| import Mathlib | |
| import OAI.Analysis.HardSphere.Fourier.CharacterDerivativeFactor | |
| import OAI.Analysis.HardSphere.Geometry.GraphLiftHeatBound | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped ENNReal Topology BigOperators | |
| open scoped BigOperators ComplexConjugate | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| open scoped ENNReal NNReal Topology BigOperators | |
| open MeasureTheory ProbabilityTheory Real Complex | |
| open scoped ENNReal ComplexConjugate | |
| namespace HardSphere | |
| open ContinuousLinearMap Filter | |
| open scoped Topology | |
| lemma freeResolvent_denseRange (N : ℕ) (L : Length) (a : ℝ) (ha : a<0) : | |
| DenseRange (formResolvent N L a) := by | |
| change Dense (Set.range (formResolvent N L a)) | |
| change Dense ((formResolvent N L a).range : Set (ScalarL2 N L)) | |
| apply Submodule.dense_iff_topologicalClosure_eq_top.mpr | |
| apply top_unique | |
| rw [← characterL2_span_dense N L] | |
| apply Submodule.topologicalClosure_mono | |
| apply Submodule.span_le.mpr | |
| rintro _ ⟨k,rfl⟩ | |
| refine ⟨(1+(fourierEnergy N L k:ℂ)) • characterL2 N L k,?_⟩ | |
| change formResolvent N L a ((1+(fourierEnergy N L k:ℂ)) • characterL2 N L k)=_ | |
| rw [map_smul,character_resolvent N L a ha k,smul_smul,mul_inv_cancel₀,one_smul] | |
| have hreal : (1:ℝ)+fourierEnergy N L k≠0 := ne_of_gt (by linarith [fourierEnergy_nonneg N L k]) | |
| exact_mod_cast hreal | |
| lemma free_graph_lift_of_heat_bound (N : ℕ) (L : Length) (a : ℝ) (ha : a<0) | |
| (u : ScalarL2 N L) (C : ℝ) | |
| (hb : ∀ n : ℕ, (1/((n:ℝ)+1))⁻¹ * | |
| (‖u‖^2-(inner ℂ u (freeTransition N L (1/((n:ℝ)+1)) u)).re)≤C) : | |
| ∃ v : hilbertDomain N L a, scalarEmbedding N L a v=u ∧ ‖v‖^2≤‖u‖^2+C := by | |
| rcases subsingleton_or_nontrivial (ScalarL2 N L) with h | h | |
| · have := h | |
| refine ⟨0,Subsingleton.elim _ _,?_⟩ | |
| have hz : u=0 := Subsingleton.elim _ _ | |
| simpa [hz] using hb 0 | |
| have := h | |
| apply exists_graph_lift_of_heat_bound (scalarEmbedding N L a) (scalarEmbedding_norm_le N L a) | |
| (freeResolvent_denseRange N L a ha) u C | |
| intro n | |
| have ht : 0<1/((n:ℝ)+1) := by positivity | |
| simpa only [formHeat,formResolvent] using | |
| (show (1/((n:ℝ)+1))⁻¹ * (‖u‖^2-(inner ℂ u | |
| (formHeat N L a (1/((n:ℝ)+1)) u)).re)≤C by | |
| rw [← freeTransition_eq_formHeat N L a ha ht] | |
| exact hb n) | |
| end HardSphere | |
| end | |
| end OAI | |