Syllabus

Starting from a precise mathematical formulation of quantum mechanics (no previous physics background is assumed), including in particular the famous uncertainty and exclusion principles, we will aim to rigorously prove the stability of ordinary matter. This is something which most of us take for granted and physics textbooks rarely discuss but which is quite subtle and requires proper analysis. Various generally useful functional inequalities such as the Hardy, Sobolev and Lieb-Thirring inequalities will be introduced and proved during the course.

Here is a preliminary plan for the lectures based on the table of contents of the lecture notes. The lecture number 1-17 is indicated in square brackets. Not all topics will be covered.

1. Introduction [1]2. Some preliminaries and notation [1,2]2.1. Hilbert spaces2.2. Lebesgue spaces2.3. Fourier transform2.4. Sobolev spaces2.5. Forms and operators3. A very brief mathematical formulation of classical and quantum mechanics [3,4]3.1. Some classical mechanics3.2. The instability of classical matter3.3. Some quantum mechanics3.4. The one-body problem3.5. The two-body problem and the hydrogenic atom3.6. The N -body problem4. Uncertainty principles [5,6,7,8]4.1. Heisenberg4.2. Hardy4.3. Sobolev4.4. Application to the stability of the hydrogen atom4.5. Poincaré4.6. Poincaré-Sobolev?4.7. IMS localization?4.8. Local uncertainty5. Exclusion principles [9,10]5.1. Fermions5.2. Repulsive bosons5.3. Local exclusion5.4. Anyons6. The Lieb–Thirring inequality [11,12,13]6.1. Some history6.2. Covering lemma6.3. Local proof of LT for fermions6.4. Local proof of LT for inverse-square repulsive bosons6.5. One-body formulations6.6. Some applications of LT6.7. Connections between Hardy–Sobolev–LT?7. The stability of matter [14,15,16]7.1. Some history7.2. Stability of the first kind7.3. Some electrostatics7.4. Proof of stability of the second kind

Examination: There will be homework assignments and, if aiming for a higher grade, an individual project. More information will become available when the course starts.