| import Mathlib |
| import OAI.Analysis.CoulombIonization.FormDomain.CoherentSchwartz |
|
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| namespace OAI |
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| noncomputable section |
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| namespace CoulombAtom |
|
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| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| section Work_CompactEigenbasis_scope |
|
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| open MeasureTheory Filter Module |
| open scoped BigOperators ComplexConjugate Topology |
|
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| universe u |
| variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
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| def compactEigenspaceIndex (T : H →L[ℂ] H) (μ : ℂ) : Set (Module.End.eigenspace T.toLinearMap μ) := |
| (exists_hilbertBasis ℂ (Module.End.eigenspace T.toLinearMap μ)).choose |
|
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| def compactEigenspaceBasis (T : H →L[ℂ] H) (μ : ℂ) : |
| HilbertBasis (compactEigenspaceIndex T μ) ℂ (Module.End.eigenspace T.toLinearMap μ) := |
| (exists_hilbertBasis ℂ (Module.End.eigenspace T.toLinearMap μ)).choose_spec.choose |
|
|
| abbrev CompactSpectralIndex (T : H →L[ℂ] H) := Σ μ : ℂ, compactEigenspaceIndex T μ |
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| def compactSpectralVector (T : H →L[ℂ] H) (i : CompactSpectralIndex T) : H := |
| compactEigenspaceBasis T i.1 i.2 |
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| lemma compactSpectralVector_eigen (T : H →L[ℂ] H) (i : CompactSpectralIndex T) : |
| T (compactSpectralVector T i) = i.1 • compactSpectralVector T i := |
| Module.End.mem_eigenspace_iff.mp (compactEigenspaceBasis T i.1 i.2).property |
|
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| lemma compactSpectralVector_orthonormal (T : H →L[ℂ] H) (hT : T.toLinearMap.IsSymmetric) : |
| Orthonormal ℂ (compactSpectralVector T) := |
| hT.orthogonalFamily_eigenspaces.orthonormal_sigma_orthonormal |
| (fun μ => (compactEigenspaceBasis T μ).orthonormal) |
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| lemma compactSpectralVector_orthogonal_eq_bot (T : H →L[ℂ] H) |
| (hTc : IsCompactOperator T) (hTs : T.toLinearMap.IsSymmetric) : |
| (Submodule.span ℂ (Set.range (compactSpectralVector T)))ᗮ = ⊥ := by |
| let S := Submodule.span ℂ (Set.range (compactSpectralVector T)) |
| have hs : (⨆ μ, Module.End.eigenspace T.toLinearMap μ) ≤ S.topologicalClosure := by |
| apply iSup_le |
| intro μ y hy |
| let v : Module.End.eigenspace T.toLinearMap μ := ⟨y,hy⟩ |
| have hh := ((compactEigenspaceBasis T μ).hasSum_repr v).mapL |
| (Module.End.eigenspace T.toLinearMap μ).subtypeL |
| apply mem_closure_of_tendsto hh |
| apply Eventually.of_forall |
| intro s |
| apply Submodule.sum_mem |
| intro i _ |
| change ((compactEigenspaceBasis T μ).repr v i) • (compactSpectralVector T ⟨μ,i⟩) ∈ S |
| exact S.smul_mem _ (Submodule.subset_span ⟨⟨μ,i⟩,rfl⟩) |
| apply le_antisymm _ bot_le |
| rw [← S.orthogonal_closure] |
| exact (Submodule.orthogonal_le hs).trans |
| (ContinuousLinearMap.orthogonalComplement_iSup_eigenspaces_eq_bot hTc hTs).le |
|
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| def compactSpectralBasis (T : H →L[ℂ] H) |
| (hTc : IsCompactOperator T) (hTs : T.toLinearMap.IsSymmetric) : |
| HilbertBasis (CompactSpectralIndex T) ℂ H := |
| HilbertBasis.mkOfOrthogonalEqBot (compactSpectralVector_orthonormal T hTs) |
| (compactSpectralVector_orthogonal_eq_bot T hTc hTs) |
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| lemma compactSpectralBasis_apply (T : H →L[ℂ] H) |
| (hTc : IsCompactOperator T) (hTs : T.toLinearMap.IsSymmetric) |
| (i : CompactSpectralIndex T) : compactSpectralBasis T hTc hTs i = compactSpectralVector T i := by |
| rw [compactSpectralBasis,HilbertBasis.coe_mkOfOrthogonalEqBot] |
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| lemma compactSpectralBasis_eigen (T : H →L[ℂ] H) |
| (hTc : IsCompactOperator T) (hTs : T.toLinearMap.IsSymmetric) |
| (i : CompactSpectralIndex T) : |
| T (compactSpectralBasis T hTc hTs i) = i.1 • compactSpectralBasis T hTc hTs i := by |
| simp only [compactSpectralBasis_apply,compactSpectralVector_eigen] |
|
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| end Work_CompactEigenbasis_scope |
|
|
| section Work_HilbertCountable_scope |
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| open scoped Topology |
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| universe u v |
| lemma orthonormal_countable {H : Type u} {ι : Type v} |
| [NormedAddCommGroup H] [InnerProductSpace ℂ H] [TopologicalSpace.SeparableSpace H] |
| {v : ι → H} (hv : Orthonormal ℂ v) : Countable ι := by |
| have hd : Pairwise (fun i j => Disjoint (Metric.ball (v i) (1/2 : ℝ)) (Metric.ball (v j) (1/2 : ℝ))) := by |
| intro i j hij |
| apply Metric.ball_disjoint_ball |
| have hs : ‖v i-v j‖^2 = 2 := by |
| rw [norm_sub_sq (𝕜 := ℂ),hv.norm_eq_one,hv.norm_eq_one,hv.inner_eq_zero hij] |
| norm_num |
| rw [dist_eq_norm] |
| nlinarith [norm_nonneg (v i-v j)] |
| exact hd.countable_of_isOpen_disjoint (fun _ => Metric.isOpen_ball) |
| (fun _ => Metric.nonempty_ball.mpr (by norm_num)) |
|
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| end Work_HilbertCountable_scope |
|
|
| section Work_HilbertTrace_scope |
|
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| open MeasureTheory Filter |
| open scoped BigOperators Topology |
|
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| universe u v w |
| variable {H : Type u} {ι : Type v} [NormedAddCommGroup H] [InnerProductSpace ℂ H] |
|
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| lemma hilbertBasis_hasSum_norm_inner (b : HilbertBasis ι ℂ H) (x : H) : |
| HasSum (fun i => ‖inner ℂ (b i) x‖^2) (‖x‖^2) := by |
| simpa only [ENNReal.toReal_ofNat,Real.rpow_two,b.repr_apply_apply, |
| LinearIsometryEquiv.norm_map] using lp.hasSum_norm (by norm_num : 0 < (2:ENNReal).toReal) (b.repr x) |
|
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| lemma hilbertBasis_frame_trace {α : Type w} [MeasurableSpace α] |
| (μ : Measure α) (b : HilbertBasis ι ℂ H) [Countable ι] (v : α → H) |
| (hv : AEStronglyMeasurable v μ) (hn : Integrable (fun x => ‖v x‖^2) μ) : |
| HasSum (fun i => ∫ x, ‖inner ℂ (b i) (v x)‖^2 ∂μ) (∫ x, ‖v x‖^2 ∂μ) := by |
| have hmeas (i : ι) : AEStronglyMeasurable (fun x => ‖inner ℂ (b i) (v x)‖^2) μ := |
| (((continuous_const.inner continuous_id).norm.pow 2).comp_aestronglyMeasurable hv) |
| apply hasSum_integral_of_dominated_convergence |
| (fun i x => ‖inner ℂ (b i) (v x)‖^2) hmeas |
| · intro i |
| exact Eventually.of_forall (fun x => (Real.norm_of_nonneg (sq_nonneg _)).le) |
| · exact Eventually.of_forall (fun x => b.orthonormal.inner_products_summable (v x)) |
| · convert hn using 1 |
| ext x |
| exact (hilbertBasis_hasSum_norm_inner b (v x)).tsum_eq |
| · exact Eventually.of_forall (fun x => hilbertBasis_hasSum_norm_inner b (v x)) |
|
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| end Work_HilbertTrace_scope |
|
|
| open MeasureTheory Filter |
| open scoped BigOperators ComplexConjugate ContDiff Topology |
|
|
| instance orbitalHilbert_secondCountable : SecondCountableTopology OrbitalHilbert := by |
| let : Fact ((2 : ENNReal) ≠ ⊤) := ⟨ENNReal.ofNat_ne_top⟩ |
| infer_instance |
|
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| abbrev CoherentIndex {g : Space → ℂ} (hg : Continuous g) (hcg : HasCompactSupport g) |
| (μ : Measure (Space × Space)) := CompactSpectralIndex (coherentOperator hg hcg μ) |
|
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| def coherentBasis {g : Space → ℂ} (hg : Continuous g) (hcg : HasCompactSupport g) |
| (μ : Measure (Space × Space)) [IsFiniteMeasure μ] : |
| HilbertBasis (CoherentIndex hg hcg μ) ℂ OrbitalHilbert := |
| compactSpectralBasis _ (coherentOperator_compact hg hcg μ) |
| (coherentOperator_positive hg hcg μ).isSymmetric |
|
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| instance coherentIndex_countable {g : Space → ℂ} (hg : Continuous g) (hcg : HasCompactSupport g) |
| (μ : Measure (Space × Space)) [IsFiniteMeasure μ] : Countable (CoherentIndex hg hcg μ) := |
| orthonormal_countable (coherentBasis hg hcg μ).orthonormal |
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| lemma coherentBasis_eigen {g : Space → ℂ} (hg : Continuous g) (hcg : HasCompactSupport g) |
| (μ : Measure (Space × Space)) [IsFiniteMeasure μ] (i : CoherentIndex hg hcg μ) : |
| coherentOperator hg hcg μ (coherentBasis hg hcg μ i) = |
| i.1 • coherentBasis hg hcg μ i := compactSpectralBasis_eigen _ _ _ i |
|
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| lemma coherentBasis_inner_eigen {g : Space → ℂ} (hg : Continuous g) (hcg : HasCompactSupport g) |
| (μ : Measure (Space × Space)) [IsFiniteMeasure μ] (i : CoherentIndex hg hcg μ) : |
| inner ℂ (coherentBasis hg hcg μ i) (coherentOperator hg hcg μ (coherentBasis hg hcg μ i)) = i.1 := by |
| rw [coherentBasis_eigen,inner_smul_right,inner_self_eq_norm_sq_to_K, |
| (coherentBasis hg hcg μ).orthonormal.norm_eq_one] |
| simp |
|
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| lemma coherentBasis_eigenvalue_eq {g : Space → ℂ} (hg : Continuous g) (hcg : HasCompactSupport g) |
| (μ : Measure (Space × Space)) [IsFiniteMeasure μ] (i : CoherentIndex hg hcg μ) : |
| i.1 = (((2*Real.pi)^3)⁻¹ * ∫ q, |
| ‖inner ℂ (coherentBasis hg hcg μ i) (coherentVector hg hcg q)‖^2 ∂μ : ℝ) := by |
| rw [← coherentBasis_inner_eigen hg hcg μ i,coherentOperator_inner] |
| congr 2 |
| apply integral_congr_ae |
| exact Eventually.of_forall (fun q => congrArg (fun t : ℝ => t^2) (norm_inner_symm _ _)) |
|
|
| def coherentWeight {g : Space → ℂ} (hg : Continuous g) (hcg : HasCompactSupport g) |
| (μ : Measure (Space × Space)) (i : CoherentIndex hg hcg μ) : ℝ := i.1.re |
|
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| lemma coherentWeight_eq {g : Space → ℂ} (hg : Continuous g) (hcg : HasCompactSupport g) |
| (μ : Measure (Space × Space)) [IsFiniteMeasure μ] (i : CoherentIndex hg hcg μ) : |
| coherentWeight hg hcg μ i = ((2*Real.pi)^3)⁻¹ * ∫ q, |
| ‖inner ℂ (coherentBasis hg hcg μ i) (coherentVector hg hcg q)‖^2 ∂μ := by |
| rw [coherentWeight,coherentBasis_eigenvalue_eq,Complex.ofReal_re] |
|
|
| lemma coherentWeight_nonneg {g : Space → ℂ} (hg : Continuous g) (hcg : HasCompactSupport g) |
| (μ : Measure (Space × Space)) [IsFiniteMeasure μ] (i : CoherentIndex hg hcg μ) : |
| 0 ≤ coherentWeight hg hcg μ i := by |
| rw [coherentWeight_eq] |
| exact mul_nonneg (by positivity) (integral_nonneg (fun _ => sq_nonneg _)) |
|
|
| lemma coherentBasis_real_eigen {g : Space → ℂ} (hg : Continuous g) (hcg : HasCompactSupport g) |
| (μ : Measure (Space × Space)) [IsFiniteMeasure μ] (i : CoherentIndex hg hcg μ) : |
| coherentOperator hg hcg μ (coherentBasis hg hcg μ i) = |
| coherentWeight hg hcg μ i • coherentBasis hg hcg μ i := by |
| rw [coherentBasis_eigen,coherentBasis_eigenvalue_eq,coherentWeight_eq] |
| rw [Complex.coe_smul] |
| rfl |
|
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| lemma coherentWeight_le_one {g : Space → ℂ} (hg : ContDiff ℝ ∞ g) |
| (hcg : HasCompactSupport g) (hgn : ∫ x : Space, ‖g x‖^2 = 1) |
| (μ : Measure (Space × Space)) [IsFiniteMeasure μ] (hμ : μ ≤ volume) |
| (i : CoherentIndex hg.continuous hcg μ) : coherentWeight hg.continuous hcg μ i ≤ 1 := by |
| have h := coherentOperator_quadratic_le hg hcg hgn μ hμ (coherentBasis hg.continuous hcg μ i) |
| rw [coherentBasis_inner_eigen,(coherentBasis hg.continuous hcg μ).orthonormal.norm_eq_one] at h |
| change i.1.re ≤ 1 |
| simpa only [one_pow] using h |
|
|
| lemma coherentWeight_hasSum {g : Space → ℂ} (hg : Continuous g) (hcg : HasCompactSupport g) |
| (μ : Measure (Space × Space)) [IsFiniteMeasure μ] : |
| HasSum (coherentWeight hg hcg μ) (((2*Real.pi)^3)⁻¹ * μ.real Set.univ * ∫ x : Space, ‖g x‖^2) := by |
| have hn : Integrable (fun q => ‖coherentVector hg hcg q‖^2) μ := by |
| simp_rw [coherentVector_norm_sq] |
| exact integrable_const _ |
| have ht := hilbertBasis_frame_trace μ (coherentBasis hg hcg μ) (coherentVector hg hcg) |
| (coherentVector_continuous hg hcg).aestronglyMeasurable hn |
| have he : (∫ q, ‖coherentVector hg hcg q‖^2 ∂μ) = μ.real Set.univ * ∫ x : Space, ‖g x‖^2 := by |
| simp only [coherentVector_norm_sq,integral_const,smul_eq_mul] |
| rw [he] at ht |
| have hsum := ht.mul_left (((2*Real.pi)^3)⁻¹) |
| have hf : coherentWeight hg hcg μ = (fun i => ((2*Real.pi)^3)⁻¹ * |
| ∫ q, ‖inner ℂ (coherentBasis hg hcg μ i) (coherentVector hg hcg q)‖^2 ∂μ) := |
| funext (coherentWeight_eq hg hcg μ) |
| rw [hf,mul_assoc] |
| exact hsum |
|
|
| end CoulombAtom |
|
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| end |
|
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| end OAI |
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