| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.HeatScalar |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory Filter ContinuousLinearMap |
| open scoped Topology ComplexOrder |
| namespace HardSphere.CompactForm |
|
|
| def cutoff (δ x : ℝ) : ℝ := Real.smoothTransition (((2 : ℕ) : ℝ)*x/δ-1) |
| lemma cutoff_smooth (δ : ℝ) : ContDiff ℝ (⊤ : ℕ∞) (cutoff δ) := by |
| exact Real.smoothTransition.contDiff.comp ((contDiff_const.mul contDiff_id).div_const δ |>.sub contDiff_const) |
| lemma cutoff_nonneg (δ x : ℝ) : 0≤cutoff δ x := Real.smoothTransition.nonneg _ |
| lemma cutoff_le_one (δ x : ℝ) : cutoff δ x≤1 := Real.smoothTransition.le_one _ |
| lemma cutoff_zero {δ x : ℝ} (hδ : 0<δ) (hx : x≤δ/2) : cutoff δ x=0 := by |
| apply Real.smoothTransition.zero_of_nonpos |
| have hh : 2*x/δ≤1 := (div_le_one hδ).mpr (by linarith) |
| linarith |
| lemma cutoff_one {δ x : ℝ} (hδ : 0<δ) (hx : δ≤x) : cutoff δ x=1 := by |
| apply Real.smoothTransition.one_of_one_le |
| have hh : 2≤2*x/δ := (le_div_iff₀ hδ).mpr (by linarith) |
| linarith |
| lemma cutoff_error {δ x : ℝ} (hδ : 0<δ) (hx : 0≤x) : |
| ‖(1-cutoff δ x)^2*x‖≤δ := by |
| have hq := cutoff_nonneg δ x |
| have hq' := cutoff_le_one δ x |
| rw [Real.norm_eq_abs,abs_of_nonneg (mul_nonneg (sq_nonneg _) hx)] |
| by_cases hh : δ≤x |
| · rw [cutoff_one hδ hh]; simpa using hδ.le |
| · have hx' : x≤δ := (lt_of_not_ge hh).le |
| have hs : (1-cutoff δ x)^2≤1 := by nlinarith |
| exact (mul_le_mul_of_nonneg_right hs hx).trans (by simpa using hx') |
|
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| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| omit [CompleteSpace H] in |
| lemma finite_range_compact (F : H →L[ℂ] H) (W : Submodule ℂ H) |
| [FiniteDimensional ℂ W] (hW : ∀x,F x∈W) : IsCompactOperator F := by |
| have hh := isCompactOperator_of_locallyCompactSpace_dom (F.codRestrict W hW) |
| exact hh.clm_comp W.subtypeL |
|
|
| lemma cutoff_compact {ι : Type*} (T : H →L[ℂ] H) (hT : IsSelfAdjoint T) |
| (v : ι→H) (r : ι→ℝ) |
| (hdense : (Submodule.span ℂ (range v)).topologicalClosure=⊤) |
| (heigen : ∀i,T (v i)=r i • v i) |
| (hfinite : ∀ε : ℝ,0<ε → {i | ε≤r i}.Finite) |
| {δ : ℝ} (hδ : 0<δ) : |
| IsCompactOperator (cfc (p:=IsSelfAdjoint) (cutoff δ) T : H →L[ℂ] H) := by |
| let S : Set ι := {i | δ/2≤r i} |
| have hS : S.Finite := hfinite _ (by positivity) |
| let W : Submodule ℂ H := Submodule.span ℂ (v '' S) |
| have : FiniteDimensional ℂ W := FiniteDimensional.span_of_finite ℂ (hS.image v) |
| apply finite_range_compact _ W |
| have hgen : Submodule.span ℂ (range v) ≤ |
| W.comap (cfc (p:=IsSelfAdjoint) (cutoff δ) T).toLinearMap := by |
| apply Submodule.span_le.mpr |
| rintro _ ⟨i,rfl⟩ |
| change cfc (p:=IsSelfAdjoint) (cutoff δ) T (v i)∈W |
| rw [cfc_eigen_apply T hT _ (cutoff_smooth δ).continuous _ |
| ((cutoff_smooth δ).differentiable (by norm_num)).differentiableAt _ (heigen i)] |
| by_cases hi : i∈S |
| · exact W.smul_mem _ (Submodule.subset_span ⟨i,hi,rfl⟩) |
| · have hr : r i≤δ/2 := (lt_of_not_ge hi).le |
| rw [cutoff_zero hδ hr,zero_smul] |
| exact W.zero_mem |
| have hclosed : IsClosed (W.comap (cfc (p:=IsSelfAdjoint) (cutoff δ) T).toLinearMap : Set H) := |
| W.closed_of_finiteDimensional.preimage (cfc (p:=IsSelfAdjoint) (cutoff δ) T).continuous |
| have hh := (Submodule.span ℂ (range v)).topologicalClosure_minimal hgen hclosed |
| rw [hdense] at hh |
| intro x |
| exact hh (Submodule.mem_top : x∈(⊤ : Submodule ℂ H)) |
|
|
| end HardSphere.CompactForm |
|
|
| namespace HardSphere.CompactForm |
| variable {G H : Type*} [NormedAddCommGroup G] [InnerProductSpace ℂ G] [CompleteSpace G] |
| [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
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| lemma cutoff_embedding_error (J : G →L[ℂ] H) (T : H →L[ℂ] H) |
| (hJT : J ∘L J.adjoint=T) (hT : T.IsPositive) (hTn : ‖T‖≤1) |
| {δ : ℝ} (hδ : 0<δ) : |
| ‖((1-cfc (p:=IsSelfAdjoint) (cutoff δ) T) : H →L[ℂ] H) ∘L J‖^2≤δ := by |
| let q : ℝ→ℝ := fun x ↦ 1-cutoff δ x |
| have hq : Continuous q := continuous_const.sub (cutoff_smooth δ).continuous |
| let Q : H →L[ℂ] H := cfc (p:=IsSelfAdjoint) q T |
| have hQ : Q=1-cfc (p:=IsSelfAdjoint) (cutoff δ) T := by |
| dsimp only [Q,q] |
| rw [cfc_sub _ _ T continuousOn_const (cutoff_smooth δ).continuous.continuousOn, |
| cfc_const 1 T hT.isSelfAdjoint,map_one] |
| have hQadj : Q.adjoint=Q := cfc_predicate q T |
| let E : G →L[ℂ] H := Q ∘L J |
| have hEE : E ∘L E.adjoint=Q*T*Q := by |
| change (Q ∘L J) ∘L (Q ∘L J).adjoint=Q*T*Q |
| rw [adjoint_comp,hQadj] |
| calc |
| _ = Q ∘L ((J ∘L J.adjoint) ∘L Q) := by simp only [comp_assoc] |
| _ = Q*T*Q := by rw [hJT]; rfl |
| have hcalc : Q*T*Q=cfc (p:=IsSelfAdjoint) (fun x : ℝ ↦ (1-cutoff δ x)^2*x) T := by |
| have hf : (fun x : ℝ ↦ (1-cutoff δ x)^2*x)=(fun x ↦ (q x*x)*q x) := by |
| funext x; dsimp only [q]; ring |
| rw [hf] |
| symm |
| calc |
| _ = cfc (p:=IsSelfAdjoint) (fun x : ℝ ↦ q x*x) T * cfc q T := |
| cfc_mul (fun x : ℝ ↦ q x*x) q T (hq.mul continuous_id).continuousOn hq.continuousOn |
| _ = (cfc q T * cfc (p:=IsSelfAdjoint) id T) * cfc q T := by |
| congr 1 |
| exact cfc_mul q id T hq.continuousOn continuous_id.continuousOn |
| _ = Q*T*Q := by |
| change cfc q T * cfc (p:=IsSelfAdjoint) (fun x : ℝ ↦ x) T * cfc q T = _ |
| rw [cfc_id' ℝ T hT.isSelfAdjoint] |
| have hb : ‖E ∘L E.adjoint‖≤δ := by |
| rw [hEE,hcalc] |
| apply norm_cfc_le hδ.le |
| intro spectralValue hmem |
| have hnonneg := spectrum_nonneg_of_nonneg |
| (ContinuousLinearMap.nonneg_iff_isPositive.mpr hT) hmem |
| by_cases hlarge : 1 ≤ δ |
| · have hbound : spectralValue ≤ 1 := (le_abs_self _).trans |
| ((spectrum.norm_le_norm_mul_of_mem hmem).trans |
| ((mul_le_of_le_one_right (norm_nonneg T) norm_id_le).trans hTn)) |
| have hcutoff : (1-cutoff δ spectralValue)^2 ≤ 1 := by |
| nlinarith [cutoff_nonneg δ spectralValue, cutoff_le_one δ spectralValue] |
| rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg (sq_nonneg _) hnonneg)] |
| exact (mul_le_of_le_one_left hnonneg hcutoff).trans (hbound.trans hlarge) |
| · exact cutoff_error hδ hnonneg |
| have hn := norm_adjoint_comp_self E.adjoint |
| rw [adjoint_adjoint,LinearIsometryEquiv.norm_map] at hn |
| rw [hn] at hb |
| change ‖Q ∘L J‖*‖Q ∘L J‖≤δ at hb |
| rw [hQ] at hb |
| simpa only [sq] using hb |
|
|
| lemma compact_embedding {ι : Type*} (J : G →L[ℂ] H) (T : H →L[ℂ] H) |
| (hJT : J ∘L J.adjoint=T) (hT : T.IsPositive) (hTn : ‖T‖≤1) |
| (v : ι→H) (r : ι→ℝ) |
| (hdense : (Submodule.span ℂ (range v)).topologicalClosure=⊤) |
| (heigen : ∀i,T (v i)=r i • v i) |
| (hfinite : ∀ε : ℝ,0<ε → {i | ε≤r i}.Finite) : IsCompactOperator J := by |
| have hclosed : IsClosed {F : G →L[ℂ] H | IsCompactOperator F} := isClosed_setOfPred_isCompactOperator |
| apply hclosed.closure_subset |
| rw [Metric.mem_closure_iff] |
| intro ε hε |
| let δ : ℝ := ε^2/4 |
| have hδ : 0<δ := by dsimp only [δ]; positivity |
| let F : H →L[ℂ] H := cfc (p:=IsSelfAdjoint) (cutoff δ) T |
| have hF : IsCompactOperator F := cutoff_compact T hT.isSelfAdjoint v r hdense heigen hfinite hδ |
| refine ⟨F ∘L J,hF.comp_clm J,?_⟩ |
| rw [dist_eq_norm] |
| have heq : J-F ∘L J=(1-F) ∘L J := by ext x; simp |
| rw [heq] |
| have hh := cutoff_embedding_error J T hJT hT hTn hδ |
| change ‖(1-F) ∘L J‖^2≤δ at hh |
| have hn := norm_nonneg ((1-F) ∘L J) |
| dsimp only [δ] at hh |
| nlinarith |
|
|
| end HardSphere.CompactForm |
| end |
|
|
| end OAI |
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|
