Quantitative methods in semiclassical spectral theory

Dr Simon Larson will receive funding from the Knut and Alice Wallenberg Foundation to recruit an international researcher for a postdoctoral position at the Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.

Spectral theory belongs to classical mathematical analysis and has many applications in physics and engineering, beyond a purely mathematical interest. The proposed project lies at the intersection of spectral theory, geometry and mathematical physics.

The connection between geometry and spectral theory was popularized by the Polish American mathematician Mark Kac. In 1966, he asked: can one hear the shape of a drum? The answer came almost 30 years later: no, one cannot hear the shape of a drum, as the same sound can come from different vibrating membranes. However, it is possible to determine some properties of a membrane, such as area and perimeter.

Mathematically, the question is formulated in terms of eigenvalues of operators (the frequencies with which the membrane oscillates) whether they can be used to recreate the underlying geometry. In the project, the focus is on the reverse problem of understanding the eigenvalues based on knowledge of the operator, especially the Schrödinger operators of quantum mechanics, where eigenvalues correspond to the energy levels of particles or systems of particles.

Of particular interest here is how the eigenvalues of the Schrödinger operators are distributed asymptotically towards infinity. The interest is motivated by the fact that it is precisely in this so-called semiclassical limit to infinity that quantum mechanics becomes classical mechanics. The proposed project aims to develop new tools for estimating the accuracy of semiclassical approximations under minimal assumptions on the underlying operators.