Lectures on the Mathematics of Quantum Mechanics II: Selected Topics

1. Lecture 1: Wigner Functions. Coherent States. Gabor Transform. Semiclassical Correlation Functions
Gianfausto Dell’Antonio

2. Lecture 2: Pseudo-differential Operators. Berezin, Kohn–Nirenberg, Born–Jordan Quantizations
Gianfausto Dell’Antonio

3. Lecture 3: Compact and Schatten Class Operators. Compactness Criteria. Bouquet of Inequalities
Gianfausto Dell’Antonio

4. Lecture 4: Periodic Potentials. Wigner–Seitz Cell and Brillouen Zone. Bloch and Wannier Functions
Gianfausto Dell’Antonio

5. Lecture 5: Connection with the Properties of a Crystal. Born–Oppenheimer Approximation. Edge States and Role of Topology
Gianfausto Dell’Antonio

6. Lecture 6: Lie–Trotter Formula, Wiener Process, Feynman–Kac Formula
Gianfausto Dell’Antonio

7. Lecture 7: Elements of Probability Theory. Construction of Brownian Motion. Diffusions
Gianfausto Dell’Antonio

8. Lecture 8: Ornstein–Uhlenbeck Process. Markov Structure. Semigroup Property. Paths Over Function Spaces
Gianfausto Dell’Antonio

9. Lecture 9: Modular Operator. Tomita–Takesaki Theory Non-commutative Integration
Gianfausto Dell’Antonio

10. Lecture 10: Scattering Theory. Time-Dependent Formalism. Wave Operators
Gianfausto Dell’Antonio

11. Lecture 11: Time Independent Formalisms. Flux-Across Surfaces. Enss Method. Inverse Scattering
Gianfausto Dell’Antonio

12. Lecture 12: The Method of Enss. Propagation Estimates. Mourre Method. Kato Smoothness, Elements of Algebraic Scattering Theory
Gianfausto Dell’Antonio

13. Lecture 13: The N-Body Quantum System: Spectral Structure and Scattering
Gianfausto Dell’Antonio

14. Lecture 14: Positivity Preserving Maps. Markov Semigropus. Contractive Dirichlet Forms
Gianfausto Dell’Antonio

15. Lecture 15: Hypercontractivity. Logarithmic Sobolev Inequalities. Harmonic Group
Gianfausto Dell’Antonio

16. Lecture 16: Measure (Gage) Spaces. Clifford Algebra, C.A.R. Relations. Fermi Field
Gianfausto Dell’Antonio

Keywords: Physics, Quantum Physics, Mathematical Physics, Mechanics