File size: 3,294 Bytes
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 | import Mathlib
import OAI.Analysis.HardSphere.Fourier.CharacterDerivativeFactor
import OAI.Analysis.HardSphere.Operators.CompactFormCutoff
namespace OAI
section
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
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
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
end HardSphere
end
end
end OAI
|