File size: 3,247 Bytes
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 | import Mathlib
import OAI.Analysis.CoulombIonization.Fermionic.InsertTest
import OAI.Analysis.CoulombIonization.Localization.FermionGradient
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 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
end OAI
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