All proofs
Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Is Essentially Self Adjoint of defect Number eq zero

LinearPMap.IsSymmetric.isEssentiallySelfAdjoint_of_defectNumber_eq_zero

Plain-language statement

The basic criterion for essential self-adjointness: a symmetric, densely-defined operator whose defect numbers at I and -I both vanish is essentially self-adjoint.

Exact Lean statement

lemma isEssentiallySelfAdjoint_of_defectNumber_eq_zero
    (hdense : T.HasDenseDomain) (hpos : T.defectNumber I = 0) (hneg : T.defectNumber (-I) = 0) :
    T.IsEssentiallySelfAdjoint

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
lemma isEssentiallySelfAdjoint_of_defectNumber_eq_zero    (hdense : T.HasDenseDomain) (hpos : T.defectNumber I = 0) (hneg : T.defectNumber (-I) = 0) :    T.IsEssentiallySelfAdjoint := by  have hrange {z : ℂ} (hz : z.im  0) (hd : T.defectNumber z = 0) :      (T.closure - z • 1).toFun.range =:= by    have hz' : z  T.regularityDomain := hT.mem_regularityDomain_of_im_ne_zero hz    have hcl : T.closure.IsClosed := (hT.isClosable hdense).closure_isClosed    rw [ hcl.defectNumber_eq_zero_iff (T.regularityDomain_closure ▸ hz'),      defectNumber_closure hz']    exact hd  rw [isEssentiallySelfAdjoint_def]  apply (hT.closure hdense).isSelfAdjoint_of_range_eq_top hdense.closure  · have hI : T.closure + I • 1 = T.closure - (-I) • 1 :=      LinearPMap.ext rfl fun x hf hg => by simp [sub_apply, add_apply, smul_apply, sub_neg_eq_add]    rw [hI]    exact hrange (by norm_num) hneg  · exact hrange (by norm_num) hpos
Project
Physlib
License
Apache-2.0
Commit
dd43e9e65791
Source
Physlib/QuantumMechanics/Operators/SpectralTheory/Symmetric.lean:229-245

Reuse this declaration

Bring the exact result into your workflow

The import identifies the source module. Your project still needs the pinned package dependency shown on this page.

What this badge means

This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.

Continue in this project

Related declarations

Project-declaredLean 4.32.0

Adiabatic relation log

adiabatic_relation_log

Plain-language statement

Adiabatic relation in logarithmic form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then c * log (Ua/Ub) + log (Va/Vb) = 0.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

View proof record
Project-declaredLean 4.32.0

Adiabatic relation Ua Ub Va Vb

adiabatic_relation_UaUbVaVb

Plain-language statement

Adiabatic relation in product form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then (Ua/Ub)^c * (Va/Vb) = 1.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

View proof record