ψ k).value s x‖^2 / ‖x i‖ := by |
| simp only [sumForm, norm_sum_sq_disjoint _ (fun k l hkl => hd.1 k l hkl s _), Finset.sum_div] |
| exact integral_finsetSum _ (fun k _ => (hψ k).nuclear_integrable s i) |
| have hp (s : Spins N) (i j : Fin N) : |
| (if i < j then ∫ x, ‖(sumForm ψ).value s x‖^2 / ‖x i-x j‖ else 0) = |
| ∑ k, if i < j then ∫ x, ‖(ψ k).value s x‖^2 / ‖x i-x j‖ else 0 := by |
| by_cases hij : i < j |
| · simp only [ite_eq_left hij, sumForm, |
| norm_sum_sq_disjoint _ (fun k l hkl => hd.1 k l hkl s _), Finset.sum_div] |
| exact integral_finsetSum _ (fun k _ => (hψ k).pair_integrable s i j hij.ne) |
| · simp only [ite_eq_right hij, Finset.sum_const_zero] |
| unfold formEnergy |
| simp_rw [hk,hn,hp] |
| rw [show (∑ s, ∑ i, ∑ a, ∑ k, ∫ x, ‖(ψ k).gradient s i a x‖^2) = |
| ∑ k, ∑ s, ∑ i, ∑ a, ∫ x, ‖(ψ k).gradient s i a x‖^2 by |
| simp_rw [Finset.sum_comm (s := Finset.univ) (t := Finset.univ) (f := fun a k => |
| ∫ x, ‖(ψ k).gradient _ _ a x‖^2)] |
| simp_rw [Finset.sum_comm (s := Finset.univ) (t := Finset.univ) (f := fun i k => |
| ∑ a, ∫ x, ‖(ψ k).gradient _ i a x‖^2)] |
| exact Finset.sum_comm, |
| show (∑ s, ∑ i, ∑ k, ∫ x, ‖(ψ k).value s x‖^2 / ‖x i‖) = |
| ∑ k, ∑ s, ∑ i, ∫ x, ‖(ψ k).value s x‖^2 / ‖x i‖ by |
| simp_rw [Finset.sum_comm (s := Finset.univ) (t := Finset.univ) (f := fun i k => |
| ∫ x, ‖(ψ k).value _ x‖^2 / ‖x i‖)] |
| exact Finset.sum_comm, |
| show (∑ s, ∑ i, ∑ j, ∑ k, if i < j then ∫ x, ‖(ψ k).value s x‖^2 / ‖x i-x j‖ else 0) = |
| ∑ k, ∑ s, ∑ i, ∑ j, if i < j then ∫ x, ‖(ψ k).value s x‖^2 / ‖x i-x j‖ else 0 by |
| simp_rw [Finset.sum_comm (s := Finset.univ) (t := Finset.univ) (f := fun j k => |
| if _ < j then ∫ x, ‖(ψ k).value _ x‖^2 / ‖x _-x j‖ else 0)] |
| simp_rw [Finset.sum_comm (s := Finset.univ) (t := Finset.univ) (f := fun i k => |
| ∑ j, if i < j then ∫ x, ‖(ψ k).value _ x‖^2 / ‖x i-x j‖ else 0)] |
| exact Finset.sum_comm] |
| simp only [Finset.mul_sum, Finset.sum_sub_distrib, Finset.sum_add_distrib] |
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| end CoulombAtom |
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| open MeasureTheory Filter |
| open scoped BigOperators |
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| end |
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| end OAI |
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