`BookProof.FriedrichsCanonical.qgOneParticleSector_friedrichs_unique` (M alpha : ℝ) (hM : 0 < M) (halpha : 0 < alpha) (V3 : Polynomial ℝ) (c : ℝ) (hV3 : ∀ t : ℝ, -c ≤ V3.eval t) {D

Proved

BookProof.FriedrichsCanonical.qgOneParticleSector_friedrichs_unique

by leonardopedro · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFriedrichsCanonical.

BookProof.FriedrichsCanonical.qgOneParticleSector_friedrichs_unique (M alpha : ℝ) (hM : 0 < M) (halpha : 0 < alpha) (V3 : Polynomial ℝ) (c : ℝ) (hV3 : ∀ t : ℝ, -c ≤ V3.eval t) {Dom' : Submodule ℂ (L2d 2)} (A' : Dom' →ₗ[ℂ] L2d 2) (hA' : IsSemiboundedSelfAdjointExtension c (hamCore (scalaronSectorPotential M alpha V3) (continuous_scalaronSectorPotential M alpha V3) (expBounded_scalaronSectorPotential M alpha hM V3)) A') (hform : Dom' ≤ formDomain (qgSectorShiftedPosSymOp M alpha hM halpha V3 c hV3)) : Dom' = friedrichsDomain (qgSectorShiftedPosSymOp M alpha hM halpha V3 c hV3) ∧ ∀ (x : L2d 2) (hx : x ∈ Dom') (hx' : x ∈ friedrichsDomain (qgSectorShiftedPosSymOp M alpha hM halpha V3 c hV3)), A' ⟨x, hx⟩ = qgSectorFriedrichsOp M alpha hM halpha V3 c hV3 ⟨x, hx'⟩

Formalization note: Lean 4 identifier BookProof.FriedrichsCanonical.qgOneParticleSector_friedrichs_unique.

Preamble

-- Generated from ChapterFriedrichsCanonical.lean — theorem BookProof.FriedrichsCanonical.qgOneParticleSector_friedrichs_unique
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterYangMillsFriedrichs
import Definitions.Def_ChapterComplexShiftCore
import Mathlib
import Definitions.Def_ChapterFriedrichsCanonical
import Definitions.Def_ChapterFriedrichsExtension
import Definitions.Def_ChapterHermiteProductCore
import Definitions.Def_ChapterQgHermiteCore
import Definitions.Def_ChapterQgHermiteFriedrichs
open BookProof.FriedrichsExtension
open BookProof.FriedrichsExtension.FormDom
open BookProof.HermiteProductCore
open BookProof.QgHermiteCore
open BookProof.QgHermiteFriedrichs
open BookProof.FriedrichsCanonical



open BookProof.FarisLavine BookProof.YangMillsFriedrichs BookProof.HashimotoShiftInvert
open BookProof.FriedrichsExtension BookProof.FriedrichsExtension.FormDom

noncomputable section

variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F]

variable {D : Submodule ℂ F}

Formal statement

theorem BookProof.FriedrichsCanonical.qgOneParticleSector_friedrichs_unique (M alpha : ℝ) (hM : 0 < M) (halpha : 0 < alpha)
    (V3 : Polynomial ℝ) (c : ℝ) (hV3 : ∀ t : ℝ, -c ≤ V3.eval t)
    {Dom' : Submodule ℂ (L2d 2)} (A' : Dom' →ₗ[ℂ] L2d 2)
    (hA' : IsSemiboundedSelfAdjointExtension c
      (hamCore (scalaronSectorPotential M alpha V3)
        (continuous_scalaronSectorPotential M alpha V3)
        (expBounded_scalaronSectorPotential M alpha hM V3)) A')
    (hform : Dom' ≤ formDomain (qgSectorShiftedPosSymOp M alpha hM halpha V3 c hV3)) :
    Dom' = friedrichsDomain (qgSectorShiftedPosSymOp M alpha hM halpha V3 c hV3) ∧
      ∀ (x : L2d 2) (hx : x ∈ Dom')
        (hx' : x ∈ friedrichsDomain (qgSectorShiftedPosSymOp M alpha hM halpha V3 c hV3)),
        A' ⟨x, hx⟩ = qgSectorFriedrichsOp M alpha hM halpha V3 c hV3 ⟨x, hx'⟩ := by sorry

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