example A.283 Momentum on the line

Rests on

• depends_on example 12.104 The Schwartz triple ¶
  • depends_on definition 12.103 Gelfand triple ¶
    • depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶
      • depends_on definition 12.45 Continuous linear functional; the dual ¶
        • depends_on definition 12.35 Bounded operator; operator norm ¶
        • depends_on definition 5.46 Functional ¶
      • 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-25 ¶
    • depends_on corollary 12.19 Double complement; the density criterion ¶
      • depends_on proposition 12.17 The complement is always a closed subspace ¶
        • depends_on corollary 12.5 Continuity of the norm and of orthogonality ¶
        • depends_on definition 12.16 Orthogonal complement ¶
        • proves proof ch:10-hilbert-spaces@proof-8 ¶
      • depends_on theorem 12.18 Projection theorem ¶ ↺
      • proves proof ch:10-hilbert-spaces@proof-10 ¶
    • depends_on definition 12.45 Continuous linear functional; the dual ¶ ↺
  • depends_on example 12.11 The function space $L^{2}$ ¶
    • depends_on definition 12.2 Hilbert space ¶
      • depends_on definition 5.18 Inner product ¶
        • depends_on definition 4.33 Vector space ¶
      • depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
        • depends_on definition 6.24 Metric ¶
      • depends_on equation 5.44 eq:lin-norm-assoc ¶
    • depends_on definition 5.18 Inner product ¶ ↺
  • depends_on proposition 12.102 Neither plane waves nor deltas are in $L^{2}$ ¶
    • depends_on example 12.11 The function space $L^{2}$ ¶ ↺
    • 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 5.18 Inner product ¶ ↺
        • proves proof ch:03-linear-algebra-representations@proof-4 ¶
      • proves proof ch:10-hilbert-spaces@proof-1 ¶
    • proves proof ch:10-hilbert-spaces@proof-48 ¶
• depends_on proposition 12.106 The plane wave is a generalized momentum eigenvector ¶
  • depends_on definition 12.105 Generalized eigenvector ¶
    • depends_on definition 12.103 Gelfand triple ¶ ↺
    • depends_on definition 12.50 Point, continuous and residual spectrum ¶
      • depends_on corollary 12.19 Double complement; the density criterion ¶ ↺
      • depends_on definition 12.49 Resolvent set; spectrum ¶
        • depends_on definition 12.35 Bounded operator; operator norm ¶ ↺
        • depends_on definition 5.47 Inverse of a linear transformation ¶
  • depends_on equation 12.54 eq:hilbert-momentum-operator ¶
  • depends_on example 12.104 The Schwartz triple ¶ ↺
  • proves proof ch:10-hilbert-spaces@proof-49 ¶
• depends_on theorem A.279 Gelfand–Maurin ¶
  • depends_on definition 12.103 Gelfand triple ¶ ↺
  • depends_on definition 12.105 Generalized eigenvector ¶ ↺
  • depends_on theorem A.238 Spectral theorem, both forms ¶
    • depends_on definition 12.58 Projection-valued measure ¶
      • depends_on definition 12.41 The operator classes ¶
        • depends_on definition 6.9 Compact set ¶
        • depends_on proposition 12.21 Characterization of orthogonal projections ¶
        • depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
      • 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 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 app:A-long-proofs@proof-151 ¶
    • proves proof app:A-long-proofs@proof-153 ¶
    • proves proof app:A-long-proofs@proof-156 ¶
  • proves proof app:A-long-proofs@proof-179 ¶

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