| import Mathlib |
| import OAI.Analysis.CoulombIonization.Localization.CoherentPackets |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
|
|
| open MeasureTheory |
| open scoped BigOperators ComplexConjugate ContDiff FourierTransform |
|
|
| namespace CoulombAtom |
|
|
| lemma coherentOverlap_continuous {g u : Space → ℂ} |
| (hg : Continuous g) (hcg : HasCompactSupport g) |
| (hu : Continuous u) (hcu : HasCompactSupport u) : |
| Continuous (fun q : Space × Space => coherentOverlap g u q.1 q.2) := by |
| obtain ⟨C,hC⟩ := hg.norm.exists_forall_ge_of_hasCompactSupport hcg.norm |
| apply continuous_of_dominated (bound := fun x => ‖g C‖ * ‖u x‖) |
| · intro q |
| unfold coherentPacket |
| fun_prop |
| · intro q |
| filter_upwards [] with x |
| rw [norm_mul,Complex.norm_conj,coherentPacket_norm] |
| exact mul_le_mul_of_nonneg_right (hC (x-q.1)) (norm_nonneg _) |
| · exact (hu.norm.integrable_of_hasCompactSupport hcu.norm).const_mul ‖g C‖ |
| · filter_upwards [] with x |
| unfold coherentPacket |
| fun_prop |
|
|
| lemma coherentOverlap_momentum_integrable {g u : Space → ℂ} |
| (hg : ContDiff ℝ ∞ g) (hu : ContDiff ℝ ∞ u) (hcu : HasCompactSupport u) (z : Space) : |
| Integrable (fun p : Space => ‖coherentOverlap g u z p‖^2) := by |
| rw [← integrable_comp_smul_iff volume (fun p : Space => ‖coherentOverlap g u z p‖^2) |
| (show 2*Real.pi ≠ 0 by positivity)] |
| simp_rw [coherentOverlap_fourier] |
| have hc : HasCompactSupport (fun x : Space => conj (g (x-z))*u x) := hcu.mul_left |
| have hd : ContDiff ℝ ∞ (fun x : Space => conj (g (x-z))*u x) := |
| (Complex.conjCLE.contDiff.comp (hg.comp (contDiff_id.sub contDiff_const))).mul hu |
| have hi := ((𝓕 (hc.toSchwartzMap hd)).memLp 2 volume).norm.integrable_sq |
| change Integrable (fun x : Space => ‖𝓕 (fun x : Space => conj (g (x-z))*u x) x‖^2) at hi |
| exact hi |
|
|
| lemma coherent_position_integrable {g u : Space → ℂ} |
| (hg : Continuous g) (hcg : HasCompactSupport g) |
| (hu : Continuous u) (hcu : HasCompactSupport u) : |
| Integrable (fun q : Space × Space => ‖g (q.2-q.1)‖^2 * ‖u q.2‖^2) := by |
| have hgs : Integrable (fun z : Space => ‖g z‖^2) := |
| (hg.memLp_of_hasCompactSupport hcg (p := 2)).norm.integrable_sq |
| have hus : Integrable (fun z : Space => ‖u z‖^2) := |
| (hu.memLp_of_hasCompactSupport hcu (p := 2)).norm.integrable_sq |
| apply (integrable_prod_iff' (by fun_prop)).mpr |
| constructor |
| · filter_upwards [] with x |
| exact (hgs.comp_sub_left x).mul_const _ |
| · have he (x : Space) : |
| (∫ z : Space, ‖‖g (x-z)‖^2 * ‖u x‖^2‖) = (∫ z : Space, ‖g z‖^2) * ‖u x‖^2 := by |
| simp_rw [norm_mul,norm_pow,Real.norm_of_nonneg (norm_nonneg _)] |
| rw [integral_mul_const,integral_sub_left_eq_self (fun t : Space => ‖g t‖^2) volume x] |
| change Integrable (fun x : Space => ∫ z : Space, ‖‖g (x-z)‖^2 * ‖u x‖^2‖) |
| simp_rw [he] |
| exact hus.const_mul _ |
| |
| lemma coherentOverlap_resolution {g u : Space → ℂ} |
| (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g) |
| (hu : ContDiff ℝ ∞ u) (hcu : HasCompactSupport u) : |
| ((2*Real.pi)^3)⁻¹ * (∫ z : Space, ∫ p : Space, ‖coherentOverlap g u z p‖^2) = |
| (∫ z : Space, ‖g z‖^2) * (∫ x : Space, ‖u x‖^2) := by |
| rw [← integral_const_mul] |
| simp_rw [coherentOverlap_momentum hg hu hcu] |
| rw [integral_integral_swap (coherent_position_integrable hg.continuous hcg hu.continuous hcu)] |
| simp_rw [integral_mul_const,integral_sub_left_eq_self (fun t : Space => ‖g t‖^2) volume] |
| rw [integral_const_mul] |
| |
| lemma coherentOverlap_integrable {g u : Space → ℂ} |
| (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g) |
| (hu : ContDiff ℝ ∞ u) (hcu : HasCompactSupport u) : |
| Integrable (fun q : Space × Space => ‖coherentOverlap g u q.1 q.2‖^2) := by |
| apply (integrable_prod_iff ((coherentOverlap_continuous hg.continuous hcg hu.continuous hcu).norm.pow 2).aestronglyMeasurable).mpr |
| constructor |
| · exact Filter.Eventually.of_forall (coherentOverlap_momentum_integrable hg hu hcu) |
| · have he (z : Space) : (∫ p : Space, ‖‖coherentOverlap g u z p‖^2‖) = |
| (2*Real.pi)^3 * (∫ x : Space, ‖g (x-z)‖^2 * ‖u x‖^2) := by |
| simp_rw [norm_pow,Real.norm_of_nonneg (norm_nonneg _)] |
| have h := coherentOverlap_momentum hg hu hcu z |
| have hn : (2*Real.pi)^3 ≠ 0 := by positivity |
| calc |
| _ = (2*Real.pi)^3 * (((2*Real.pi)^3)⁻¹ * _) := by field_simp |
| _ = _ := by rw [h] |
| change Integrable (fun z : Space => ∫ p : Space, ‖‖coherentOverlap g u z p‖^2‖) |
| simp_rw [he] |
| exact (coherent_position_integrable hg.continuous hcg hu.continuous hcu).integral_prod_left.const_mul _ |
| |
| end CoulombAtom |
| |
| end |
| |
| end OAI |
| |
