lemma A.257 Integrated form of the equation of motion

open in the book · appendices/A-long-proofs.tex:12788 · p. 2917

Rests on

depends_on lemma A.254 Riemann integral of a continuous curve ¶
- depends_on definition 12.2 Hilbert space ¶
  - depends_on definition 5.18 Inner product ¶
    - depends_on definition 4.33 Vector space ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
    - depends_on definition 6.24 Metric ¶
      - depends_on definition 3.43 Map ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - depends_on equation 5.44 eq:lin-norm-assoc ¶
- depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
  - depends_on definition 12.2 Hilbert space ¶ ↺
  - depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶
    - depends_on definition 5.18 Inner product ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-4 ¶
  - proves proof ch:10-hilbert-spaces@proof-1 ¶
- proves proof app:A-long-proofs@proof-157 ¶
depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
- 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 proposition 12.21 Characterization of orthogonal projections ¶
      - depends_on definition 12.20 Orthogonal projection operator ¶
      - 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 proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
      - depends_on theorem 12.46 Riesz representation ¶
      - 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 6.1 Topological space ¶ ↺
- 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 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 definition 12.41 The operator classes ¶ ↺
      - depends_on theorem 12.54 The spectrum is compact and non-empty ¶
      - 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 ¶
proves proof app:A-long-proofs@proof-160 ¶

Supports

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.280 Direct-integral form of the spectral theorem ¶
  - 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`	→	Riemann integral of a continuous curve	declared	appendices/A-long-proofs.tex:12794
`depends_on`	→	The generator is symmetric, and generates the motion	declared	appendices/A-long-proofs.tex:12794
`depends_on`	←	The generator is closed	declared	appendices/A-long-proofs.tex:12816
`proves`	←	app:A-long-proofs@proof-160	declared	appendices/A-long-proofs.tex:12797