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67aee1d | 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 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | 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
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