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9eeb9c7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 | import Mathlib
import OAI.Analysis.HardSphere.FormDomain.AeSelfGradientSquare
import OAI.Analysis.HardSphere.FormDomain.AeGradientPhase
import OAI.Analysis.HardSphere.Conditioning.LintegralReferenceMeasure
import OAI.Analysis.HardSphere.Bridges.AccessibleAuxiliary
import OAI.Analysis.HardSphere.PathBounds.RegularVisitors
import OAI.Analysis.HardSphere.Collisions.InsertionProducer
import OAI.Analysis.HardSphere.Kernels.KernelPi
import OAI.Analysis.HardSphere.Kernels.ParametricTrial
import OAI.Analysis.HardSphere.PathBounds.UniformVisitMGF
import OAI.Analysis.HardSphere.Density.GroundProjection
namespace OAI
section
noncomputable section
open MeasureTheory Set Filter
open scoped ENNReal BigOperators Topology
namespace HardSphere
theorem condensation_all_ground_states :
∃ε₀ c₀ : ℝ,0<ε₀ ∧ 0<c₀ ∧
∀(a ρ : ℝ) (ha : 0<a),0<ρ → ρ*a^3<ε₀ →
∀(N : ℕ→ℕ) (L : ℕ→Length) (hN : ∀k,0<N k),
Tendsto N atTop atTop → Tendsto (fun k ↦ (L k).val) atTop atTop →
Tendsto (fun k ↦ (N k:ℝ)/(L k).val^3) atTop (𝓝 ρ) →
(∀Ψ : (k : ℕ)→Jet (N k) (L k),
(∀k,IsGroundVector (N k) (L k) a (Ψ k)) →
c₀≤liminf (fun k ↦ occupation (hN k) (Ψ k).1) atTop) ∧
(∀T : (k : ℕ)→Density.Operator (ScalarL2 (N k) (L k)),
(∀k,GroundSupported a (T k)) →
c₀≤liminf (fun k ↦ mixedOccupation (hN k) (T k)) atTop) ∧
(∀hG : ∀k,groundSpace (N k) (L k) a≠⊥,
c₀≤liminf (fun k ↦ mixedOccupation (hN k)
(normalizedGroundProjection (N k) (L k) a ha (hG k))) atTop) := by
obtain ⟨ε,c,hε,hc,h⟩ := condensation_with_mixed
refine ⟨ε,c,hε,hc,?_⟩
intro a ρ ha hρ hd N L hN hNtop hLtop hdensity
obtain ⟨hp,hm⟩ := h a ρ ha hρ hd N L hN hNtop hLtop hdensity
refine ⟨hp,hm,?_⟩
intro hG
exact hm (fun k ↦ normalizedGroundProjection (N k) (L k) a ha (hG k))
(fun k ↦ normalizedGroundProjection_supported (N k) (L k) a ha (hG k))
end HardSphere
end
end
end OAI
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