import Mathlib import OAI.Analysis.CoulombIonization.Variational.RotationCoreKernelIntegrable import OAI.Analysis.CoulombIonization.Localization.PacketCloud 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 open MeasureTheory Filter open scoped Topology BigOperators open MeasureTheory Filter open scoped Topology BigOperators open MeasureTheory Filter open scoped Topology BigOperators open MeasureTheory Filter Set Metric TopologicalSpace open scoped Topology BigOperators open MeasureTheory Filter Set Metric TopologicalSpace open scoped Topology BigOperators open MeasureTheory Filter Set Metric TopologicalSpace open scoped Topology BigOperators open MeasureTheory Filter Set Metric TopologicalSpace open scoped Topology BigOperators open MeasureTheory Filter Set Metric open scoped Topology BigOperators open MeasureTheory Filter Set Metric open scoped Topology BigOperators open MeasureTheory Filter Set Metric open scoped Topology BigOperators open MeasureTheory Filter Set Metric open scoped Topology BigOperators namespace CoulombAtom def gridPacketRadius (B : ℝ) (n : ℕ) : ℝ := B/(20*(n:ℝ)) def packetKineticConstant : ℝ := 400*radialPacketKinetic def packetRepulsionConstant : ℝ := 128000*packetDensityConstant*unitCoulombIntegral lemma packetKineticConstant_nonneg : 0 ≤ packetKineticConstant := by unfold packetKineticConstant exact mul_nonneg (by norm_num) radialPacketKinetic_nonneg lemma packetRepulsionConstant_nonneg : 0 ≤ packetRepulsionConstant := by unfold packetRepulsionConstant positivity [packetDensityConstant_pos,unitCoulombIntegral_nonneg] lemma gridPacketRadius_pos {B : ℝ} (hB : 0 < B) {n : ℕ} (hn : 0 < n) : 0 < gridPacketRadius B n := div_pos hB (by positivity) lemma gridPacketRadius_mul {B : ℝ} {n : ℕ} (hn : 0 < n) : gridPacketRadius B n * n = B/20 := by have hn' : (n:ℝ) ≠ 0 := (Nat.cast_pos.mpr hn).ne' unfold gridPacketRadius field_simp lemma gridPacketRadius_le {B : ℝ} (hB : 0 < B) {n : ℕ} (hn : 0 < n) : gridPacketRadius B n ≤ B/20 := by have hn' : (1:ℝ) ≤ n := Nat.one_le_cast.mpr hn have hr := gridPacketRadius_pos hB hn have he := gridPacketRadius_mul (B := B) hn nlinarith lemma packetGrid_ball {B : ℝ} (hB : 0 < B) {n : ℕ} (hn : 0 < n) (i : ℕ) : ‖packetGrid n (gridPacketRadius B n) i‖ ≤ 9*B/20 := by calc _ ≤ 9*gridPacketRadius B n*n := packetGrid_norm n (gridPacketRadius_pos hB hn) i _ = _ := by rw [mul_assoc,gridPacketRadius_mul hn]; ring lemma packetGrid_repulsion {B : ℝ} (hB : 0 < B) {n : ℕ} (hn : 0 < n) : packetPairEnergy (packetGrid n (gridPacketRadius B n)) (n^3) ≤ packetRepulsionConstant*(n:ℝ)^6/B := by let r := gridPacketRadius B n have hr : 0 < r := gridPacketRadius_pos hB hn have hrB : r ≤ B := by have := gridPacketRadius_le hB hn; dsimp [r]; linarith have hp (i : ℕ) (_hi : i < n^3) : ‖packetGrid n r i‖ ≤ B := by have := packetGrid_ball hB hn i dsimp only [r] linarith have hb := packetPairEnergy_bound (packetGrid n r) hr hB.le (n^3) (packetGrid_separated n hr) hp have hs : (2*B+2*r)^2 ≤ 16*B^2 := by nlinarith have hn' : (n:ℝ) ≠ 0 := (Nat.cast_pos.mpr hn).ne' calc _ ≤ (n^3:ℕ)*(packetDensityConstant/r^3*(2*B+2*r)^2*unitCoulombIntegral) := hb _ ≤ (n^3:ℕ)*(packetDensityConstant/r^3*(16*B^2)*unitCoulombIntegral) := by gcongr exact unitCoulombIntegral_nonneg exact div_nonneg packetDensityConstant_pos.le (pow_nonneg hr.le _) _ = _ := by dsimp [r,gridPacketRadius,packetRepulsionConstant] push_cast field_simp ring theorem grid_packet_screening {N : ℕ} {ψ : FormVector N} (hψ : SobolevFermion ψ) (y : Space) {B : ℝ} (hB : 0 < B) (hnuc : B ≤ ‖y‖) (hc : FormAvoids ψ (ball y B)) {n : ℕ} (hn : 0 < n) {Z lam : ℝ} (hZ : 0 ≤ Z) (hlam : 0 < lam) : (n:ℝ)^3*(Z*formMass ψ/‖y‖-coreCoulombAt ψ y-lam*formMass ψ) ≤ (formEnergy Z ψ + lam*N*formMass ψ-priceEnergy (energy Z) lam*formMass ψ) + formMass ψ*(packetKineticConstant*(n:ℝ)^5/B^2+packetRepulsionConstant*(n:ℝ)^6/B) := by let r := gridPacketRadius B n have hr : 0 < r := gridPacketRadius_pos hB hn have hrB : r ≤ B/20 := gridPacketRadius_le hB hn have hp (i : ℕ) (_hi : i < n^3) : ‖packetGrid n r i‖+2*r ≤ B := by have := packetGrid_ball hB hn i dsimp only [r] at * linarith have hnp (i : ℕ) (_hi : i < n^3) : ‖packetGrid n r i‖+r < ‖y‖ := by have := packetGrid_ball hB hn i dsimp only [r] at * linarith have hh := averaged_many_packet_screening hψ y (packetGrid n r) hr (n^3) hc hp (packetGrid_separated n hr) hnp hZ hlam have hpair := packetGrid_repulsion hB hn have hm := formMass_nonneg ψ have hh' := mul_le_mul_of_nonneg_left hpair hm have he : (n:ℝ)^3*(radialPacketKinetic/r^2) = packetKineticConstant*(n:ℝ)^5/B^2 := by dsimp [r,gridPacketRadius,packetKineticConstant] field_simp ring push_cast at hh hh' simp only [r] at hh hh' he nlinarith end CoulombAtom open MeasureTheory Filter Set Metric open scoped Topology BigOperators end end OAI