proposition A.592 The vacuum of the Schrödinger system

Rests on

• depends_on example 12.110 The Schrödinger system ¶
  • depends_on definition 12.109 Weyl system ¶
    • 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 proposition 12.21 Characterization of orthogonal projections ¶
        • depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
      • 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 theorem 12.75 The canonical commutation relation admits no bounded solution ¶
      • depends_on definition 12.35 Bounded operator; operator norm ¶
        • depends_on definition 5.37 Linear transformation ¶
        • depends_on definition 5.19 Norm ¶
      • 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 ¶
      • proves proof ch:10-hilbert-spaces@proof-37 ¶
    • 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 ¶
        • 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 proposition 12.43 Norm of a self-adjoint operator ¶
        • depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
        • proves proof ch:10-hilbert-spaces@prooflink-2 ¶
      • proves proof ch:10-hilbert-spaces@prooflink-3 ¶
  • 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 lemma A.586 Gaussian integral with a complex linear term ¶
  • depends_on equation 7.151 eq:ana-iterated ¶
  • depends_on equation 17.42 eq:ft-gaussian ¶
  • depends_on equation 17.35 eq:ft-translation ¶
  • proves proof app:A-long-proofs@proof-351 ¶
• depends_on theorem A.587 The Gaussian average is a non-zero projector that absorbs the Weyl operators ¶
  • depends_on definition A.585 The Gaussian average of the Weyl operators ¶
    • depends_on definition A.580 Weyl operator ¶
      • depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
      • depends_on definition 12.109 Weyl system ¶ ↺
      • depends_on equation 12.71 eq:hilbert-weyl-relation ¶
    • depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶
      • depends_on lemma A.254 Riemann integral of a continuous curve ¶
        • depends_on definition 12.2 Hilbert space ¶ ↺
        • depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
        • proves proof app:A-long-proofs@proof-157 ¶
      • depends_on lemma A.582 Joint strong continuity ¶
        • depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
        • depends_on lemma A.581 Composition law ¶
        • proves proof app:A-long-proofs@proof-348 ¶
      • depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶ ↺
      • proves proof app:A-long-proofs@proof-349 ¶
  • depends_on lemma A.584 Fourier injectivity in $n$ variables ¶
    • depends_on definition 17.29 Fourier transform; the treatise convention ¶
    • depends_on equation 17.42 eq:ft-gaussian ¶ ↺
    • depends_on theorem 17.32 Fourier inversion ¶
      • depends_on definition 17.29 Fourier transform; the treatise convention ¶ ↺
      • depends_on equation 17.42 eq:ft-gaussian ¶ ↺
      • proves proof ch:15-fourier-integral-transforms@proof-20 ¶
    • proves proof app:A-long-proofs@proof-350 ¶
  • depends_on lemma A.586 Gaussian integral with a complex linear term ¶ ↺
  • proves proof app:A-long-proofs@proof-352 ¶
• proves proof app:A-long-proofs@proof-356 ¶

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