| import Mathlib |
| import OAI.Analysis.CoulombIonization.FormDomain.UnboundedSpectrum |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace Convolution |
| namespace CoulombAtom |
| section ShiftedFormBottom |
| variable {V H : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V] |
| [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| theorem shifted_form_spectral_infimum [Nontrivial H] (j : V →L[ℂ] H) |
| (hj : Function.Injective j) (hd : DenseRange j) (A : V →L[ℂ] V) |
| (hs : IsSelfAdjoint A) {c : ℝ} (hc : 0 < c) |
| (hA : ∀ u, c * ‖u‖ ^ 2 ≤ (⟪u, A u⟫_ℂ).re) (b : ℝ) : |
| sInf {r : ℝ | (r : ℂ) ∈ unboundedSpectrum |
| (inverseShift (formResolvent j A hc hA) b)} = |
| sInf {e : ℝ | ∃ u : V, ‖j u‖ ^ 2 = 1 ∧ (⟪u, A u⟫_ℂ).re - b = e} := by |
| let S := {e : ℝ | ∃ u : V, ‖j u‖ ^ 2 = 1 ∧ (⟪u, A u⟫_ℂ).re = e} |
| have hne : S.Nonempty := by |
| obtain ⟨u, hu⟩ := denseRange_exists_nonzero j hd |
| let v : V := (‖j u‖⁻¹ : ℂ) • u |
| refine ⟨(⟪v, A v⟫_ℂ).re, v, ?_, rfl⟩ |
| simp [v, norm_smul, hu] |
| have hbb : BddBelow S := ⟨0, fun e ⟨u, _, he⟩ => he ▸ coercive_nonneg A hc hA u⟩ |
| have he : (OrderIso.addRight (-b)) '' S = |
| {e : ℝ | ∃ u : V, ‖j u‖ ^ 2 = 1 ∧ (⟪u, A u⟫_ℂ).re - b = e} := by |
| ext e |
| constructor |
| · rintro ⟨a, ⟨u, hu, rfl⟩, rfl⟩ |
| exact ⟨u, hu, rfl⟩ |
| · rintro ⟨u, hu, rfl⟩ |
| exact ⟨(⟪u, A u⟫_ℂ).re, ⟨u, hu, rfl⟩, rfl⟩ |
| have hi := (OrderIso.addRight (-b)).map_csInf' hne hbb |
| rw [he] at hi |
| change sInf S - b = _ at hi |
| rw [form_infimum_eq_reciprocal j hj hd A hs hc hA] at hi |
| rw [inverseShift_spectral_bottom _ |
| (ContinuousLinearMap.nonneg_iff_isPositive.mpr (formResolvent_positive j A hs hc hA)) |
| (formResolvent_injective j hj hd A hc hA)] |
| exact hi |
|
|
| end ShiftedFormBottom |
|
|
| variable {V H : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V] |
| [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
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| theorem shifted_representedOperator_graph (j : V →L[ℂ] H) (hj : Function.Injective j) |
| (hd : DenseRange j) (A B : V →L[ℂ] V) (b : ℝ) |
| (hAB : ∀ u v, ⟪v, A u⟫_ℂ = ⟪v, B u⟫_ℂ + (b : ℂ) * ⟪j v, j u⟫_ℂ) |
| {c : ℝ} (hc : 0 < c) (hA : ∀ u, c * ‖u‖ ^ 2 ≤ (⟪u, A u⟫_ℂ).re) (x y : H) : |
| (x, y) ∈ (inverseShift (formResolvent j A hc hA) b).graph ↔ |
| ∃ u : V, j u = x ∧ ∀ v : V, ⟪v, B u⟫_ℂ = ⟪j v, y⟫_ℂ := by |
| have hi := formResolvent_injective j hj hd A hc hA |
| have hg := representedOperator_graph j hj hd A hc hA x (y + (b : ℂ) • x) |
| rw [representedOperator, inverseOperator_graph hi] at hg |
| rw [inverseShift_graph _ hi, hg] |
| constructor |
| · rintro ⟨u, hu, he⟩ |
| refine ⟨u, hu, fun v => ?_⟩ |
| have hh := he v |
| rw [hAB, inner_add_right, inner_smul_right, hu] at hh |
| exact add_right_cancel hh |
| · rintro ⟨u, hu, he⟩ |
| refine ⟨u, hu, fun v => ?_⟩ |
| rw [hAB, he, inner_add_right, inner_smul_right, hu] |
|
|
| end CoulombAtom |
|
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| end |
|
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| end OAI |
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|
