# Pseudo-differential operators

Prof. Bernd Ammann, Prof. Helmut Abels
Veranstaltungsnummer

# Aktuelles

Letzte Vorlesung, Mo 15-17, 30.7.2012, in Physik 7.1.21

## Description

The class of pseudo-differential operators is a natural extension of the class differential operators, and contains many more important operators. For example, if an elliptic differential operator has an inverse, then this inverse is a pseudo-differential operator as well. Inside the class of pseudo-differential operators one can also take complex powers, and expression like $\sqrt{1-\Delta}$ are well-defined pseudo-differential operators.

The lecture will start with an introduction of pseudo-differential operators on euclidean space. Here Fourier transformation is an important technique, and we will see how one can use the symbol calculus to invert many pseudo-differential operators up to smoothing operators. Applications to partial differential equations will be developed, we discuss adjoints and composition of operators, and we discuss regularity theory.

The definition of the operators on euclidean spaces is then used to define pseudo-differential operators on manifolds. We study the behavior of the wave front set of solutions of hyperbolic equations and study the relation to geodesics on an associated Lorentzian manifold. We get a very conceptual description of the propagation of waves in space-time.

In the last part of the lecture we intend to give an overview over the Boutet-de-Monvel calculus, which extends the class of usual pseudodifferential operators to model boundary value problems and their solution operators.

## Place and time

Tuesday 14-16, M101

## Exercises

Tuesday 16-18, M101

### Exercises

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## Literature

1. Helmut Abels, Pseudodifferential and singular integral operators, de Gruyter
2. Gerd Grubb, Distributions and operators, Graduate Texts in Mathematics 252, Springer
3. H. Kumano-Go, Pseudo-differential operators, MIT press, Cambridge Massachusetts
4. X. Saint Raymond, Elementary introduction to the theory of pseudodifferential operators, Studies in Advanced Mathematics, CRC Press, Boca Raton, Ann Arbor, Boston, London, 1991
5. Fabio Nicola, Luigi Rodino, Global Pseudo-differential calculus on euclidean Spaces, Birkhäuser
6. Blaine Lawson, Marie-Louise Michelsohn, Spin geometry, Chapter III Section 3, Princeton Math. Series
7. Alexander Strohmeier, Microlocal Analysis in Bär, Fredenhagen (ed.) Quantum Field Theory on Curved Spacetimes, Lecture Notes in Physics 786, Springer
8. Lars Hörmander, The Analysis of Linear partial Differential operators III, Pseudo-differential operators, Springer
Recommendations: For obtaining a first impression on what pseudodifferential operators are and what kind of results can be done with them, have a look in the book by Abels or by Nicola and Rodino. The book by Hörmander is an excellent reference if you want to go into the details. To adapt the PDOs to (compact) manifolds, we recommend the short summary in the book by Lawson and Michelsohn. Consult the summary of Strohmaier for applications to quantum field theory.

## Modulteilprüfung

Die Modulteilprüfung ist mündlich. Voraussetzung zur Zulassung ist die erfolgreiche aktive Teilnahme an den Übungen.

Bernd Ammann, 2.2.2012 oder später