## Advanced Mathematical Physics (with Jérémy Sok), Copenhagen 2017

- July 6: Seminar notes "Van der Waals Force" available. You can pick up your corrected assignments in office 04.2.11 until July 14.

**Lecture Notes**:

Lectures Niels:

- April 24: quantum mechanics, basic functional analysis, definition of spectrum
- April 25: resolvent identities, Neumann series, analyticity of the resolvent, adjoint operator, self-adjointness
- April 28: Hellinger-Töplitz, Schwartz space, basic criterion for self-adjointness, Kato-Rellich, uniqueness for the time-dependent Schrödinger equation
- May 2: self-adjoint extensions, Fourier transform, Sobolev spaces, Weyl criterion, multiplication operators, the Laplacian
- May 5: Schrödinger operators with Coulomb potential, Sobolev inequalities, strongly continuous unitary groups
- May 8: the generator of a strongly continuous unitary group and its properties
- May 9: translations & rotations, existence of SCUGs, commuting operators, Noether theorem, scattering theory: the free evolution
- May 15: Wave operators
- May 16: Absence of bound states, asymptotic completeness
- May 19: Stationary scattering theory (heuristic), exponential localization of eigenstates

Lectures Jérémy:

- The Spectral Theorem
- Self-Adjoint Extensions
- Quadratic forms
- Rollnik potentials
- Spectral Analysis of (some) Schrödinger Operators I
- Spectral Analysis of (some) Schrödinger Operators II
- Hartree-Fock theory

**Assignments** (to be handed in at the beginning of the Friday seminar!):

- Assignment 1 (Deadline: May 5)
- Assignment 2 (Deadline: May 19)
- Assignment 3 (Deadline: June 2)
- Assignment 4 (Deadline: June 23, strictly before the seminar!)

**Summer School** on Current Topics in Mathematic Physics at the University of Zurich, Switzerland, July 17 - July 21:

**Seminar:** All talks have to be about 40 minutes long (not more!). Schedule is subject to changes depending on our progression in the lecture. The summary is due on Monday after the seminar. Please remember to provide a list of references. Contact Jérémy or me at least two weeks before your talk for a briefing. Topics and summaries:

- May 5: Compact operators
- May 5: Fredholm alternative
- May 26: Solutions for Problems 3-5 of Assigment 2
- May 26: Trotter product formula & BCH formula
- June 2: Uniqueness of the ground state (positivity improving operators)
- June 2: Lieb's estimate on maximum ionization
- June 9: Solution of Assigment 3, Problems 3 and 4
- June 9: Tensor products; fermions and bosons
- June 16: Fock space & creation/annihilation operators
- June 16: Perturbation theory
- June 23: new Derivation of the Van-der-Waals force
- June 23: Discussion of Assigment 4

**Criteria for passing the course:**

Reach approximately 50% of the points averaged over the four assignments.

Give a seminar talk and produce a summary of your talk for the other participants.

**Literature:** The lecture notes should be self-contained for most of the course. If you are looking for additional reading, here are some recommendations.

- Gerald Teschl: Mathematical Methods in Quantum Mechanics. Online version
Contains essentially all topics of the lecture, but in a different arrangement. Sometimes details are missing.

- Stephen J. Gustafson, Israel Michael Sigal: Mathematical Concepts of Quantum Mechanics. Online version
Nice selection of material, but sometimes sketchy. Good overview.

- Michael Reed, Barry Simon: Methods of Modern Mathematical Physics, Volumes I-IV.
Recommended as a reference work, not as a text book. Concise but rather dense.

- Elliott H. Lieb, Michael Loss: Analysis. Errata
A functional analysis book aimed at applications in quantum mechanics. Contains many useful explicit estimates.

If you want to read up on quantum mechanics from the physicist point of view (which is not the focus of this course):

- Gordon Baym: Lectures on Quantum Mechanics
- Leslie E. Ballentine: Quantum Mechanics, A Modern Development
- Steven Weinberg: Lectures on Quantum Mechanics

## Advanced Mathematical Physics (with Jérémy Sok), Copenhagen 2016

Thank you all for following the course! New materials for the 2017 course will appear above.

## Teaching assistant, Bonn 2011-2014

I assisted Prof. Benjamin Schlein at the University of Bonn for the courses

- Analysis 2, spring 2013
- Analysis 1, fall 2012
- Vorkurs Mathematik, fall 2012
- Functional Analysis and Partial Differential Equations, fall 2011.

and for a Summer School at the University of Heidelberg

- Mathematical Physics, Analysis and Stochastics, 21.-26. July 2014.