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import OAI.Analysis.CoulombIonization.RadialBounds.Radial
import OAI.Analysis.CoulombIonization.Ionization.InsertionTermsDisjointSet
import OAI.Analysis.CoulombIonization.Localization.RadialPacketBase
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff InnerProductSpace Convolution
open Filter
open scoped Topology InnerProductSpace
open MeasureTheory Complex Filter
open scoped Topology InnerProductSpace
open MeasureTheory Complex Filter
open scoped Topology InnerProductSpace ContDiff
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff InnerProductSpace Convolution
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff InnerProductSpace
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff InnerProductSpace ENNReal
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open Set Filter Topology InnerProductSpace Laplacian
open MeasureTheory Filter
open scoped Topology
open MeasureTheory Filter
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric
open scoped Topology ENNReal
open MeasureTheory Filter
open scoped Topology BigOperators InnerProductSpace
open MeasureTheory Filter Set Metric
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric
open scoped Topology ENNReal
open MeasureTheory Filter
open scoped Topology BigOperators Pointwise
open MeasureTheory Filter Set Metric
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian
open scoped Convolution
open scoped RealInnerProductSpace
open MeasureTheory Filter Set Metric
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian
open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian
open MeasureTheory Filter Set Metric Topology
open MeasureTheory Set Filter Metric Topology InnerProductSpace Laplacian
open MeasureTheory Set Filter Metric Topology InnerProductSpace Laplacian
open MeasureTheory Filter Set Metric Topology
open MeasureTheory Filter Set Metric Topology
open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian
open Filter Set Metric Topology InnerProductSpace Laplacian
open MeasureTheory Filter Set Metric Topology
open MeasureTheory Filter Set Metric Topology
open MeasureTheory Filter Set Metric Topology
open MeasureTheory Filter Set Metric Topology
open Filter
open scoped Topology
open MeasureTheory Filter Set Metric Topology
open MeasureTheory Filter Set Metric Topology
open MeasureTheory Complex Filter
open scoped Topology InnerProductSpace ContDiff BigOperators
open MeasureTheory Filter Set
open scoped Topology BigOperators
open MeasureTheory Filter
open scoped Topology BigOperators InnerProductSpace
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open scoped BigOperators
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open MeasureTheory Filter
open scoped Topology ContDiff
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open MeasureTheory Filter
open scoped BigOperators
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open MeasureTheory Filter
open scoped BigOperators
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open MeasureTheory Filter
open scoped Topology BigOperators
open MeasureTheory Filter
open scoped Topology BigOperators
open MeasureTheory Filter
open scoped Topology BigOperators
open MeasureTheory Filter
open scoped Topology BigOperators
open MeasureTheory Filter
open scoped Topology BigOperators
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
namespace CoulombAtom
lemma integral_one_position (f : Space → ℝ) :
(∫ x : Configuration 1, f (x 0)) = ∫ y, f y := by
exact (volume_preserving_funUnique (Fin 1) Space).integral_comp
(MeasurableEquiv.funUnique (Fin 1) Space).measurableEmbedding f
lemma integral_one_configuration (f : Configuration 1 → ℝ) :
(∫ y : Space, f (fun _ => y)) = ∫ x, f x := by
exact ((volume_preserving_funUnique (Fin 1) Space).symm
(MeasurableEquiv.funUnique (Fin 1) Space)).integral_comp
(MeasurableEquiv.funUnique (Fin 1) Space).symm.measurableEmbedding f
def packetDensity (y : Space) (R : ℝ) (x : Space) : ℝ :=
∑ s : Spins 1, ‖(radialLocalOrbital y R).value s (fun _ => x)‖^2
lemma packetDensity_formula (y : Space) (R : ℝ) (x : Space) :
packetDensity y R x = (Fintype.card (Spins 1) : ℝ) *
(Real.sqrt (formMass (radialScaledRaw R)))⁻¹ ^ 2 *
‖radialPacketBase (R⁻¹ • (x-y))‖^2 := by
simp only [packetDensity,radialLocalOrbital,scaleForm,norm_real_mul_sq,radialLocalRaw_mass]
simp only [radialLocalRaw,smoothForm,radialLocalSeed,radialScaledSeed,
radialPacketSeed,radialCenterShift,Pi.smul_apply,Pi.add_apply,Finset.sum_const,
Finset.card_univ,nsmul_eq_mul,sub_eq_add_neg,mul_assoc]
lemma packetDensity_continuous (y : Space) (R : ℝ) : Continuous (packetDensity y R) := by
change Continuous (fun x => packetDensity y R x)
simp_rw [packetDensity_formula]
have hc : Continuous (fun x : Space => R⁻¹ • (x-y)) :=
(continuous_id.sub continuous_const).const_smul R⁻¹
have hn : Continuous (fun x : Space => ‖radialPacketBase (R⁻¹ • (x-y))‖^2) :=
(radialPacketBase_smooth.continuous.comp hc).norm.pow 2
exact continuous_const.mul hn
lemma packetDensity_support (y : Space) {R : ℝ} (hR : 0 < R) {x : Space}
(hx : packetDensity y R x ≠ 0) : ‖x-y‖ ≤ R := by
by_contra hh
have hz := radialLocalOrbital_zero y hR (x := fun _ => x) (not_le.mp hh)
apply hx
unfold packetDensity
apply Finset.sum_eq_zero
intro s _
rw [(hz s).1,norm_zero,zero_pow (by decide : (2:ℕ) ≠ 0)]
lemma packetDensity_compact (y : Space) {R : ℝ} (hR : 0 < R) :
HasCompactSupport (packetDensity y R) := by
apply HasCompactSupport.of_support_subset_isCompact (isCompact_closedBall y R)
intro x hx
exact (Metric.mem_closedBall).mpr (packetDensity_support y hR hx)
lemma packetDensity_integrable (y : Space) {R : ℝ} (hR : 0 < R) :
Integrable (packetDensity y R) :=
(packetDensity_continuous y R).integrable_of_hasCompactSupport (packetDensity_compact y hR)
lemma packetDensity_memLp (y : Space) {R : ℝ} (hR : 0 < R) :
MemLp (packetDensity y R) (5/3) :=
(packetDensity_continuous y R).memLp_of_hasCompactSupport (packetDensity_compact y hR)
lemma packetDensity_mass (y : Space) {R : ℝ} (hR : 0 < R) :
(∫ x, packetDensity y R x) = 1 := by
have hm := (radialLocalOrbital_admissible y hR).2.2.2.2.1
change formMass (radialLocalOrbital y R) = 1 at hm
rw [← hm]
unfold packetDensity formMass
rw [integral_finsetSum]
· apply Finset.sum_congr rfl
intro s _
exact integral_one_configuration (fun x => ‖(radialLocalOrbital y R).value s x‖^2)
· intro s _
have hh := ((radialLocalOrbital_admissible y hR).1 s).integrable_norm_pow (by norm_num)
exact ((volume_preserving_funUnique (Fin 1) Space).symm
(MeasurableEquiv.funUnique (Fin 1) Space)).integrable_comp_of_integrable hh
lemma packetDensity_translate (y : Space) (R : ℝ) (x : Space) :
packetDensity y R x = packetDensity 0 R (x-y) := by
simp only [packetDensity_formula,sub_zero]
lemma packetDensity_radial (R : ℝ) : CoulombAnalysis.IsRadial (packetDensity 0 R) := by
intro x y hxy
simp only [packetDensity_formula,sub_zero,radialPacketBase,norm_smul,hxy]
theorem packetDensity_potential (y : Space) {R : ℝ} (hR : 0 < R)
{z : Space} (hz : R < ‖z-y‖) :
(∫ x, packetDensity y R x / ‖z-x‖) = 1 / ‖z-y‖ := by
have ht := integral_add_right_eq_self (μ := volume)
(fun x : Space => packetDensity y R x / ‖z-x‖) y
rw [← ht]
have he : (fun x => packetDensity y R (x+y) / ‖z-(x+y)‖) =
(fun x => packetDensity 0 R x / ‖(z-y)-x‖) := by
funext x
rw [packetDensity_translate,add_sub_cancel_right]
congr 2
abel
rw [he]
have hn := CoulombAnalysis.tfPotential_newton_exterior
(packetDensity_integrable 0 hR) (packetDensity_memLp 0 hR) (packetDensity_radial R)
hR.le (fun x hx => by simpa only [sub_zero] using packetDensity_support 0 hR hx) hz
rw [packetDensity_mass 0 hR] at hn
exact hn
lemma packetDensity_eq_two (y : Space) (R : ℝ) (x : Space) (s : Spins 1) :
packetDensity y R x = 2 * ‖(radialLocalOrbital y R).value s (fun _ => x)‖^2 := by
simp only [packetDensity,radialLocalOrbital,scaleForm,radialLocalRaw,smoothForm,
Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
congr 1
lemma radialLocalOrbital_value_config (y : Space) (R : ℝ) (s : Spins 1)
(x : Configuration 1) :
(radialLocalOrbital y R).value s x =
(radialLocalOrbital y R).value s (fun _ => x 0) := by
congr 1
funext i
exact congrArg x (Subsingleton.elim i 0)
lemma radialLocalOrbital_potential (y : Space) {R : ℝ} (hR : 0 < R)
{z : Space} (hz : R < ‖z-y‖) (s : Spins 1) :
(∫ x : Configuration 1, ‖(radialLocalOrbital y R).value s x‖^2 / ‖z-x 0‖) =
(1/2 : ℝ) / ‖z-y‖ := by
have ht := packetDensity_potential y hR hz
simp_rw [packetDensity_eq_two y R _ s,mul_div_assoc] at ht
rw [integral_const_mul] at ht
have he : (∫ x : Configuration 1, ‖(radialLocalOrbital y R).value s x‖^2 / ‖z-x 0‖) =
∫ v : Space, ‖(radialLocalOrbital y R).value s (fun _ => v)‖^2 / ‖z-v‖ := by
rw [← integral_one_configuration]
rw [he]
calc
_ = (1 / ‖z-y‖) / 2 := by linarith
_ = _ := by ring
lemma radialLocalOrbital_nuclear (y : Space) {R : ℝ} (hR : 0 < R)
(hy : R < ‖y‖) : formNuclear (radialLocalOrbital y R) = 1 / ‖y‖ := by
have hh (s : Spins 1) := radialLocalOrbital_potential y hR
(z := 0) (by simpa only [zero_sub,norm_neg] using hy) s
simp only [zero_sub,norm_neg] at hh
unfold formNuclear
simp only [Fin.sum_univ_one,hh,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
norm_num [Spins]
ring
lemma integral_join_left {N M : ℕ} {f : Configuration (N+M) → ℝ}
(hf : Integrable f) :
(∫ x : Configuration N, ∫ y : Configuration M, f (joinLists x y)) = ∫ z, f z := by
have hj := integrable_join (N := N) (M := M) hf
have he := (configurationJoin_preserving N M).integral_comp
(configurationJoin N M).toHomeomorph.measurableEmbedding f
simp only [configurationJoin_apply_prod] at he
rw [Measure.volume_eq_prod] at hj he
exact (integral_prod _ hj).symm.trans he
def coreCoulombAt {N : ℕ} (ψ : FormVector N) (y : Space) : ℝ :=
∑ s : Spins N, ∑ i : Fin N, ∫ x, ‖ψ.value s x‖^2 / ‖x i-y‖
lemma tensor_packet_pair {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ)
(y : Space) {R B : ℝ} (hR : 0 < R) (hRB : R < B)
(hc : ∀ x i, ‖x i-y‖ < B → FormZeroAt ψ x)
(s : Spins N) (t : Spins 1) (i : Fin N) :
(∫ z, ‖(tensorForm ψ (radialLocalOrbital y R)).value (joinLists s t) z‖^2 /
‖z i.castSucc-z (Fin.last N)‖) =
(1/2 : ℝ) * ∫ x, ‖ψ.value s x‖^2 / ‖x i-y‖ := by
have hφ := (radialLocalOrbital_admissible y hR).sobolevFermion.sobolevVector
have hT := hψ.tensor hφ
have hi := pair_integrable i.castSucc (Fin.last N) (Fin.castSucc_ne_last i)
(hT.1 (joinLists s t)) (hT.2.1 (joinLists s t) _) (hT.2.2 (joinLists s t) _)
rw [← integral_join_left hi]
change (∫ x : Configuration N, ∫ v : Configuration 1,
‖(tensorForm ψ (radialLocalOrbital y R)).value (joinLists s t) (joinLists x v)‖^2 /
‖joinLists x v (finSumFinEquiv (Sum.inl i)) -
joinLists x v (finSumFinEquiv (Sum.inr (0 : Fin 1)))‖) = _
simp only [tensorForm_value_join,joinLists_left,joinLists_right,norm_mul,mul_pow,
mul_div_assoc,integral_const_mul]
rw [← integral_const_mul]
apply integral_congr_ae
apply Eventually.of_forall
intro x
by_cases hz : ψ.value s x = 0
· simp only [hz,norm_zero,zero_pow (by decide : (2:ℕ) ≠ 0),zero_mul,zero_div,mul_zero]
· have hd : R < ‖x i-y‖ := hRB.trans_le (le_of_not_gt fun hh => hz ((hc x i hh s).1))
dsimp only
rw [radialLocalOrbital_potential y hR hd]
ring
lemma tensorCross_packet {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ)
(y : Space) {R B : ℝ} (hR : 0 < R) (hRB : R < B)
(hc : ∀ x i, ‖x i-y‖ < B → FormZeroAt ψ x) :
tensorCross ψ (radialLocalOrbital y R) = coreCoulombAt ψ y := by
unfold tensorCross coreCoulombAt
simp_rw [tensor_packet_pair hψ y hR hRB hc,← Finset.mul_sum]
simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
norm_num [Spins]
simp only [← Finset.mul_sum]
ring
theorem local_point_field {N : ℕ} {ψ : FormVector N} (hψ : SobolevFermion ψ)
(y : Space) {R B : ℝ} (hR : 0 < R) (hRB : R < B) (hy : R < ‖y‖)
(hc : ∀ x i, ‖x i-y‖ < B → FormZeroAt ψ x)
{Z lam : ℝ} (hZ : 0 ≤ Z) (hlam : 0 < lam) :
Z * formMass ψ / ‖y‖ - coreCoulombAt ψ y ≤
(formEnergy Z ψ + lam*N*formMass ψ - priceEnergy (energy Z) lam * formMass ψ) +
formMass ψ * (radialPacketKinetic / R^2 + lam) := by
have hh := local_screened_insertion hψ (radialLocalOrbital_admissible y hR)
(Metric.ball y B) (fun x i hi => hc x i (Metric.mem_ball.mp hi))
(fun x hx => radialLocalOrbital_zero y hR (hRB.trans_le (le_of_not_gt
(fun hh => hx (Metric.mem_ball.mpr hh))))) hZ hlam
rw [radialLocalOrbital_nuclear y hR hy,tensorCross_packet hψ.sobolevVector y hR hRB hc,
radialLocalOrbital_kinetic y hR] at hh
simpa only [mul_one_div] using hh
end CoulombAtom
open MeasureTheory Filter
open scoped Topology BigOperators
end
end OAI
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