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159357f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | import Mathlib
import OAI.Analysis.CoulombIonization.Fermionic.Spin
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
def manyTest {N : ℕ} (φ : Test (Configuration N)) (s : Spins N) : Many N :=
pull (splitSpin N) (splitSpin_preserving N) (tensor φ.toL2 (countVector s))
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
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
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
end CoulombAtom.Pauli
end
end OAI
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