Vorlesung (Block-kurs): Geometry and bound states of magnetic Schrödinger operators (N. Raymond) (SoSe 2016)
Lecturer: Nicolas Raymond (IRMAR, University of Rennes 1).
Time, room: This is a 'Block-kurs' consisting of 10 lectures each of
30 May - 03 June 2016 (Mo-Fr) 18-20Uhr & 06 - 10 June 2016 (Mo-Fri) 18-20Uhr.
First meeting: Monday 30 May 18.15Uhr in A 027.
Exercises (Übungen): There will be a number of homeworks for students wishing to acquire a 'Schein'.
This course will address the spectral theory of magnetic Schrödinger operators, in the semiclassical limit. We will introduce the methods and ideas related to the accurate description of the low-lying spectrum. We will first recall the basics of spectral theory and deal with many examples. In particular, we will explain what the quantum harmonic approximation (for electric potentials) is and justify it, with the idea to extend it to the magnetic situation. For that purpose, we will discuss the methods related to partitions of unity and to localization properties of eigenfunctions (with applications to some non-linear situations). In this course, we will also meet the strategy of dimensional reduction called "Born-Oppenheimer approximation" and apply it, for instance, to the eigenvalue-counting problem. If time permits, we will also talk about the famous Birkhoff normal form and about complex WKB constructions (with examples).
Overview of topics:
0. Informal introduction
1. Elements of spectral theory (reminders)
- Min-max and spectral theorems and examples
- Simplicity of electric groundstates: Harnack inequality
- Harmonic oscillator
- De Gennes operator
- Elements of analytic perturbation theory
3. Semiclassical examples
- A Weyl estimate in dimension one
- Electric harmonic approximation (in dimension one)
- Magnetic harmonic approximation
4. From local models to global estimates
- A localization formula
- Harmonic approximation (bis)
- Agmon-Persson estimates
- Harmonic approximation (ter)
5. Birkhoff normal form in dimension one
- Reminders of pseudo-differential calculus
- Birkhoff normal form
- Microlocalization and spectral estimates
6. About the Born-Oppenheimer approximation
- Electric case
- Magnetic case
If time permits:
7. Examples of complex WKB constructions
- Electric case in dimension one
- Some magnetic examples
Audience (Hörerkreis): Master students of Mathematics and Physics, TMP-Master.
Credits: 2 ECTS.
Exam (Prüfung): There will be a number of homeworks for students wishing to acquire a 'Schein'. The exact modus will be discussed by Prof. Raymond in the lectures.
Prerequisites: A basic knowledge of Mathematical Quantum Mechanics (corresponding to the course 'Mathematical Quantum Mechanics 1' (MQM1)) is an advantage.
Language (Sprache): English.
Literature: The lectures will be based on
[R] N. Raymond, Bound States of the Magnetic Schrödinger Operator, EMS Tracts in Mathematics (2016, forthcoming).
Letzte Änderung: 27 July 2016 (no more updates).
Thomas Østergaard Sørensen