File size: 10,211 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 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 | import Mathlib
import OAI.Analysis.CoulombIonization.RadialBounds.Radial
import OAI.Analysis.CoulombIonization.Variational.SpinFinite
namespace OAI
noncomputable section
namespace CoulombAtom
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff
open MeasureTheory Filter
open scoped BigOperators ContDiff
lemma compact_potential_bounded {d : Space → ℝ} (hd : Continuous d) (hcd : HasCompactSupport d) :
∃ C : ℝ, ∀ x : Space, |CoulombAnalysis.tfPotential d x| ≤ C := by
have h1 : Integrable d := hd.integrable_of_hasCompactSupport hcd
have hp : MemLp d (5/3 : ENNReal) := hd.memLp_of_hasCompactSupport hcd
obtain ⟨R,_,hR⟩ := hcd.isBounded.exists_pos_norm_le
have hs (y : Space) (hy : d y ≠ 0) : ‖y‖ ≤ R := hR y (subset_tsupport d hy)
obtain ⟨S,_,hS⟩ := CoulombAnalysis.tfPotential_compact_decay h1 hp hs 1 zero_lt_one
obtain ⟨C,hC⟩ := (isCompact_closedBall (0 : Space) S).exists_bound_of_continuousOn
(CoulombAnalysis.tfPotential_continuous h1 hp).continuousOn
refine ⟨max C 1,fun x => ?_⟩
by_cases hx : ‖x‖ ≤ S
· exact (hC x (by simpa using hx)).trans (le_max_left _ _)
· exact (hS x (le_of_not_ge hx)).trans (le_max_right _ _)
lemma core_bounded_potential_integrable {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ)
{V : Space → ℝ} (hV : Continuous V) {C : ℝ} (hC : ∀ x, |V x| ≤ C)
(s : Spins N) (j : Fin N) :
Integrable (fun x : Configuration N => ‖ψ.value s x‖^2 * V (x j)) := by
apply ((hψ.1 s).norm.integrable_sq.mul_const C).mono'
(((hψ.1 s).aestronglyMeasurable.norm.pow 2).mul
(hV.comp (continuous_apply j)).aestronglyMeasurable)
filter_upwards [] with x
change ‖‖ψ.value s x‖^2 * V (x j)‖ ≤ _
rw [norm_mul,Real.norm_of_nonneg (sq_nonneg _),Real.norm_eq_abs]
exact mul_le_mul_of_nonneg_left (hC (x j)) (sq_nonneg _)
lemma core_compact_potential_integrable {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ)
{d : Space → ℝ} (hd : Continuous d) (hcd : HasCompactSupport d) (s : Spins N) (j : Fin N) :
Integrable (fun x : Configuration N => ‖ψ.value s x‖^2 * CoulombAnalysis.tfPotential d (x j)) := by
obtain ⟨C,hC⟩ := compact_potential_bounded hd hcd
exact core_bounded_potential_integrable hψ
(CoulombAnalysis.tfPotential_continuous (hd.integrable_of_hasCompactSupport hcd)
(hd.memLp_of_hasCompactSupport hcd)) hC s j
def coreDensityInteraction {N : ℕ} (ψ : FormVector N) (d : Space → ℝ) : ℝ :=
∑ s : Spins N, ∑ j : Fin N, ∫ x : Configuration N,
‖ψ.value s x‖^2 * CoulombAnalysis.tfPotential d (x j)
lemma spinLift_pole (u : Space → ℂ) (σ : Fin 2) (a : Space) :
orbitalPole (spinLift u σ) a = CoulombAnalysis.tfPotential (fun y => ‖u y‖^2) a := by
unfold orbitalPole CoulombAnalysis.tfPotential
simpa only [div_eq_mul_inv,mul_comm] using spinLift_weight u σ (fun x => ‖a-x‖⁻¹)
lemma spinLift_core_interaction {N : ℕ} (ψ : FormVector N) (u : Space → ℂ) (σ : Fin 2) :
orbitalCoreInteraction ψ (spinLift u σ) = coreDensityInteraction ψ (fun y => ‖u y‖^2) := by
simp only [orbitalCoreInteraction,coreDensityInteraction,spinLift_pole]
section Work_CompactDirect_scope
open MeasureTheory Filter
open scoped BigOperators ContDiff
lemma continuous_compact_coulomb_integrable {f g : Space → ℝ}
(hf : Continuous f) (hg : Continuous g) (hcf : HasCompactSupport f) (hcg : HasCompactSupport g) :
Integrable (fun r : Space × Space => f r.1 * g r.2 / ‖r.1-r.2‖) := by
have hm : Measurable (fun r : Space × Space => f r.1 * g r.2 / ‖r.1-r.2‖) :=
((hf.measurable.comp measurable_fst).mul (hg.measurable.comp measurable_snd)).div
((measurable_fst.sub measurable_snd).norm)
have hg1 : Integrable g := hg.integrable_of_hasCompactSupport hcg
have hgp : MemLp g (5/3 : ENNReal) := hg.memLp_of_hasCompactSupport hcg (p := (5/3 : ENNReal))
have hgn1 : Integrable (fun x => ‖g x‖) := hg.norm.integrable_of_hasCompactSupport hcg.norm
have hgnp : MemLp (fun x => ‖g x‖) (5/3 : ENNReal) := hg.norm.memLp_of_hasCompactSupport hcg.norm (p := (5/3 : ENNReal))
apply (integrable_prod_iff hm.aestronglyMeasurable).2
refine ⟨Eventually.of_forall (fun x => ?_),?_⟩
· simpa only [mul_div_assoc] using (CoulombAnalysis.tfPotential_integrable hg1 hgp x).const_mul (f x)
· have hi : Integrable (fun x => ‖f x‖ * CoulombAnalysis.tfPotential (fun y => ‖g y‖) x) := (hf.norm.mul (CoulombAnalysis.tfPotential_continuous hgn1 hgnp)).integrable_of_hasCompactSupport
hcf.norm.mul_right
apply hi.congr
filter_upwards [] with x
simp only [norm_div,norm_mul,norm_norm,CoulombAnalysis.tfPotential,← integral_const_mul]
congr 1
funext y
ring
lemma spatialFiniteDensity_continuous {ι : Type*} [Fintype ι] (p : ι → ℝ) {u : ι → Space → ℂ}
(hu : ∀ i, Continuous (u i)) : Continuous (spatialFiniteDensity p u) := by
unfold spatialFiniteDensity
fun_prop
lemma spatialFiniteDensity_compact {ι : Type*} [Fintype ι] (p : ι → ℝ) {u : ι → Space → ℂ}
(hu : ∀ i, HasCompactSupport (u i)) : HasCompactSupport (spatialFiniteDensity p u) := by
unfold spatialFiniteDensity
classical
induction (Finset.univ : Finset ι) using Finset.induction_on with
| empty =>
simp only [Finset.sum_empty]
exact HasCompactSupport.zero
| @insert a s ha ih =>
simp only [Finset.sum_insert ha]
exact ((compact_norm_sq (hu a)).mul_left).add ih
lemma spatialFiniteDensity_nonneg {ι : Type*} [Fintype ι] {p : ι → ℝ} {u : ι → Space → ℂ}
(hp : ∀ i, 0 ≤ p i) (x : Space) : 0 ≤ spatialFiniteDensity p u x :=
Finset.sum_nonneg (fun i _ => mul_nonneg (hp i) (sq_nonneg _))
lemma compact_direct_mono {f g : Space → ℝ} (hf : Continuous f) (hg : Continuous g)
(hcf : HasCompactSupport f) (hcg : HasCompactSupport g)
(hn : ∀ x, 0 ≤ f x) (hb : ∀ x, f x ≤ g x) :
(∫ r : Space × Space, f r.1 * f r.2 / ‖r.1-r.2‖) ≤
∫ r : Space × Space, g r.1 * g r.2 / ‖r.1-r.2‖ := by
apply integral_mono (continuous_compact_coulomb_integrable hf hf hcf hcf)
(continuous_compact_coulomb_integrable hg hg hcg hcg)
intro r
exact div_le_div_of_nonneg_right (mul_le_mul (hb r.1) (hb r.2) (hn r.2)
((hn r.1).trans (hb r.1))) (norm_nonneg _)
end Work_CompactDirect_scope
open MeasureTheory Filter Set Metric
open scoped BigOperators ContDiff
lemma bounded_compact_memLp {f : Space → ℝ} (hf : Measurable f)
{J : Set Space} (hJ : IsCompact J) (hs : Function.support f ⊆ J)
{B : ℝ} (hb : ∀ x, |f x| ≤ B) (p : ENNReal) : MemLp f p := by
have hi := memLp_indicator_const (μ := volume) p hJ.measurableSet B (Or.inr hJ.measure_lt_top.ne)
apply hi.mono' hf.aestronglyMeasurable
filter_upwards [] with x
by_cases hx : x ∈ J
· simpa only [Set.indicator_of_mem hx,Real.norm_eq_abs] using hb x
· have hz : f x = 0 := by by_contra hn; exact hx (hs hn)
simp only [Set.indicator_of_notMem hx,hz,norm_zero,le_refl]
lemma bounded_compact_integrable {f : Space → ℝ} (hf : Measurable f)
{J : Set Space} (hJ : IsCompact J) (hs : Function.support f ⊆ J)
{B : ℝ} (hb : ∀ x, |f x| ≤ B) : Integrable f :=
memLp_one_iff_integrable.mp (bounded_compact_memLp hf hJ hs hb 1)
lemma compact_support_potential_bounded {f : Space → ℝ} (h1 : Integrable f)
(hp : MemLp f (5/3 : ENNReal)) {J : Set Space} (hJ : IsCompact J)
(hs : Function.support f ⊆ J) :
∃ C : ℝ, ∀ x : Space, |CoulombAnalysis.tfPotential f x| ≤ C := by
obtain ⟨R,_,hR⟩ := hJ.isBounded.exists_pos_norm_le
have hr (y : Space) (hy : f y ≠ 0) : ‖y‖ ≤ R := hR y (hs hy)
obtain ⟨S,_,hS⟩ := CoulombAnalysis.tfPotential_compact_decay h1 hp hr 1 zero_lt_one
obtain ⟨C,hC⟩ := (isCompact_closedBall (0 : Space) S).exists_bound_of_continuousOn
(CoulombAnalysis.tfPotential_continuous h1 hp).continuousOn
refine ⟨max C 1,fun x => ?_⟩
by_cases hx : ‖x‖ ≤ S
· exact (hC x (by simpa using hx)).trans (le_max_left _ _)
· exact (hS x (le_of_not_ge hx)).trans (le_max_right _ _)
lemma compact_support_coulomb_integrable {f g : Space → ℝ}
(hfm : Measurable f) (hgm : Measurable g) (hf : Integrable f) (hg : Integrable g)
(hgp : MemLp g (5/3 : ENNReal)) {J : Set Space} (hJ : IsCompact J)
(hs : Function.support g ⊆ J) :
Integrable (fun r : Space × Space => f r.1*g r.2/‖r.1-r.2‖) := by
have hm : Measurable (fun r : Space × Space => f r.1*g r.2/‖r.1-r.2‖) := by fun_prop
apply (integrable_prod_iff hm.aestronglyMeasurable).2
refine ⟨Eventually.of_forall (fun x => ?_),?_⟩
· simpa only [mul_div_assoc] using (CoulombAnalysis.tfPotential_integrable hg hgp x).const_mul (f x)
· obtain ⟨C,hC⟩ := compact_support_potential_bounded hg.norm hgp.norm hJ
(fun x hx => hs (by intro hz; exact hx (by simp [hz])))
have hi : Integrable (fun x => ‖f x‖*CoulombAnalysis.tfPotential (fun y => ‖g y‖) x) := by
apply (hf.norm.mul_const C).mono'
((hfm.norm.mul (CoulombAnalysis.tfPotential_continuous hg.norm hgp.norm).measurable).aestronglyMeasurable)
filter_upwards [] with x
change ‖‖f x‖*CoulombAnalysis.tfPotential (fun y => ‖g y‖) x‖ ≤ ‖f x‖*C
rw [norm_mul,norm_norm,Real.norm_eq_abs]
exact mul_le_mul_of_nonneg_left (hC x) (abs_nonneg _)
apply hi.congr
filter_upwards [] with x
simp only [norm_div,norm_mul,norm_norm,CoulombAnalysis.tfPotential,← integral_const_mul]
congr 1
funext y
ring
lemma bounded_compact_coulomb_integrable {f g : Space → ℝ}
(hfm : Measurable f) (hgm : Measurable g)
{J K : Set Space} (hJ : IsCompact J) (hK : IsCompact K)
(hsf : Function.support f ⊆ J) (hsg : Function.support g ⊆ K)
{B C : ℝ} (hb : ∀ x, |f x| ≤ B) (hc : ∀ x, |g x| ≤ C) :
Integrable (fun r : Space × Space => f r.1*g r.2/‖r.1-r.2‖) :=
compact_support_coulomb_integrable hfm hgm (bounded_compact_integrable hfm hJ hsf hb)
(bounded_compact_integrable hgm hK hsg hc) (bounded_compact_memLp hgm hK hsg hc _) hK hsg
end CoulombAtom
end
end OAI
|