| import Mathlib |
| import OAI.Analysis.CoulombIonization.Fermionic.Spin |
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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 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 |
|
|
| def manyTest {N : ℕ} (φ : Test (Configuration N)) (s : Spins N) : Many N := |
| pull (splitSpin N) (splitSpin_preserving N) (tensor φ.toL2 (countVector s)) |
|
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| lemma manyTest_ae {N : ℕ} (φ : Test (Configuration N)) (s : Spins N) : |
| manyTest φ s =ᵐ[configMeasure N oneMeasure] (fun x : Fin N → OneParticle => |
| (φ.fn (fun i => (x i).1) : ℂ) * (if (fun i => (x i).2) = s then 1 else 0)) := by |
| have hh : tensor φ.toL2 (countVector s) =ᵐ[volume.prod Measure.count] |
| (fun p : Configuration N × Spins N => (φ.fn p.1 : ℂ) * (if p.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 p h1 h2 h3 |
| rw [h1,h2,h3] |
| by_cases h : p.2 = s <;> simp [h] |
| exact (pull_ae _ _ _).trans ((splitSpin_preserving N).quasiMeasurePreserving.ae hh) |
|
|
| lemma toMany_eq_pull {N : ℕ} (ψ : FormVector N) (hψ : ∀ s, MemLp (ψ.value s) 2) : |
| toMany ψ hψ = pull (splitSpin N) (splitSpin_preserving N) |
| ((memLp_spin ψ.value hψ).toLp _) := by |
| apply Lp.ext |
| exact (toMany_ae ψ hψ).trans ((pull_ae _ _ _).trans |
| ((splitSpin_preserving N).quasiMeasurePreserving.ae (memLp_spin ψ.value hψ).coeFn_toLp)).symm |
|
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| lemma manyTest_inner {N : ℕ} (ψ : FormVector N) (hψ : ∀ s, MemLp (ψ.value s) 2) |
| (φ : Test (Configuration N)) (s : Spins N) : |
| inner ℂ (manyTest φ s) (toMany ψ hψ) = ∫ x, ψ.value s x * (φ.fn x : ℂ) := by |
| rw [manyTest, toMany_eq_pull, pull_inner] |
| exact spin_test_inner ψ.value hψ φ s |
|
|
| lemma manyTest_total {N : ℕ} (ψ : Many N) |
| (hψ : ∀ (φ : Test (Configuration N)) (s : Spins N), inner ℂ (manyTest φ s) ψ = 0) : ψ = 0 := by |
| let e := splitSpin N |
| let he := splitSpin_preserving N |
| have hz : pull e.symm (he.symm e) ψ = 0 := by |
| apply spin_test_total |
| intro φ s |
| have hh := hψ φ s |
| unfold manyTest at hh |
| rw [← pull_symm e he ψ] at hh |
| rw [pull_inner] at hh |
| exact hh |
| have hh := congrArg (pull e he) hz |
| simpa only [pull_symm, map_zero] using hh |
|
|
| variable {X : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X] [FiniteDimensional ℝ X] |
| def coordinateSplit {N : ℕ} (i : Fin (N+1)) : (Fin (N+1) → X) ≃L[ℝ] X × (Fin N → X) := |
| ({ toEquiv := (Fin.insertNthEquiv (fun _ : Fin (N+1) => X) i).symm |
| map_add' := fun _ _ => rfl |
| map_smul' := fun _ _ => rfl } : (Fin (N+1) → X) ≃ₗ[ℝ] X × (Fin N → X)).toContinuousLinearEquiv |
|
|
| lemma coordinateSplit_apply {N : ℕ} (i : Fin (N+1)) (x : Fin (N+1) → X) : |
| coordinateSplit i x = (x i, fun j => x (i.succAbove j)) := rfl |
|
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| lemma coordinateSplit_symm_apply {N : ℕ} (i : Fin (N+1)) (z : X × (Fin N → X)) : |
| (coordinateSplit i).symm z = Fin.insertNth i z.1 z.2 := rfl |
|
|
| lemma coordinateTest {N : ℕ} (i : Fin (N+1)) (φ : Test Space) (η : Test (Configuration N)) : |
| HasCompactSupport (fun x : Configuration (N+1) => φ.fn (x i) * η.fn (fun j => x (i.succAbove j))) := by |
| have hp : HasCompactSupport (fun z : Space × Configuration N => φ.fn z.1 * η.fn z.2) := by |
| apply HasCompactSupport.of_support_subset_isCompact (φ.compact.prod η.compact) |
| intro z hz |
| simp only [Function.mem_support, ne_eq, mul_eq_zero, not_or] at hz |
| exact ⟨subset_tsupport _ hz.1, subset_tsupport _ hz.2⟩ |
| exact hp.comp_homeomorph (coordinateSplit i).toHomeomorph |
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| end CoulombAtom.Pauli |
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| end |
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| end OAI |
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