| import Mathlib |
| import OAI.Analysis.CoulombIonization.RadialBounds.TruncationRadius |
| import OAI.Analysis.CoulombIonization.FormDomain.Form |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace CoulombAtom |
|
|
| 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 |
|
|
| def diffuseSeed (R : ℝ) (x : Configuration 1) : ℂ := trialSeed 1 (R⁻¹ • x) |
|
|
| def diffuseRaw (R : ℝ) : FormVector 1 := smoothForm (diffuseSeed R) |
|
|
| lemma diffuseSeed_smooth (R : ℝ) : ContDiff ℝ ∞ (diffuseSeed R) := |
| (trialSeed_smooth 1).comp (R⁻¹ • ContinuousLinearMap.id ℝ (Configuration 1)).contDiff |
|
|
| lemma diffuseSeed_compact {R : ℝ} (hR : 0 < R) : HasCompactSupport (diffuseSeed R) := |
| (trialSeed_compact 1).comp_smul (inv_ne_zero hR.ne') |
| |
| lemma diffuseRaw_sobolev (R : ℝ) (hR : 0 < R) : SobolevFermion (diffuseRaw R) := by |
| refine ⟨fun _ => (diffuseSeed_smooth R).continuous.memLp_of_hasCompactSupport |
| (diffuseSeed_compact hR),fun s i a => ?_,fun s i a φ hφ hcφ => ?_,?_⟩ |
| · exact (smooth_derivative_continuous (diffuseSeed_smooth R) _).memLp_of_hasCompactSupport |
| (compact_derivative (diffuseSeed_compact hR) _) |
| · exact smooth_weak_gradient (diffuseSeed_smooth R) (diffuseSeed_compact hR) hφ hcφ _ |
| · intro π s |
| have hp : π = 1 := Subsingleton.elim _ _ |
| subst π |
| exact Eventually.of_forall fun x => by simp [diffuseRaw,smoothForm] |
| |
| lemma diffuseRaw_mass_pos {R : ℝ} (hR : 0 < R) : 0 < formMass (diffuseRaw R) := by |
| have hn : diffuseSeed R (R • trialCenter 1) = 1 := by |
| simp only [diffuseSeed,smul_smul,inv_mul_cancel₀ hR.ne',one_smul,trialSeed_center] |
| have hi : 0 < ∫ x, ‖diffuseSeed R x‖^2 := |
| ((diffuseSeed_smooth R).continuous.norm.pow 2).integral_pos_of_hasCompactSupport_nonneg_nonzero |
| (compact_norm_sq (diffuseSeed_compact hR)) (fun x => sq_nonneg _) |
| (x := R • trialCenter 1) (by rw [hn]; norm_num) |
| change 0 < ∑ _s : Spins 1, ∫ x, ‖diffuseSeed R x‖^2 |
| simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul] |
| exact mul_pos (Nat.cast_pos.mpr Fintype.card_pos) hi |
|
|
| lemma diffuseRaw_gradient (R : ℝ) (s : Spins 1) (i : Fin 1) (a : Fin 3) (x : Configuration 1) : |
| (diffuseRaw R).gradient s i a x = R⁻¹ • |
| (smoothForm (trialSeed 1)).gradient s i a (R⁻¹ • x) := by |
| change fderiv ℝ ((trialSeed 1) ∘ ⇑(R⁻¹ • ContinuousLinearMap.id ℝ (Configuration 1))) x (direction i a) = _ |
| rw [fderiv_comp x ((trialSeed_smooth 1).differentiable (by simp)).differentiableAt |
| (R⁻¹ • ContinuousLinearMap.id ℝ (Configuration 1)).differentiableAt] |
| simp only [ContinuousLinearMap.comp_apply,ContinuousLinearMap.fderiv, |
| smul_apply,ContinuousLinearMap.id_apply,map_smul,smoothForm] |
|
|
| lemma diffuseRaw_mass (R : ℝ) (hR : 0 < R) : |
| formMass (diffuseRaw R) = R ^ Module.finrank ℝ (Configuration 1) * |
| formMass (smoothForm (trialSeed 1)) := by |
| unfold formMass |
| simp only [diffuseRaw,smoothForm,diffuseSeed] |
| have hi := Measure.integral_comp_inv_smul_of_nonneg (volume : Measure (Configuration 1)) |
| (fun x => ‖trialSeed 1 x‖^2) hR.le |
| simp only [smul_eq_mul] at hi |
| simp_rw [hi,Finset.mul_sum] |
|
|
| lemma diffuseRaw_kinetic (R : ℝ) (hR : 0 < R) : |
| formKinetic (diffuseRaw R) = R⁻¹^2 * R ^ Module.finrank ℝ (Configuration 1) * |
| formKinetic (smoothForm (trialSeed 1)) := by |
| unfold formKinetic |
| simp_rw [diffuseRaw_gradient,norm_smul,Real.norm_eq_abs,mul_pow,sq_abs,integral_const_mul] |
| have hi (s : Spins 1) (i : Fin 1) (a : Fin 3) := |
| Measure.integral_comp_inv_smul_of_nonneg (volume : Measure (Configuration 1)) |
| (fun x => ‖(smoothForm (trialSeed 1)).gradient s i a x‖^2) hR.le |
| simp only [smul_eq_mul] at hi |
| simp_rw [hi,← Finset.mul_sum] |
| ring |
|
|
| lemma formKinetic_scale {N : ℕ} (c : ℝ) (ψ : FormVector N) : |
| formKinetic (scaleForm c ψ) = c^2*formKinetic ψ := by |
| unfold formKinetic |
| simp only [scaleForm,norm_real_mul_sq,integral_const_mul,← Finset.mul_sum] |
| ring |
|
|
| def diffuseOrbital (R : ℝ) : FormVector 1 := |
| scaleForm (Real.sqrt (formMass (diffuseRaw R)))⁻¹ (diffuseRaw R) |
|
|
| lemma diffuseOrbital_admissible {R : ℝ} (hR : 0 < R) : FormAdmissible (diffuseOrbital R) := |
| (diffuseRaw_sobolev R hR).normalize (diffuseRaw_mass_pos hR) |
|
|
| def diffuseKinetic : ℝ := |
| formKinetic (smoothForm (trialSeed 1)) / formMass (smoothForm (trialSeed 1)) |
|
|
| lemma diffuseKinetic_nonneg : 0 ≤ diffuseKinetic := |
| div_nonneg (formKinetic_nonneg _) (by unfold formMass; positivity) |
|
|
| lemma diffuseOrbital_kinetic {R : ℝ} (hR : 0 < R) : |
| formKinetic (diffuseOrbital R) = diffuseKinetic / R^2 := by |
| have hmpos : 0 < formMass (smoothForm (trialSeed 1)) := by |
| have hh := diffuseRaw_mass_pos (by norm_num : (0:ℝ)<1) |
| rw [diffuseRaw_mass 1 (by norm_num), one_pow, one_mul] at hh |
| exact hh |
| rw [diffuseOrbital,formKinetic_scale,inv_pow,Real.sq_sqrt (diffuseRaw_mass_pos hR).le, |
| diffuseRaw_mass R hR,diffuseRaw_kinetic R hR,diffuseKinetic] |
| field_simp |
|
|
| lemma diffuseSeed_zero {R : ℝ} (hR : 0 < R) {x : Configuration 1} (hx : ‖x 0‖ < R) : |
| diffuseSeed R x = 0 := by |
| by_contra hn |
| have hh := trialSeed_nucleus hn (0 : Fin 1) |
| change 1 ≤ ‖R⁻¹ • x 0‖ at hh |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hR)] at hh |
| have hl : R⁻¹ * ‖x 0‖ < 1 := by |
| rw [← div_eq_inv_mul,div_lt_one hR] |
| exact hx |
| exact not_lt_of_ge hh hl |
|
|
| lemma diffuseRaw_zero {R : ℝ} (hR : 0 < R) {x : Configuration 1} (hx : ‖x 0‖ < R) : |
| FormZeroAt (diffuseRaw R) x := by |
| have he : diffuseSeed R =ᶠ[𝓝 x] fun _ => (0:ℂ) := |
| (((continuous_apply (0 : Fin 1)).norm.tendsto x).eventually (eventually_lt_nhds hx)).mono |
| fun _ hy => diffuseSeed_zero hR hy |
| intro s |
| refine ⟨diffuseSeed_zero hR hx,fun i a => ?_⟩ |
| change fderiv ℝ (diffuseSeed R) x (direction i a) = 0 |
| rw [he.fderiv_eq] |
| simp |
|
|
| lemma diffuseOrbital_zero {R : ℝ} (hR : 0 < R) {x : Configuration 1} (hx : ‖x 0‖ < R) : |
| FormZeroAt (diffuseOrbital R) x := by |
| intro s |
| have hh := diffuseRaw_zero hR hx s |
| constructor |
| · change _ * (diffuseRaw R).value s x = 0 |
| rw [hh.1,mul_zero] |
| · intro i a |
| change _ * (diffuseRaw R).gradient s i a x = 0 |
| rw [hh.2,mul_zero] |
|
|
|
|
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
|
|
|
|
| lemma tensorCross_le {N : ℕ} {ψ : FormVector N} {φ : FormVector 1} |
| (hψ : SobolevVector ψ) (hφ : SobolevVector φ) {R d : ℝ} (hd : 0 < d) |
| (hc : ∀ s x i, R ≤ ‖x i‖ → ψ.value s x = 0) |
| (he : ∀ s x, ‖x 0‖ < R+d → φ.value s x = 0) : |
| tensorCross ψ φ ≤ (N : ℝ)/d * formMass ψ * formMass φ := by |
| have hT := hψ.tensor hφ |
| have hi (s : Spins N) (t : Spins 1) (i : Fin N) : |
| (∫ z, ‖(tensorForm ψ φ).value (joinLists s t) z‖^2 / ‖z i.castSucc-z (Fin.last N)‖) ≤ |
| d⁻¹ * (∫ x, ‖ψ.value s x‖^2) * (∫ y, ‖φ.value t y‖^2) := by |
| have hn : i.castSucc ≠ Fin.last N := Fin.castSucc_ne_last i |
| calc |
| _ ≤ ∫ z, d⁻¹ * ‖(tensorForm ψ φ).value (joinLists s t) z‖^2 := by |
| apply integral_mono |
| (pair_integrable _ _ hn (hT.1 _) (hT.2.1 _ _) (hT.2.2 _ _)) |
| (((hT.1 _).integrable_norm_pow (by norm_num)).const_mul d⁻¹) |
| intro z |
| dsimp only |
| by_cases hz : (tensorForm ψ φ).value (joinLists s t) z = 0 |
| · simp only [hz,norm_zero,zero_pow (by norm_num : 2 ≠ 0),zero_div,mul_zero,le_refl] |
| · have hz' : ψ.value s (leftList z) ≠ 0 ∧ φ.value t (rightList z) ≠ 0 := by |
| simpa only [tensorForm,leftList_join,rightList_join,mul_ne_zero_iff] using hz |
| have hin : ‖z i.castSucc‖ < R := lt_of_not_ge fun h => hz'.1 (hc _ _ i h) |
| have hout : R+d ≤ ‖z (Fin.last N)‖ := le_of_not_gt fun h => hz'.2 (he _ _ h) |
| have hg : d ≤ ‖z i.castSucc-z (Fin.last N)‖ := by |
| have ht := norm_sub_norm_le (z (Fin.last N)) (z i.castSucc) |
| rw [norm_sub_rev] at ht |
| linarith |
| rw [← div_eq_inv_mul] |
| exact div_le_div_of_nonneg_left (sq_nonneg _) hd hg |
| _ = _ := by |
| rw [integral_const_mul] |
| have hh : (∫ z, ‖(tensorForm ψ φ).value (joinLists s t) z‖^2) = |
| (∫ x, ‖ψ.value s x‖^2)*(∫ y, ‖φ.value t y‖^2) := by |
| simp only [tensorForm,leftList_join,rightList_join,norm_mul,mul_pow] |
| exact integral_product_join (N := N) (M := 1) |
| (fun x => ‖ψ.value s x‖^2) (fun y => ‖φ.value t y‖^2) |
| rw [hh,mul_assoc] |
| calc |
| tensorCross ψ φ ≤ ∑ s : Spins N, ∑ t : Spins 1, ∑ _i : Fin N, |
| d⁻¹ * (∫ x, ‖ψ.value s x‖^2) * (∫ y, ‖φ.value t y‖^2) := by |
| exact Finset.sum_le_sum fun s _ => Finset.sum_le_sum fun t _ => Finset.sum_le_sum fun i _ => hi s t i |
| _ = _ := by |
| simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] |
| unfold formMass |
| simp_rw [mul_assoc,← Finset.mul_sum,← Finset.sum_mul] |
| ring |
| |
| lemma exterior_insertion_trial {N : ℕ} {ψ : FormVector N} (hψ : FormAdmissible ψ) |
| {Z R r : ℝ} (hZ : 0 ≤ Z) (hR : 0 < R) (hr : 2*R ≤ r) |
| (hc : ∀ x i, R ≤ ‖x i‖ → FormZeroAt ψ x) : |
| ∃ F : FormVector (N+1), FormAdmissible F ∧ |
| formEnergy Z F ≤ formEnergy Z ψ + diffuseKinetic/r^2 + 2*(N : ℝ)/r := by |
| have hrp : 0 < r := lt_of_lt_of_le (mul_pos (by norm_num) hR) hr |
| have hφ := diffuseOrbital_admissible hrp |
| obtain ⟨F,hF,hFE⟩ := normalized_wedge_trial hψ hφ R hc |
| (fun x hx => diffuseOrbital_zero hrp (lt_of_lt_of_le hx (by linarith))) Z |
| refine ⟨F,hF,?_⟩ |
| have hcross := tensorCross_le hψ.sobolevFermion.sobolevVector |
| hφ.sobolevFermion.sobolevVector (R := R) (d := r/2) (half_pos hrp) |
| (fun s x i hx => (hc x i hx s).1) |
| (fun s x hx => (diffuseOrbital_zero hrp (lt_of_lt_of_le hx (by linarith)) s).1) |
| have hmψ : formMass ψ = 1 := hψ.2.2.2.2.1 |
| have hmφ : formMass (diffuseOrbital r) = 1 := hφ.2.2.2.2.1 |
| rw [hmψ,hmφ,mul_one,mul_one] at hcross |
| rw [hFE,formEnergy_tensor_one,hmψ,hmφ,mul_one,one_mul,diffuseOrbital_kinetic hrp] |
| have hn := mul_nonneg hZ (formNuclear_nonneg (diffuseOrbital r)) |
| have hc' : (N : ℝ)/(r/2) = 2*(N : ℝ)/r := by ring |
| rw [hc'] at hcross |
| linarith |
| |
| theorem quantum_energy_succ_le {Z : ℝ} (hZ : 0 ≤ Z) (N : ℕ) : |
| energy Z (N+1) ≤ energy Z N := by |
| apply le_of_forall_pos_le_add |
| intro ε hε |
| obtain ⟨R,hR,ψ,hψ,hene,hc⟩ := quantum_sector_compact_near_minimizer Z N (half_pos hε) |
| have ht : Tendsto (fun r : ℝ => diffuseKinetic/r^2 + 2*(N : ℝ)/r) atTop (𝓝 0) := by |
| have hh := (tendsto_const_nhds (x := diffuseKinetic)).div_atTop |
| (tendsto_pow_atTop (by norm_num : 2 ≠ 0)) |
| have hk := (tendsto_const_nhds (x := 2*(N : ℝ))).div_atTop tendsto_id |
| simpa only [zero_add, id_eq] using hh.add hk |
| have he : ∀ᶠ r : ℝ in atTop, diffuseKinetic/r^2 + 2*(N : ℝ)/r < ε/2 := |
| ht.eventually (eventually_lt_nhds (half_pos hε)) |
| obtain ⟨r,hlarge,herror⟩ := ((eventually_ge_atTop (2*R)).and he).exists |
| obtain ⟨F,hF,hbound⟩ := exterior_insertion_trial hψ hZ hR hlarge |
| (fun x i hx s => hc s x i hx) |
| have hl := energy_le_form hZ hF |
| linarith |
| |
| theorem quantum_energy_antitone {Z : ℝ} (hZ : 0 ≤ Z) : Antitone (energy Z) := |
| antitone_nat_of_succ_le (quantum_energy_succ_le hZ) |
| |
| |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| |
| end CoulombAtom |
| |
| end |
| |
| end OAI |
| |
