Institute of Mathematics Jena

Summer School

Spectral Theory
Schrödinger Operators

Jena, July 31 - August 3, 2018

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Tanya Christiansen (University of Missouri)
Resonances and Schrödinger operators

This minicourse provides an introduction to the theory of resonances via the particular case of Schrödinger operators on ℝd. Physically, resonances may correspond to decaying waves. From a mathematical point of view, resonances can provide a replacement for discrete spectral data for a class of operators with continuous spectrum. When the dimension d is odd, resonances are isolated points in the complex plane.
We will explore some of what is known about the distribution of resonances for Schrödinger operators, and outline some proofs. For example, one can show that there are regions of the complex plane which have no resonances. We talk about the problem of bounding, both from above and below, the counting function for the number of resonances in a disk of radius r. Lower bounds in particular are not well-understood in dimension greater than one. The proofs of these results use tools from both functional analysis and complex analysis.