Welcome to Green's Fuction Methods ind PDE and Physics

**Description: **The course has two distinct parts as described below.

**1. Horia Cornean:** “The general mathematical theory behind Green's functions”.

**2. Thomas Søndergaard:** “Green’s function integral equation methods for solving the Helmholtz equation”.

A “Green function” is a generic name given by physicists to the integral kernel of the resolvent of certain PDEs generated by the Laplace, Helmholtz, Maxwell, and D'Alembert operators. In a physics language the Greens function essentially describes the response to a point source.

1. The first part of the course will cover how to find this integral kernel based on solving linear integral equations. We will also discuss several mathematical properties of the integral kernels of resolvents of elliptic partial differential operators acting on (un)bounded domains. In particular, we will be interested in self-adjointness and regularity issues, the Lippmann-Schwinger equation and the Limiting Absorption Principle.

2. The second part of the course covers Green’s function integral equation methods for solving the Helmholtz equation in relation to the type of scattering problems found in electromagnetics / optics. The methods require finding self-consistent solutions to integral equations that express e.g. the total electric field in terms of an incident field and a scattered field, where the latter is given as an overlap integral between a Green’s function and the field itself on or inside a scattering object.

For the purpose of these methods this part of the course will first cover the construction of Green’s functions using eigenmode expansion techniques with radiating (open) or periodic boundary conditions depending on the type of scattering problem. In the case of layered reference structures the Green’s function will be expressed in terms of Sommerfeld integrals, and details of their evaluation in terms of poles of Fresnel reflection and transmission coefficients corresponding to bound modes will be discussed. In addition, details of the handling of singularities in Green’s functions will be covered. The course will cover a range of integral equation methods for theoretical research within the area of nano optics.

**Prerequisites: ** advanced knowledge in mathematical analysis and/or theoretical physics.

**Evaluation:** During each lecture the participants are evaluated through exercises, which provide and prove an understanding of the presented theory. In order to pass the course, at least the exercises related to one of the topics should be satisfactory evaluated.

**Organizers and lecturers: **Professor Horia Cornean, e-mail: cornean@math.aau.dk and Associate Professor Thomas Søndergaard, e-mail: ts@mp.aau.dk

**ECTS: **4 **Time: **First part of the course: Full days (8:15-16:15) on 26 February to 2 March 2018.

Second part of the course: Full days (8:15-16:15) on 5 March to 9 March 2018.

**Place: **Aalborg University, Skjernvej 4A, Room 3.102.**Zip code: **9220

**Aalborg Øst**

City:

City:

**Number of seats:**40

**Deadline:**12 February 2018

**Important information concerning PhD courses **We have over some time experienced problems with no-show for both project and general courses. It has now reached a point where we are forced to take action. Therefore, the Doctoral School has decided to introduce a no-show fee of DKK 5,000 for each course where the student does not show up. Cancellations are accepted no later than 2 weeks before start of the course. Registered illness is of course an acceptable reason for not showing up on those days. Furthermore, all courses open for registration approximately three months before start. This can hopefully also provide new students a chance to register for courses during the year. We look forward to your registrations.

- Teacher: Horia Cornean
- Teacher: Thomas Søndergaard