proposition 12.67 The generator is symmetric, and generates the motion

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:1835 · p. 433

Rests on

depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
- depends_on definition 12.41 The operator classes ¶
  - depends_on definition 6.9 Compact set ¶
    - depends_on definition 6.5 Open cover ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
  - depends_on proposition 12.21 Characterization of orthogonal projections ¶
    - depends_on definition 12.20 Orthogonal projection operator ¶
      - depends_on theorem 12.18 Projection theorem ¶
    - depends_on theorem 12.18 Projection theorem ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-11 ¶
  - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
    - depends_on definition 5.41 Adjoint ¶
      - depends_on definition 5.18 Inner product ¶
      - depends_on definition 5.37 Linear transformation ¶
    - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
      - depends_on definition 12.35 Bounded operator; operator norm ¶
      - depends_on proposition 12.8 Absolutely convergent series test ¶
      - proves proof ch:10-hilbert-spaces@proof-19 ¶
    - depends_on theorem 12.46 Riesz representation ¶
      - depends_on definition 12.45 Continuous linear functional; the dual ¶
      - depends_on theorem 12.18 Projection theorem ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-24 ¶
    - proves proof ch:10-hilbert-spaces@proof-20 ¶
- depends_on definition 6.6 Continuous map ¶
  - depends_on definition 3.53 Preimage ¶
  - depends_on definition 6.2 Open set ¶ ↺
  - depends_on definition 6.1 Topological space ¶
    - depends_on definition 3.31 Empty set ¶
    - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.8 Conjunction ¶
      - depends_on definition 3.9 Disjunction ¶
      - depends_on definition 3.7 Negation ¶
    - depends_on definition 3.30 Subset ¶
    - depends_on equation 3.51 eq:set-indexed ¶ ↺
depends_on theorem 12.66 Stone ¶
- depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
- depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
  - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
  - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
  - depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶
    - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
    - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-21 ¶
  - proves proof ch:10-hilbert-spaces@proof-33 ¶
- depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
  - depends_on definition 12.58 Projection-valued measure ¶
    - depends_on definition 12.41 The operator classes ¶ ↺
    - depends_on proposition 12.21 Characterization of orthogonal projections ¶ ↺
  - depends_on proposition 12.43 Norm of a self-adjoint operator ¶
    - depends_on proposition 12.42 Elementary consequences ¶
      - depends_on definition 12.41 The operator classes ¶ ↺
      - depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-22 ¶
    - depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-23 ¶
  - depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
    - depends_on corollary 12.19 Double complement; the density criterion ¶
      - depends_on proposition 12.17 The complement is always a closed subspace ¶
      - depends_on theorem 12.18 Projection theorem ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-10 ¶
    - depends_on definition 12.41 The operator classes ¶ ↺
    - depends_on theorem 12.54 The spectrum is compact and non-empty ¶
      - depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶
      - depends_on proposition 12.53 The resolvent set is open, the resolvent analytic ¶
      - depends_on theorem 8.18 Liouville ¶
      - proves proof ch:10-hilbert-spaces@proof-29 ¶
    - proves proof ch:10-hilbert-spaces@proof-30 ¶
  - proves proof ch:10-hilbert-spaces@prooflink-2 ¶
- proves proof ch:10-hilbert-spaces@prooflink-3 ¶
proves proof ch:10-hilbert-spaces@proof-34 ¶

Supports

depends_on lemma A.257 Integrated form of the equation of motion ¶
- depends_on proposition A.258 The generator is closed ¶
  - depends_on proposition A.259 The generator is self-adjoint ¶
    - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
      - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶
    - depends_on proposition A.263 The two constructions are inverse ¶
depends_on proposition A.263 The two constructions are inverse ¶ ↺
depends_on proposition A.259 The generator is self-adjoint ¶ ↺
depends_on proposition 12.111 The Weyl relation is a covariance statement ¶
- depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶
depends_on theorem A.253 Stone ¶
- depends_on proposition A.263 The two constructions are inverse ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Strongly continuous one-parameter unitary group	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1849
`depends_on`	→	Stone	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1849
`depends_on`	←	Integrated form of the equation of motion	declared	appendices/A-long-proofs.tex:12794
`depends_on`	←	The two constructions are inverse	declared	appendices/A-long-proofs.tex:13194
`depends_on`	←	The generator is self-adjoint	declared	appendices/A-long-proofs.tex:12845
`depends_on`	←	The Weyl relation is a covariance statement	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:3051
`depends_on`	←	Stone	declared	appendices/A-long-proofs.tex:12645
`proves`	←	ch:10-hilbert-spaces@proof-34	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1852