This 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.

**X: THE FOURIER TRANSFORM**

1. The Fourier transform on 9°(R”) and 9”(R”), convolutions

2. The range of the Fourier transform: Classical spaces

3. The range of the Fourier transform: Analyticity

4. Lp Estimates

Appendix Abstract interpolation

5. Fundamental solutions of partial differential equations with constant coefficients

6. Elliptic regularity

7. The free Hamiltonian for nonrelativistic quantum mechanics

8. The Garding-Wightman axioms

Appendix Lorentz invariant measures

9. Restriction to submanifolds

10. Products of distributions, wave front sets, and oscillatory integrals

**X: SELF-ADJOINTNESS AND THE EXISTENCE OF DYNAMICS**

1. Extensions of symmetric operators

Appendix Motion on a half-line, limit point-limit circle methods

2. Perturbations of self-adjoint operators

3. Positivity and self-adjointness I: Quadratic forms

4. Positivity and self-adjointness II: Pointwise positivity

5. The commutator theorem

6. Analytic vectors

7. Free quantum fields

Appendix The Weyl relations for the free field

8. Semigroups and their generators

9. Hypercontractive semigroups

10. Graph Limits

11. The Feynman-Kac formula

12. Time-dependent H amiltonians

13. Classical nonlinear wave equations

14. The Hilbert space approach to classical mechanics

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