| import Mathlib |
| import OAI.Analysis.CoulombIonization.Fermionic.InsertTest |
| import OAI.Analysis.CoulombIonization.Localization.FermionGradient |
|
|
| namespace OAI |
|
|
| noncomputable section |
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|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| open MeasureTheory Filter Complex TopologicalSpace |
| open scoped Topology InnerProductSpace ENNReal |
| open MeasureTheory Filter Complex |
| open scoped Topology BigOperators ComplexConjugate FourierTransform SchwartzMap ENNReal |
| open MeasureTheory Filter |
| open scoped Topology ContDiff SchwartzMap FourierTransform ENNReal |
| open MeasureTheory Filter |
| open scoped ContDiff InnerProductSpace Topology |
| open MeasureTheory Filter |
| open scoped ENNReal |
| namespace CoulombAtom.Pauli |
| open CoulombPauli CoulombSobolev CoulombPackets |
|
|
| def gradientVector {N : ℕ} (ψ : FormVector N) (i : Fin N) (a : Fin 3) : FormVector N where |
| value s := ψ.gradient s i a |
| gradient _ _ _ _ := 0 |
|
|
| def gradientMany {N : ℕ} (ψ : FormVector N) (hψ : FormAdmissible ψ) |
| (i : Fin N) (a : Fin 3) : Many N := |
| toMany (gradientVector ψ i a) (fun s => hψ.2.1 s i a) |
|
|
| lemma weak_test_identity {N : ℕ} (ψ : FormVector N) (hψ : FormAdmissible ψ) |
| (i : Fin N) (a : Fin 3) (φ : Test (Configuration N)) (s : Spins N) : |
| inner ℂ (manyTest (φ.deriv (direction i a)) s) (toMany ψ hψ.1) + |
| inner ℂ (manyTest φ s) (gradientMany ψ hψ i a) = 0 := by |
| have h2 : inner ℂ (manyTest φ s) (gradientMany ψ hψ i a) = |
| ∫ x, ψ.gradient s i a x * (φ.fn x : ℂ) := |
| manyTest_inner (gradientVector ψ i a) (fun s => hψ.2.1 s i a) φ s |
| rw [manyTest_inner, h2] |
| have hw := hψ.2.2.1 s i a φ.fn φ.smooth φ.compact |
| have he : (∫ x, ψ.value s x * ((φ.deriv (direction i a)).fn x : ℂ)) = |
| ∫ x, ψ.value s x * Complex.ofReal (lineDeriv ℝ φ.fn x (direction i a)) := by |
| apply integral_congr_ae |
| filter_upwards [] with x |
| rw [(φ.smooth.differentiable (by simp) x).lineDeriv_eq_fderiv] |
| rfl |
| rw [he,hw] |
| simp |
|
|
| lemma form_hasFermionGradient {N : ℕ} (ψ : FormVector (N+1)) (hψ : FormAdmissible ψ) : |
| HasFermionGradient (EuclideanSpace.basisFun (Fin 3) ℝ) (toMany ψ hψ.1) |
| (gradientMany ψ hψ) := by |
| intro i a φ s |
| apply manyTest_total |
| intro η τ |
| rw [inner_add_right] |
| change inner ℂ (manyTest η τ) |
| ((tensorLeft (tensor (φ.deriv ((EuclideanSpace.basisFun (Fin 3) ℝ) a)).toL2 (countVector s))).adjoint |
| (splitFermion i (toMany ψ hψ.1))) + |
| inner ℂ (manyTest η τ) ((tensorLeft (tensor φ.toL2 (countVector s))).adjoint |
| (splitFermion i (gradientMany ψ hψ i a))) = 0 |
| rw [← insert_adjoint_split, ← insert_adjoint_split, |
| ContinuousLinearMap.adjoint_inner_right, ContinuousLinearMap.adjoint_inner_right, |
| insertOrbital_manyTest, insertOrbital_manyTest] |
| rw [EuclideanSpace.basisFun_apply, ← insertTest_deriv] |
| exact weak_test_identity ψ hψ i a (insertTest i φ η) (Fin.insertNth i s τ) |
|
|
| lemma gradientMany_norm_sq {N : ℕ} (ψ : FormVector N) (hψ : FormAdmissible ψ) |
| (i : Fin N) (a : Fin 3) : |
| ‖gradientMany ψ hψ i a‖^2 = ∑ s : Spins N, ∫ x, ‖ψ.gradient s i a x‖^2 := by |
| exact toMany_norm_sq (gradientVector ψ i a) (fun s => hψ.2.1 s i a) |
|
|
| end CoulombAtom.Pauli |
|
|
| end |
|
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| end OAI |
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|
