Fachbereich Mathematik

# Approximation Methods in Mathematical Physics

## Objectives

In the course the students will learn how to apply rigorous approximation methods to problems arising from physics and applied analysis. The aim is to provide a set of useful analytical tools that simplify the analysis of physical systems, and often constitute the only way to deal with complicated structures. At the end of the course the students should be familiar with basic tools of the asymptotic expansion of integrals, perturbation theory in quantum mechanics, and iteration schemes for the solution of nonlinear differential equations. In addition, they should be able to apply the aforementioned tools to specific problems arising in Applied Mathematics, and to distinguish the rigorous methods of approximation theory from the less rigorous ones.

## Program

1. Approximation methods for integrals and series

• Laplace's Method
• Stationary phase
• Steepest Descents
• Asymptotic evaluation of Sums

2. Perturbation theory and WKB approximation

• Perturbation of spectra
• WKB Approximation
• Product Formulas (Baker-Campbell-Hausdorff, Trotter-Kato)

3. Iteration schemes and fixed point theorems in differential equations

• Duhamel's formula
• Picard's iteration scheme / Banach's fixed point theorem

4. Applications to Quantum Mechanics (time permitting)

• The Anharmonic oscillator
• Feynman path integrals
• Semiclassical analysis

## Suggested Bibliography

Lecture Notes

Lecture notes for the course (in english) will be provided.

Books

1. T. Kato. Perturbation Theory for Linear Operators. 1966
2. C. Bender, S. Orszag. Advanced Mathematical Methods for Scientists and Engineers. 1999
3. M. Reed, B. Simon. Methods of Modern Mathematical Physics. 1972-73

## Organization

• Exercise Classes and Homework

One exercise class per week. The homework assignments will be given once each two weeks. Half of the total homework points are required to be admitted to the final exam.

• Final Exam

The final examination typology will be decided in the first month of course together with the students.