## II: Fourier Analysis, Self-AdjointnessThis volume will serve several purposes: to provide an introduction for graduate students not previously acquainted with the material, to serve as a reference for mathematical physicists already working in the field, and to provide an introduction to various advanced topics which are difficult to understand in the literature. Not all the techniques and application are treated in the same depth. In general, we give a very thorough discussion of the mathematical techniques and applications in quatum mechanics, but provide only an introduction to the problems arising in quantum field theory, classical mechanics, and partial differential equations. Finally, some of the material developed in this volume will not find applications until Volume III. For all these reasons, this volume contains a great variety of subject matter. To help the reader select which material is important for him, we have provided a "Reader's Guide" at the end of each chapter. |

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### Contents

The Garding Wightman axioms | 61 |

Products of distributions wave front sets and oscillatory | 87 |

Von Neumann Algebras XIX Applications to Quantum Field Theory | 114 |

SELFADJOINTNESS AND THE EXISTENCE OF DYNAMICS Extensions of symmetric operators | 135 |

Applications to Statistical Mechanics | 139 |

Appendix Motion on a halfline limit pointlimit circle methods | 146 |

Perturbations of selfadjoint operators | 162 |

Quadratic forms | 176 |

Semigroups and their generators | 235 |

Hypercontractive semigroups | 258 |

Graph Limits | 268 |

The FeynmanKac formula | 274 |

Timedependent Hamiltonians | 282 |

Classical nonlinear wave equations | 293 |

The Hilbert space approach to classical mechanics | 313 |

Notes | 318 |

Pointwise positivity | 182 |

The commutator theorem | 191 |

Analytic vectors | 200 |

Free quantum fields | 207 |

Appendix The Weyl relations for the free field | 231 |

Problems | 338 |

Reader s Guide | 349 |

353 | |

355 | |

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adjoint Amer analytic vector Appendix to Section argument Banach space bounded map bounded operator classical commutation compact support conclude cone constant continuous function contraction semigroup converges convolution Corollary deficiency indices define denote differential operators discussion domain essentially self-adjoint Example exists finite follows formula Fourier transform Friedrichs extension Hamiltonian Hilbert space hypotheses implies inequality infinity integral interpolation theorem invariant Kato-Rellich theorem lemma linear Math measure norm p e D(A partial differential equations perturbation Phys Plancherel theorem pointwise polynomially bounded positivity preserving Problem Proof Let proof of Theorem properties Proposition prove quadratic form quantum field theory quantum mechanics reader real-valued representation Riesz-Thorin theorem satisfies Schrodinger self-adjoint extension self-adjoint operator semibounded semigroup singular Sobolev's solution spectral subset subspace supp Suppose symmetric operator tempered distribution Theorem X.36 uniformly bounded unique unitary group WF(T Wightman axioms zero