Spectral theory of Partial Differential Equations
(PhD course, Spring 2022)

  • Problem sheet 3 is online!.
Brief description

We learn some fundamental properties of the spectra, in particular eigenvalues and eigenfunctions, of partial differential equations such as the Laplace or Schrödinger equations on Euclidean domains and manifolds. After a thorough introduction we focus on the relation between the spectrum and the geometry of the underlying space and go through classical results such as the Faber-Krahn inequality or the counter example to Mark Kac' famous question "Can one hear the shape of a drum?". The course does not require any preknowledge on spectral theory or any deeper prerequisites on partial differential equations. Everything will be introduced on the spot as it is needed.

Prerequisites: Good knowledge of standard analysis and linear algebra; some basic knowledge of functional analysis and operator theory is useful.

  • Mondays, 10:15 - 12:00, starting from January 31, Kräftriket 5, room 22, and on Zoom, ID 61021079366.
  • Introductory meeting Friday, January 21.
Date Contents
31 January Computable spectra
7 February Discrete spectral theorem
14 February Partial differential operators with discrete spectra
21 February Variational characterization of EV and consequences
28 February Weyl's law, Faber-Krahn inequality
7 March Nodal domains
14 March Can one hear the shape of a drum?
21 March The Laplacian on ℝd
4 April Schrödinger operators

Three problem sets to be handed in will be published during the course.

Sheet 1, Sheet 2, Sheet 3.


Examination takes place through problem sheets and a presentation. Both can be done in groups of two or alone. In order to pass the course it is required to
  • reach at least 50% of all points of the problem sets and
  • give a decent presentation of the results, methods and ideas of a selected research article related to spectral theory of PDEs.
  • A nice introductory survey on the topic is provided in these lecture notes by Richard Laugesen from the University of Illinois
  • Further literature TBA

Jonathan Rohleder