| import Mathlib |
| import OAI.Analysis.HardSphere.Density.GroundSpace |
| import OAI.Analysis.HardSphere.Fourier.FreeCompactForm |
| import OAI.Analysis.HardSphere.Thermodynamics.MixedCondensation |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter ContinuousLinearMap |
| open scoped Topology ComplexOrder ENNReal |
| namespace HardSphere |
|
|
| lemma groundSpace_finiteDimensional (N : ℕ) (L : Length) {a : ℝ} (ha : 0<a) : |
| FiniteDimensional ℂ (groundSpace N L a) := by |
| classical |
| by_cases hg : ∃u : Jet N L,IsGroundVector N L a u |
| · obtain ⟨u,hu⟩ := hg |
| have hE : 0≤energy u := Finset.sum_nonneg fun _ _ ↦ sq_nonneg _ |
| have hc : (1+(energy u:ℂ))⁻¹≠0 := by |
| apply inv_ne_zero |
| exact_mod_cast (show (1:ℝ)+energy u≠0 by linarith) |
| have := ContinuousLinearMap.finite_dimensional_eigenspace |
| (formResolvent_compact N L ha) ((1+(energy u:ℂ))⁻¹) hc |
| exact Submodule.finiteDimensional_of_le ((groundSpace_le_resolvent hu).trans inf_le_left) |
| · rw [groundSpace_eq_bot_of_no_ground hg] |
| infer_instance |
|
|
| lemma groundSupported_iff_support_le {N : ℕ} {L : Length} {a : ℝ} |
| (T : Density.Operator (ScalarL2 N L)) : |
| GroundSupported a T ↔ T.toCLM.kerᗮ≤groundSpace N L a := by |
| constructor |
| · intro hT ψ hψ |
| by_cases hz : ψ=0 |
| · rw [hz]; exact (groundSpace N L a).zero_mem |
| · have hn : ‖ψ‖≠0 := norm_ne_zero_iff.mpr hz |
| let z : ℂ := (‖ψ‖:ℂ)⁻¹ |
| have hzn : ‖z‖=‖ψ‖⁻¹ := by |
| simp only [z,norm_inv,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (norm_nonneg ψ)] |
| have hunit : ‖z • ψ‖=1 := by rw [norm_smul,hzn,inv_mul_cancel₀ hn] |
| obtain ⟨u,hu,hg⟩ := hT (z • ψ) ((T.toCLM.kerᗮ).smul_mem z hψ) hunit |
| have hm : z • ψ∈groundSpace N L a := hu ▸ hg.mem_groundSpace |
| have hh := (groundSpace N L a).smul_mem (‖ψ‖:ℂ) hm |
| have hc : (‖ψ‖:ℂ)≠0 := by exact_mod_cast hn |
| simpa only [z,smul_smul,mul_inv_cancel₀ hc,one_smul] using hh |
| · intro hT ψ hψ hn |
| exact (groundSpace_normalized_iff hn).mp (hT hψ) |
|
|
| attribute [local instance] scalarL2_norm_exponent |
| attribute [local instance] scalarL2_normedAddCommGroup |
| attribute [local instance] scalarL2_innerProductSpace |
| attribute [local instance] scalarL2_module |
| attribute [local instance] scalarL2_completeSpace |
|
|
| lemma groundSpace_nontrivial_of_ne_bot (N : ℕ) (L : Length) (a : ℝ) |
| (hG : @Ne (@Submodule ℂ (ScalarL2 N L) Complex.instSemiring |
| (scalarL2_normedAddCommGroup N L).toAddCommMonoid (scalarL2_module N L)) |
| (groundSpace N L a) (@Bot.bot |
| (@Submodule ℂ (ScalarL2 N L) Complex.instSemiring |
| (scalarL2_normedAddCommGroup N L).toAddCommMonoid (scalarL2_module N L)) |
| (@Submodule.instBot ℂ (ScalarL2 N L) Complex.instSemiring |
| (scalarL2_normedAddCommGroup N L).toAddCommMonoid (scalarL2_module N L)))) : |
| Nontrivial (groundSpace N L a) := |
| Submodule.nontrivial_iff_ne_bot.mpr hG |
|
|
| /-- Precisely Π/Tr Π for the orthogonal projection on the entire bosonic ground space. |
| The nonzero condition is exactly the domain of normalization, not a finite-rank assumption. -/ |
| abbrev normalizedGroundProjection (N : ℕ) (L : Length) (a : ℝ) |
| (ha : @LT.lt ℝ Real.instLT (@Zero.zero ℝ Real.instZero) a) |
| (hG : @Ne (@Submodule ℂ (ScalarL2 N L) Complex.instSemiring |
| (scalarL2_normedAddCommGroup N L).toAddCommMonoid (scalarL2_module N L)) |
| (groundSpace N L a) (@Bot.bot |
| (@Submodule ℂ (ScalarL2 N L) Complex.instSemiring |
| (scalarL2_normedAddCommGroup N L).toAddCommMonoid (scalarL2_module N L)) |
| (@Submodule.instBot ℂ (ScalarL2 N L) Complex.instSemiring |
| (scalarL2_normedAddCommGroup N L).toAddCommMonoid (scalarL2_module N L)))) : |
| @Density.Operator (ScalarL2 N L) (scalarL2_normedAddCommGroup N L) |
| (scalarL2_innerProductSpace N L) (scalarL2_completeSpace N L) := |
| @Density.normalizedProjection.{0} (ScalarL2 N L) |
| (scalarL2_normedAddCommGroup N L) (scalarL2_innerProductSpace N L) |
| (scalarL2_completeSpace N L) |
| (groundSpace N L a) (@groundSpace_finiteDimensional N L a ha) |
| (groundSpace_nontrivial_of_ne_bot N L a hG) |
|
|
| lemma normalizedGroundProjection_formula (N : ℕ) (L : Length) (a : ℝ) (ha : 0<a) |
| (hG : groundSpace N L a≠⊥) : |
| let := groundSpace_finiteDimensional N L ha |
| (normalizedGroundProjection N L a ha hG).toCLM = |
| ((Module.finrank ℂ (groundSpace N L a):ℝ)⁻¹) • (groundSpace N L a).starProjection := by |
| rfl |
|
|
| lemma normalizedGroundProjection_supported (N : ℕ) (L : Length) (a : ℝ) (ha : 0<a) |
| (hG : groundSpace N L a≠⊥) : |
| GroundSupported a (normalizedGroundProjection N L a ha hG) := by |
| let := groundSpace_finiteDimensional N L ha |
| let := Submodule.nontrivial_iff_ne_bot.mpr hG |
| rw [groundSupported_iff_support_le] |
| change (Density.normalizedProjection (groundSpace N L a)).toCLM.kerᗮ≤groundSpace N L a |
| rw [Density.normalizedProjection_support] |
|
|
| lemma IsGroundVector.groundSpace_ne_bot {N : ℕ} {L : Length} {a : ℝ} |
| {u : Jet N L} (hu : IsGroundVector N L a u) : groundSpace N L a≠⊥ := |
| Submodule.nontrivial_iff_ne_bot.mp (groundSpace_nontrivial hu) |
|
|
| end HardSphere |
| end |
| end |
|
|
| end OAI |
|
|
