| import Mathlib |
| import OAI.Analysis.CoulombIonization.Fermionic.ManyTest |
| import OAI.Analysis.CoulombIonization.FormDomain.CompCL |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| 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 |
|
|
| lemma coordinateSplit_direction {N : ℕ} (i : Fin (N+1)) (a : Fin 3) : |
| coordinateSplit i (direction i a) = (EuclideanSpace.single a 1, 0) := by |
| rw [coordinateSplit_apply] |
| apply Prod.ext |
| · simp [direction] |
| · funext j |
| simp [direction, Fin.succAbove_ne] |
|
|
| def insertTest {N : ℕ} (i : Fin (N+1)) (φ : Test Space) (η : Test (Configuration N)) : |
| Test (Configuration (N+1)) := (φ.tensor η).compCL (coordinateSplit i) |
|
|
| lemma insertTest_deriv {N : ℕ} (i : Fin (N+1)) (a : Fin 3) |
| (φ : Test Space) (η : Test (Configuration N)) : |
| (insertTest i φ η).deriv (direction i a) = |
| insertTest i (φ.deriv (EuclideanSpace.single a 1)) η := by |
| rw [insertTest, Test.compCL_deriv, coordinateSplit_direction, Test.tensor_deriv] |
| rfl |
|
|
| lemma insertTest_apply {N : ℕ} (i : Fin (N+1)) (φ : Test Space) (η : Test (Configuration N)) |
| (x : Configuration (N+1)) : |
| (insertTest i φ η).fn x = φ.fn (x i) * η.fn (fun j => x (i.succAbove j)) := rfl |
|
|
| lemma insertOrbital_ae {N : ℕ} (i : Fin (N+1)) (u : Lp ℂ 2 oneMeasure) (ψ : Many N) : |
| CoulombPauli.insertOrbital i u ψ =ᵐ[configMeasure (N+1) oneMeasure] |
| (fun x : Fin (N+1) → OneParticle => u (x i) * ψ (fun j => x (i.succAbove j))) := by |
| dsimp only [CoulombPauli.insertOrbital, ContinuousLinearMap.comp_apply, tensorLeft_apply] |
| exact (pull_ae _ _ _).trans ((splitAt_preserving i).quasiMeasurePreserving.ae (tensor_ae u ψ)) |
|
|
| lemma oneTest_ae (φ : Test Space) (s : Fin 2) : |
| tensor φ.toL2 (countVector s) =ᵐ[oneMeasure] |
| (fun x : OneParticle => (φ.fn x.1 : ℂ) * (if x.2 = s then 1 else 0)) := by |
| filter_upwards [tensor_ae φ.toL2 (countVector s), |
| Measure.quasiMeasurePreserving_fst.ae φ.coe_toL2, |
| Measure.quasiMeasurePreserving_snd.ae (countVector_ae s)] with x h1 h2 h3 |
| rw [h1,h2,h3] |
| by_cases h : x.2 = s < |
|
|
| lemma insertOrbital_manyTest {N : ℕ} (i : Fin (N+1)) (φ : Test Space) (η : Test (Configuration N)) |
| (s : Fin 2) (τ : Spins N) : |
| CoulombPauli.insertOrbital i (tensor φ.toL2 (countVector s)) (manyTest η τ) = |
| manyTest (insertTest i φ η) (Fin.insertNth i s τ) := by |
| apply Lp.ext |
| have h1 := (splitAt_preserving i).quasiMeasurePreserving.ae |
| (Measure.quasiMeasurePreserving_fst.ae (oneTest_ae φ s)) |
| have h2 := (splitAt_preserving i).quasiMeasurePreserving.ae |
| (Measure.quasiMeasurePreserving_snd.ae (manyTest_ae η τ)) |
| filter_upwards [insertOrbital_ae i (tensor φ.toL2 (countVector s)) (manyTest η τ), |
| manyTest_ae (insertTest i φ η) (Fin.insertNth i s τ), h1, h2] with x hx hy hφ hη |
| rw [hx,hy] |
| change (tensor φ.toL2 (countVector s)) (x i) = |
| (φ.fn (x i).1 : ℂ) * (if (x i).2 = s then 1 else 0) at hφ |
| change manyTest η τ (fun j => x (i.succAbove j)) = |
| (η.fn (fun j => (x (i.succAbove j)).1) : ℂ) * |
| (if (fun j => (x (i.succAbove j)).2) = τ then 1 else 0) at hη |
| rw [hφ,hη,insertTest_apply, Complex.ofReal_mul] |
| have he : ((fun j => (x j).2) = Fin.insertNth i s τ) ↔ |
| (x i).2 = s ∧ (fun j => (x (i.succAbove j)).2) = τ := Fin.eq_insertNth_iff |
| by_cases hs : (x i).2 = s < |
| simp [he, hs, ht] |
| end CoulombAtom.Pauli |
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|
| end |
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| end OAI |
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