| import Mathlib |
| import OAI.Analysis.HardSphere.Fourier.CharacterDerivativeFactor |
| import OAI.Analysis.HardSphere.Operators.CompactFormCutoff |
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| namespace OAI |
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| section |
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| noncomputable section |
| open Set MeasureTheory Filter ContinuousLinearMap |
| open scoped Topology ComplexOrder |
| namespace HardSphere |
|
|
| lemma finite_fourier_resolvent (N : ℕ) (L : Length) (ε : ℝ) (hε : 0<ε) : |
| {k : Direction N→ℤ | ε≤(1+fourierEnergy N L k)⁻¹}.Finite := by |
| have hL := L.property |
| let b : ℝ := 2*Real.pi/L.val |
| have hb : 0<b := by dsimp only [b]; positivity |
| let C : ℝ := 1/(ε*b^2) |
| have hC : 0<C := by dsimp only [C]; positivity |
| obtain ⟨M,hM⟩ := exists_nat_gt (Real.sqrt C) |
| apply (Set.Finite.pi (fun p : Direction N ↦ Set.finite_Icc (-(M:ℤ)) M)).subset |
| intro k hk p _ |
| change ε≤(1+fourierEnergy N L k)⁻¹ at hk |
| have hE := fourierEnergy_nonneg N L k |
| have he : ε*(1+fourierEnergy N L k)≤1 := |
| (le_div_iff₀ (by linarith : 0<1+fourierEnergy N L k)).mp (by simpa only [one_div] using hk) |
| have hp : (2*Real.pi*k p/L.val)^2≤fourierEnergy N L k := by |
| unfold fourierEnergy |
| exact Finset.single_le_sum (fun q _ ↦ sq_nonneg (2*Real.pi*k q/L.val)) (Finset.mem_univ p) |
| have hx : (k p:ℝ)^2≤C := by |
| apply (le_div_iff₀ (by positivity : 0<ε*b^2)).mpr |
| calc |
| (k p:ℝ)^2*(ε*b^2) = ε*(2*Real.pi*k p/L.val)^2 := by dsimp only [b]; ring |
| _ ≤ ε*fourierEnergy N L k := mul_le_mul_of_nonneg_left hp hε.le |
| _ ≤ 1 := by nlinarith |
| have ha : |(k p:ℝ)|≤Real.sqrt C := by |
| nlinarith [Real.sq_sqrt hC.le,Real.sqrt_nonneg C,sq_abs (k p:ℝ),abs_nonneg (k p:ℝ)] |
| have hh := abs_le.mp (ha.trans hM.le) |
| constructor |
| · exact_mod_cast hh.1 |
| · exact_mod_cast hh.2 |
|
|
| lemma free_scalarEmbedding_compact (N : ℕ) (L : Length) : |
| IsCompactOperator (scalarEmbedding N L (-1)) := by |
| apply CompactForm.compact_embedding (scalarEmbedding N L (-1)) (formResolvent N L (-1)) |
| rfl (formResolvent_isPositive N L (-1)) (formResolvent_norm_le_one N L (-1)) |
| (characterL2 N L) (fun k ↦ (1+fourierEnergy N L k)⁻¹) |
| (characterL2_span_dense N L) |
| · intro k |
| change formResolvent N L (-1) (characterL2 N L k) = |
| ((1+fourierEnergy N L k)⁻¹:ℝ) • characterL2 N L k |
| rw [←IsScalarTower.algebraMap_smul ℂ ((1+fourierEnergy N L k)⁻¹:ℝ) (characterL2 N L k)] |
| change formResolvent N L (-1) (characterL2 N L k) = |
| (((1+fourierEnergy N L k)⁻¹:ℝ):ℂ) • characterL2 N L k |
| simpa only [Complex.ofReal_inv,Complex.ofReal_add,Complex.ofReal_one] using |
| character_resolvent N L (-1) (by norm_num) k |
| · exact finite_fourier_resolvent N L |
|
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| lemma scalarEmbedding_compact (N : ℕ) (L : Length) {a : ℝ} (ha : 0<a) : |
| IsCompactOperator (scalarEmbedding N L a) := by |
| let ι : hilbertDomain N L a →L[ℂ] hilbertDomain N L (-1) := |
| ((hilbertDomain N L a).subtypeL).codRestrict (hilbertDomain N L (-1)) |
| (fun u ↦ hilbertDomain_antitone N L (by linarith : -1≤a) u.property) |
| have hh := (free_scalarEmbedding_compact N L).comp_clm ι |
| exact hh |
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| lemma formResolvent_compact (N : ℕ) (L : Length) {a : ℝ} (ha : 0<a) : |
| IsCompactOperator (formResolvent N L a) := |
| (scalarEmbedding_compact N L ha).comp_clm (scalarEmbedding N L a).adjoint |
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| end HardSphere |
| end |
| end |
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| end OAI |
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