| import Mathlib |
| import OAI.Analysis.CoulombIonization.Fermionic.Basis |
|
|
| 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 |
| namespace CoulombPackets |
| variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] |
| section Cut |
| variable {α : Type*} [MeasurableSpace α] {μ : Measure α} |
|
|
| def cutLinear (s : Set α) (hs : MeasurableSet s) : Lp ℂ 2 μ →ₗ[ℂ] Lp ℂ 2 μ where |
| toFun u := ((Lp.memLp u).indicator hs).toLp _ |
| map_add' u v := by |
| apply Lp.ext |
| filter_upwards [((Lp.memLp (u+v)).indicator hs).coeFn_toLp, |
| ((Lp.memLp u).indicator hs).coeFn_toLp, ((Lp.memLp v).indicator hs).coeFn_toLp, |
| Lp.coeFn_add (((Lp.memLp u).indicator hs).toLp _) |
| (((Lp.memLp v).indicator hs).toLp _), Lp.coeFn_add u v] with x h1 h2 h3 h4 h5 |
| simp only [h1, h4, Pi.add_apply, h2, h3] |
| by_cases hx : x ∈ s |
| · simpa only [Set.indicator_of_mem hx, Pi.add_apply] using h5 |
| · simp only [Set.indicator_of_notMem hx, add_zero] |
| map_smul' c u := by |
| apply Lp.ext |
| filter_upwards [((Lp.memLp (c • u)).indicator hs).coeFn_toLp, |
| ((Lp.memLp u).indicator hs).coeFn_toLp, |
| Lp.coeFn_smul c (((Lp.memLp u).indicator hs).toLp _), Lp.coeFn_smul c u] |
| with x h1 h2 h3 h4 |
| simp only [RingHom.id_apply, h1, h3, Pi.smul_apply, h2] |
| by_cases hx : x ∈ s |
| · simpa only [Set.indicator_of_mem hx, Pi.smul_apply] using h4 |
| · simp only [Set.indicator_of_notMem hx, smul_zero] |
|
|
| lemma cutLinear_norm_le (s : Set α) (hs : MeasurableSet s) (u : Lp ℂ 2 μ) : |
| ‖cutLinear (μ := μ) s hs u‖ ≤ ‖u‖ := by |
| apply Lp.norm_le_norm_of_ae_le |
| filter_upwards [((Lp.memLp u).indicator hs).coeFn_toLp] with x hx |
| change ‖((Lp.memLp u).indicator hs).toLp _ x‖ ≤ _ |
| rw [hx] |
| by_cases h : x ∈ s <;> simp [h] |
| |
| def cut (s : Set α) (hs : MeasurableSet s) : Lp ℂ 2 μ →L[ℂ] Lp ℂ 2 μ := |
| (cutLinear s hs).mkContinuous 1 (fun u => by simpa using cutLinear_norm_le s hs u) |
|
|
| lemma cut_norm_sq (s : Set α) (hs : MeasurableSet s) (u : Lp ℂ 2 μ) : |
| ‖cut (μ := μ) s hs u‖^2 = ∫ x in s, ‖u x‖^2 ∂μ := by |
| rw [l2_norm_sq, ← integral_indicator hs] |
| apply integral_congr_ae |
| filter_upwards [((Lp.memLp u).indicator hs).coeFn_toLp] with x hx |
| change ‖((Lp.memLp u).indicator hs).toLp _ x‖^2 = _ |
| rw [hx] |
| by_cases h : x ∈ s <;> simp [h] |
| |
| end Cut |
| |
| open scoped InnerProductSpace |
| |
| def selectedFrame (g : 𝓢(V, ℝ)) (s : Set (V × V)) (hs : MeasurableSet s) : |
| Lp ℂ 2 (volume : Measure V) →L[ℂ] |
| Lp ℂ 2 ((volume : Measure V).prod (volume : Measure V)) := |
| cut s hs ∘L frame g |
|
|
| def densityMatrix (g : 𝓢(V, ℝ)) (s : Set (V × V)) (hs : MeasurableSet s) : |
| Lp ℂ 2 (volume : Measure V) →L[ℂ] Lp ℂ 2 (volume : Measure V) := |
| (selectedFrame g s hs).adjoint ∘L selectedFrame g s hs |
|
|
| lemma densityMatrix_positive (g : 𝓢(V, ℝ)) (s : Set (V × V)) (hs : MeasurableSet s) : |
| (densityMatrix g s hs).IsPositive := |
| ContinuousLinearMap.isPositive_adjoint_comp_self _ |
|
|
| lemma selectedFrame_norm_le (g : 𝓢(V, ℝ)) (hg : ∫ x : V, g x^2 = 1) |
| (s : Set (V × V)) (hs : MeasurableSet s) (u : Lp ℂ 2 (volume : Measure V)) : |
| ‖selectedFrame g s hs u‖ ≤ ‖u‖ := by |
| change ‖cut s hs (frame g u)‖ ≤ _ |
| exact (cutLinear_norm_le s hs (frame g u)).trans (frame_norm g hg u).le |
|
|
| lemma densityMatrix_le_one (g : 𝓢(V, ℝ)) (hg : ∫ x : V, g x^2 = 1) |
| (s : Set (V × V)) (hs : MeasurableSet s) : |
| (1 - densityMatrix g s hs).IsPositive := by |
| refine ⟨ContinuousLinearMap.isPositive_one.isSymmetric.sub |
| (densityMatrix_positive g s hs).isSymmetric, fun u => ?_⟩ |
| change 0 ≤ (inner ℂ ((1 - densityMatrix g s hs) u) u).re |
| rw [sub_apply, one_apply_eq_self, inner_sub_left, |
| Complex.sub_re, inner_self_eq_norm_sq_to_K] |
| simp only [← RCLike.ofReal_pow] |
| change 0 ≤ ‖u‖^2 - (inner ℂ (densityMatrix g s hs u) u).re |
| rw [show (inner ℂ (densityMatrix g s hs u) u).re = ‖selectedFrame g s hs u‖^2 from |
| (ContinuousLinearMap.apply_norm_sq_eq_inner_adjoint_left (selectedFrame g s hs) u).symm] |
| exact sub_nonneg.mpr (pow_le_pow_left₀ (norm_nonneg _) (selectedFrame_norm_le g hg s hs u) 2) |
|
|
| lemma densityMatrix_schwartz_form (g : 𝓢(V, ℝ)) (hg : ∫ x : V, g x^2 = 1) |
| (s : Set (V × V)) (hs : MeasurableSet s) (v : 𝓢(V, ℂ)) : |
| (inner ℂ (densityMatrix g s hs (v.toLp 2 volume)) (v.toLp 2 volume)).re = |
| ∫ p in s, ‖frameFunction g v p‖^2 ∂((volume : Measure V).prod volume) := by |
| rw [show (inner ℂ (densityMatrix g s hs (v.toLp 2 volume)) (v.toLp 2 volume)).re = |
| ‖selectedFrame g s hs (v.toLp 2 volume)‖^2 from |
| (ContinuousLinearMap.apply_norm_sq_eq_inner_adjoint_left (selectedFrame g s hs) (v.toLp 2 volume)).symm] |
| change ‖cut s hs (frame g (v.toLp 2 volume))‖^2 = _ |
| rw [frame_toLp g hg, cut_norm_sq] |
| apply integral_congr_ae |
| filter_upwards [ae_restrict_of_ae (frameFunction_memLp g v).coeFn_toLp] with p hp |
| change ‖(frameFunction_memLp g v).toLp _ p‖^2 = _ |
| rw [hp] |
|
|
| def packetFunction (g : 𝓢(V, ℝ)) (p : V × V) (x : V) : ℂ := |
| conj ((Real.fourierChar (-inner ℝ x p.2) : Circle) : ℂ) * (g (x-p.1) : ℂ) |
|
|
| omit [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] in |
| lemma packetFunction_norm (g : 𝓢(V, ℝ)) (p : V × V) (x : V) : |
| ‖packetFunction g p x‖ = ‖g (x-p.1)‖ := by |
| simp [packetFunction, Circle.norm_coe] |
|
|
| lemma packetFunction_memLp (g : 𝓢(V, ℝ)) (p : V × V) : |
| MemLp (packetFunction g p) 2 (volume : Measure V) := by |
| have h : Continuous (packetFunction g p) := by |
| unfold packetFunction |
| fun_prop |
| exact ((g.compSubConstCLM ℝ p.1).memLp 2 volume).congr_norm h.aestronglyMeasurable |
| (Filter.Eventually.of_forall fun x => by |
| simpa only [SchwartzMap.compSubConstCLM_apply] using (packetFunction_norm g p x).symm) |
|
|
| def packet (g : 𝓢(V, ℝ)) (p : V × V) : Lp ℂ 2 (volume : Measure V) := |
| (packetFunction_memLp g p).toLp _ |
|
|
| lemma packet_norm_sq (g : 𝓢(V, ℝ)) (p : V × V) : |
| ‖packet g p‖^2 = ∫ x : V, g x^2 := by |
| rw [l2_norm_sq] |
| calc |
| _ = ∫ x : V, g (x-p.1)^2 := by |
| apply integral_congr_ae |
| filter_upwards [(packetFunction_memLp g p).coeFn_toLp] with x hx |
| change ‖(packetFunction_memLp g p).toLp _ x‖^2 = _ |
| rw [hx, packetFunction_norm, Real.norm_eq_abs, sq_abs] |
| _ = _ := integral_sub_right_eq_self (fun x : V => g x^2) p.1 |
|
|
| lemma packet_frameFunction (g : 𝓢(V, ℝ)) (p : V × V) (v : 𝓢(V, ℂ)) : |
| inner ℂ (packet g p) (v.toLp 2 volume) = frameFunction g v p := by |
| rw [L2.inner_def] |
| simp only [frameFunction, SchwartzMap.fourier_coe, Real.fourier_eq] |
| apply integral_congr_ae |
| filter_upwards [(packetFunction_memLp g p).coeFn_toLp, v.coeFn_toLp 2 volume] with x h1 h2 |
| change inner ℂ ((packetFunction_memLp g p).toLp _ x) _ = _ |
| rw [h1, h2, RCLike.inner_apply, packetFunction, windowed_apply] |
| simp only [map_mul, Complex.conj_conj, Complex.conj_ofReal, Circle.smul_def, smul_eq_mul] |
| ring |
|
|
| lemma frame_ae_packet (g : 𝓢(V, ℝ)) (hg : ∫ x : V, g x^2 = 1) |
| (v : Lp ℂ 2 (volume : Measure V)) : |
| (frame g v : V × V → ℂ) =ᵐ[(volume : Measure V).prod volume] |
| fun p => inner ℂ (packet g p) v := by |
| have hd := SchwartzMap.denseRange_toLpCLM (μ := (volume : Measure V)) |
| (F := ℂ) (p := 2) ENNReal.ofNat_ne_top |
| obtain ⟨u, hu, hv⟩ := mem_closure_iff_seq_limit.1 (hd v) |
| choose w hw using hu |
| have hfr := (frame g).continuous.tendsto v |>.comp hv |
| obtain ⟨ns, hns, hae⟩ := (tendstoInMeasure_of_tendsto_Lp hfr).exists_seq_tendsto_ae |
| have heq (n : ℕ) : (frame g (u n) : V × V → ℂ) =ᵐ[(volume : Measure V).prod volume] |
| fun p => inner ℂ (packet g p) (u n) := by |
| rw [← hw n, SchwartzMap.toLpCLM_apply, frame_toLp g hg] |
| filter_upwards [(frameFunction_memLp g (w n)).coeFn_toLp] with p hp |
| exact hp.trans (packet_frameFunction g p (w n)).symm |
| filter_upwards [hae, ae_all_iff.2 heq] with p hp hp' |
| have hlim : Tendsto (fun n => inner ℂ (packet g p) (u (ns n))) atTop |
| (𝓝 (inner ℂ (packet g p) v)) := tendsto_const_nhds.inner (hv.comp hns.tendsto_atTop) |
| exact tendsto_nhds_unique (hp.congr' (Filter.Eventually.of_forall fun n => hp' (ns n))) hlim |
|
|
| end CoulombPackets |
|
|
| end |
|
|
| end OAI |
|
|
