Random matrices reveal universal characteristics

Jonathan Breuer is a professor at the Hebrew University of Jerusalem, Israel. Thanks to a grant from Knut and Alice Wallenberg Foundation, he will be a visiting professor at the Department of Mathematics, KTH Royal Institute of Technology.

Common patterns can be found in apparently very different complex systems, both in mathematics and physics. Such patterns are said to be universal – they do not depend on precise details, only on the general properties of the system. One theory which captures such universal characteristics is random matrix theory. Random matrices describe large complex systems where it is typically impossible to exactly determine the values of all the matrix elements, which are then instead replaced by random variables.

Characteristic numbers, eigenvalues, can be calculated for each matrix. When the matrix elements are random, the eigenvalues are as well. Eigenvalues are crucial to a wide range of applications, such as statistical analyses of large data sets They also determine the energy levels of atoms and molecules in quantum physics, known as spectra. Random matrices come into play when trying to understand complicated spectra.

Quantum physics’ Schrödinger operators are of vital importance in all mathematical calculations of quantum systems’ dynamics and spectra. The aim of the planned project is to investigate aspects of the deep and multifaceted connections between random matrix theory and Schrödinger operators. The fact that these two different fields of mathematics are connected at all suggests that there are universal properties to be discovered.