The mathematics of dilute quantum gases (MAT684.1)

Imortant notice: Due to the Corona virus epedemic, the lecture continues in the format of a video lecture with two question sessions per week via Skype. A link to the video of the weekly lecture is send to the participants on Thursdays during the day. If you are interested in participating please write me an email and I will send you a link to the video files. I will also inform you about the schedule of the question sessions.

Time: Thursday from 15:15 - 17:45 h (from 20.02.2020 to 28.05.2020)
Room: 27-H-28

General description: In our quantum mechanical description of nature we encouter two sorts of particles, bosons (e.g. photons, certain atoms and molecules) and fermions (e.g. electrons, protons, neutrons, certain atoms and molecules). The dilute Bose and the dilute Fermi gas, that is, a system of bosons/fermions where the range of the interaction between the particles is much shorter than their typical distance, are therefore two of the most fundamental models in quantum statistical mechanics. Of particular interest are their ground state energies in the thermodynamic limit (infinite volume limit at constant density), which we consider in the first part of the lecture. In the second part we study the ground state energy and the phenomenon of Bose-Einstein condensation (all particles do approximately the ``same thing’’) for a dilute trapped Bose gas (as prepared in modern experiments with cold alkali gases), and in the third and final part we give a proof of the Bose-Einstein condensation phase transition for an optical lattice model in the thermodynamic limit. The techniques we introduce in this lecture are relevant for the mathematical analysis of quantum many-particle systems and models in quantum statistical mechanics.

Prior Knowledge: Knowledge of a course in either Partial Differential Equations, Functional Analysis or Quantum Mechanics (from a mathematics of physics point of view). The lecture will be adapted to the background of the audience.


a) E. H. Lieb, R. Seiringer, J. P. Solovej, J. Yngvason, The mathematics of the Bose gas and its condensation, Birkhäuser, 2005. (Several parts of the lecture follow this book.)
b) G. Teschl, Mathematical methods in quantum mechanics, AMS 2009. (Introduction to Banach spaces, Hilbert spaces, (unbounded) self-adjoint operators, spectral theorem, Schrödinger operators etc.)
c) Many body quantum mechanics, lecture notes by J. P. Solovej with corrections and additions of P. T. Nam, 2014. (Hilbert spaces, principles of quantum mechanics, (unbounded) self-adjoint operators, Schrödinger operators, second quantization etc.)
d) Reed and Simon, Methods of modern mathematical physics, Volume I-IV. (The bible of mathematical quantum mechanics. First volume contains a beautiful introduction to abstract functional analysis and (unbounded) self-adjoint operators.)
e) S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, 2nd Ed., Springer, 2011.
f) E. H. Lieb, M. Loss: Analysis, Amer. Math. Soc. 2001. (Contains an introduction to measure theory, integral inequalities, distributions, Sobolev spaces, variational approach to the Schrödinger equation and much more.)
g) Evans: Partial differential equations, Amer. Math. Soc. 1997. (Chapter 5 contains an introduction to Sobolev spaces, Chapter 6 to weak solutions of elliptic PDE.)
h) E. H. Lieb, R. Seiringer, The stability of matter in quantum mechanics, Cambridge University press (Contains a short chapter on spin)
i) B. Helffer, Spectral theory and its applications, Cambridge University press (Contains a construction of the operator domains of the Dirichlet and the Neumann Laplacian)
j) L. D. Landau, E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory (Quantum mechanics from a Physics point of view.)
k) E. H. Lieb, R. Seiringer, J. P. Solovej, Ground state energy of the low density Fermi gas, Phys. Rev. A 71, 053605 (2005) (Main reference for third chapter of lecture)

Lecture notes (Disclaimer: The notes are not self-contained and certainly full of minor mistakes.):

  1. Introduction
  2. The dilute Bose gas
  3. The dilute Fermi gas
  4. Bose-Einstein Condensation and Superfluidity