TWELVE TALES IN MATHEMATICAL PHYSICS: AN EXPANDED HEINEMAN PRIZE LECTURE BARRY SIMON1;2 Abstract. This is an extended version of my 2018 Heineman prize lecture describing the work for which I got the prize. The citation is very broad, so this describes virtually all my work prior to 1995 and some afterwards. It discusses work in non-relativistic quantum mechanics, constructive quantum field theory and statis- tical mechanics. Contents 0. Introduction 2 1. Summability of Divergent Eigenvalue Perturbation Series 2 2. Complex Scaling Theory of Resonances 12 3. Statistical Mechanical Methods in EQFT 19 4. Thomas{Fermi Theory 25 5. Infrared Bounds and Continuous Symmetry Breaking 32 6. N{Body quantum mechanics 45 7. Magnetic Fields in NRQM 80 8. Quasi-classical and Non{quasi-classical limits 86 9. Almost Periodic and Ergodic Schr¨odinger Operators 103 10. Topological Methods in Condensed Matter Physics 122 11. Anderson Localization: The Simon-Wolff Criterion 133 12. Generic Singular Continuous Spectrum 144 13. Further Remarks 152 arXiv:2011.12335v2 [math-ph] 5 May 2021 References 155 Date: May 7, 2021. 2010 Mathematics Subject Classification. Primary: 81Q10, 81T08, 82B24; Sec- ondary: 47A55, 81Q15, 81Q20, 81Q70, 81T25, 81U24. Key words and phrases. Simon, Schr¨odingeroperators, quantum mechanics, quantum field theory, statistical mechanics. 1 Departments of Mathematics and Physics, Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125. E-mail:
[email protected]. 2 Research supported in part by NSF grants DMS-1265592 and DMS-1665526 and in part by Israeli BSF Grant No. 2014337. 1 2 B. SIMON 0. Introduction The citation for my 2018 Dannie Heineman prize for Mathematical Physics reads: for his fundamental contributions to the mathematical physics of quantum mechanics, quantum field theory, and statistical mechanics, including spectral theory, phase transitions, and geometric phases, and his many books and monographs that have deeply influenced generations of researchers.