Research project New spectral inequalities for domains and graphs

Eigenvalues of differential equations play an important roll in the description of numerous processes in science and engineering. Their mathematical investigation goes back to Lord Rayleigh's famous book The Theory of Sound, but the area of mathematics dealing with them – so-called spectral theory – is very active by now and is all the time inspired by new problems from physics, engineering and other fields.

This research project focuses on eigenvalues of Laplace and Schrödinger differential equations. The eigenvalues of the Laplace equation describe frequencies of vibrating strings or membranes as for instance a drum. The eigenvalues of the Schrödinger equation correspond to the energies of quantum mechanical particles under the influence of an electric potential. As eigenvalue equations can only be solved exactly in very simple geometric situations, the quantitative and qualitative investigation of eigenvalues with theoretical methods from mathematical analysis and operator theory is of particular significance.

The aim of this project is to establish inequalities between eigenvalues in different physical constellations. For instance we compare eigenvalues for the Laplace equation with Dirichlet boundary values with eigenvalues for the Laplace equation with mixed Dirichlet-Neumann boundary values. This corresponds to the frequencies of vibrations of a membrane that is fixed along the whole boundary or just on a part. Such comparison principles have been applied recently for instance in the proof of the so-called hot spots conjecture in certain geometric situations. The latter conjecture deals with the long term temperature distribution in a heat-insulated flat medium.

Another aspect is the development of spectral inequalities for so-called quantum graphs, which serve as models in the description of quantum physical processes in network-like structures such as quantum cables or photonic crystals. The mathematical investigation of quantum graphs is a lively and active field in modern mathematical physics. It includes special mathematical challenges because of the large influence of the topological and combinatorical properties of graphs.