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Twelve Tales in Mathematical Physics: An Expanded Heinemann Prize Lecture

Barry Simon

Abstract

This is an extended version of my 2018 Heinemann 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 statistical mechanics.

Twelve Tales in Mathematical Physics: An Expanded Heinemann Prize Lecture

Abstract

This is an extended version of my 2018 Heinemann 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 statistical mechanics.

Paper Structure

This paper contains 14 sections, 55 theorems, 295 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Summability of Divergent Eigenvalue Perturbation Series
  3. Complex Scaling Theory of Resonances
  4. Statistical Mechanical Methods in EQFT
  5. Thomas--Fermi Theory
  6. Infrared Bounds and Continuous Symmetry Breaking
  7. $N$--Body quantum mechanics
  8. Magnetic Fields in NRQM
  9. Quasi-classical and Non--quasi-classical limits
  10. Almost Periodic and Ergodic Schrödinger Operators
  11. Topological Methods in Condensed Matter Physics
  12. Anderson Localization: The Simon-Wolff Criterion
  13. Generic Singular Continuous Spectrum
  14. Further Remarks

Key Result

Theorem \oldthetheorem

If $A(\beta)$ is a family of type A and $E_0$ is an isolated eigenvalue of $A_0$ of finite multiplicity, $\ell$, then there exist $\ell$ analytic functions, $\{E_j(\beta)\}_{j=1}^\ell$, near $\beta=0$ which are all the eigenvalues of $A(\beta)$ near $E_0$ when $\beta$ is small. Moreover, there exist

Figures (3)

  • Figure 1: Spectrum of a Complex Scaled Hamiltonian
  • Figure 2: The Hofstadter Butterfly
  • Figure 3: Zeros of an OPUC

Theorems & Definitions (60)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem: FSS1FSS2
  • Remark
  • Theorem \oldthetheorem: DLS2
  • Theorem \oldthetheorem: SimonLSSimonLSAnon
  • ...and 50 more