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159357f | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | import Mathlib
import OAI.Analysis.CoulombIonization.FormDomain.FormRawPotential
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff
open MeasureTheory Filter Set
open scoped BigOperators
namespace CoulombAtom
lemma rawPotential_weighted_integrable {N : ℕ} {ψ : FormVector N}
(hψ : SobolevVector ψ) (s : Spins N) (y : Space) :
Integrable (fun x => rawPotential y x*‖ψ.value s x‖^2) := by
simp only [rawPotential,Finset.sum_mul]
apply integrable_finsetSum
intro i _
simpa only [one_div,mul_comm,←div_eq_mul_inv] using hψ.shifted_nuclear_integrable s i y
lemma integral_rawPotential_formRawLaw {N : ℕ} {ψ : FormVector N}
(hψ : SobolevVector ψ) (y : Space) :
(∫ x, rawPotential y x ∂formRawLaw ψ) = coreCoulombAt ψ y := by
rw [formRawLaw_integral hψ]
simp only [formRawDensity,Finset.sum_mul]
rw [integral_finsetSum _ (fun s _ => by
simpa only [mul_comm] using rawPotential_weighted_integrable hψ s y)]
unfold coreCoulombAt
congr 1
ext s
simp only [rawPotential,Finset.mul_sum,mul_one_div]
exact integral_finsetSum _ (fun i _ => hψ.shifted_nuclear_integrable s i y)
lemma coreSlice_coreCoulombAt_integrable {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) (t : Spins M) (y : Space) :
Integrable (fun u : Configuration M => coreCoulombAt (coreSlice ψ t u) y) := by
unfold coreCoulombAt
apply integrable_finsetSum
intro s _
apply integrable_finsetSum
intro i _
have hh := (integrable_join (N := N) (M := M)
(hψ.shifted_nuclear_integrable (joinLists s t) (finSumFinEquiv (Sum.inl i)) y)).integral_prod_right
simpa only [coreSlice,joinLists_left] using hh
lemma rawPotential_coreSlice_integrable {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) (t : Spins M) (y : Space) :
Integrable (fun u : Configuration M => rawPotential y u*formMass (coreSlice ψ t u)) := by
simp only [rawPotential,Finset.sum_mul]
apply integrable_finsetSum
intro i _
simpa only [one_div,mul_comm,←div_eq_mul_inv] using coreSlice_mass_shifted_integrable hψ t i y
lemma integral_coreSlice_coreCoulombAt {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) (y : Space) :
(∑ t : Spins M, ∫ u : Configuration M, coreCoulombAt (coreSlice ψ t u) y) =
∑ s : Spins (N+M), ∑ i : Fin N, ∫ x, ‖ψ.value s x‖^2/‖x (finSumFinEquiv (Sum.inl i))-y‖ := by
have hi (s : Spins N) (t : Spins M) (i : Fin N) : Integrable (fun u : Configuration M =>
∫ v : Configuration N, ‖ψ.value (joinLists s t) (joinLists v u)‖^2/‖v i-y‖) := by
simpa only [joinLists_left] using (integrable_join (N := N) (M := M)
(hψ.shifted_nuclear_integrable (joinLists s t) (finSumFinEquiv (Sum.inl i)) y)).integral_prod_right
simp only [coreCoulombAt,coreSlice]
simp_rw [integral_finsetSum _ (fun s _ => integrable_finsetSum _ (fun i _ => hi s _ i)),
integral_finsetSum _ (fun i _ => hi _ _ i)]
have he (s : Spins N) (t : Spins M) (i : Fin N) :
(∫ u : Configuration M, ∫ v : Configuration N,
‖ψ.value (joinLists s t) (joinLists v u)‖^2/‖v i-y‖) =
∫ x, ‖ψ.value (joinLists s t) x‖^2/‖x (finSumFinEquiv (Sum.inl i))-y‖ := by
simpa only [joinLists_left] using integral_join (N := N) (M := M)
(hψ.shifted_nuclear_integrable (joinLists s t) (finSumFinEquiv (Sum.inl i)) y)
simp_rw [he]
exact sum_spin_join (fun s : Spins (N+M) => ∑ i : Fin N,
∫ x, ‖ψ.value s x‖^2/‖x (finSumFinEquiv (Sum.inl i))-y‖)
lemma integral_rawPotential_coreSlice {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) (y : Space) :
(∑ t : Spins M, ∫ u : Configuration M, rawPotential y u*formMass (coreSlice ψ t u)) =
∑ s : Spins (N+M), ∑ i : Fin M, ∫ x, ‖ψ.value s x‖^2/‖x (finSumFinEquiv (Sum.inr i))-y‖ := by
have hi (s : Spins (N+M)) : Integrable (fun x =>
rawPotential y (outProjection x)*‖ψ.value s x‖^2) := by
simp only [rawPotential,Finset.sum_mul,outProjection]
apply integrable_finsetSum
intro i _
simpa only [one_div,mul_comm,←div_eq_mul_inv] using
hψ.shifted_nuclear_integrable s (finSumFinEquiv (Sum.inr i)) y
rw [weighted_coreSlice_eq_full_integrable _ _ hi]
apply Finset.sum_congr rfl
intro s _
simp only [rawPotential,outProjection]
simp only [Finset.sum_mul,div_mul_eq_mul_div,one_mul]
exact integral_finsetSum _ (fun i _ => hψ.shifted_nuclear_integrable s (finSumFinEquiv (Sum.inr i)) y)
lemma fresh_core_outer_potential_tower {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) (y : Space) :
(∑ t : Spins M, ∫ u : Configuration M,
coreCoulombAt (coreSlice ψ t u) y+rawPotential y u*formMass (coreSlice ψ t u)) =
coreCoulombAt ψ y := by
simp_rw [integral_add (coreSlice_coreCoulombAt_integrable hψ _ y)
(rawPotential_coreSlice_integrable hψ _ y)]
rw [Finset.sum_add_distrib,integral_coreSlice_coreCoulombAt hψ y,
integral_rawPotential_coreSlice hψ y,←Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro s _
change (∑ i : Fin N, ∫ x, ‖ψ.value s x‖^2/‖x (finSumFinEquiv (Sum.inl i))-y‖) +
(∑ i : Fin M, ∫ x, ‖ψ.value s x‖^2/‖x (finSumFinEquiv (Sum.inr i))-y‖) =
∑ i : Fin (N+M), ∫ x, ‖ψ.value s x‖^2/‖x i-y‖
simpa only [Fintype.sum_sum_type] using
(Equiv.sum_comp finSumFinEquiv (fun i : Fin (N+M) => ∫ x, ‖ψ.value s x‖^2/‖x i-y‖))
end CoulombAtom
end
end OAI
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