Higher order spectral shift and Callias operatorsOberseminar Global Analysis and Operator Algebras

by Oliver Fürst (Uni Bonn)

Tuesday Jun 3, 2025, 2:15 PM → 3:15 PM Europe/Berlin

Seminar room 1.008 (Mathezentrum)

The spectral shift function is a central object in mathematical scattering theory, which was developed in the 50's by M. Krein. In the late 80's it reemerged in the context of index theory, both as a replacement of Fredholm index and spectral flow for non-Fredholm operators. Due to the unstable nature of spectral shift, it took until 2008 to show that the "Index=Spectral Flow"-theorem also holds in this more general context, which was done by A. Pushnitski. Since then, many improvements have been developed in this essentially one-dimensional case.

However, the higher dimensional case, which naturally corresponds to Callias operators (Dirac operators with an operator-valued potential), was open until recently.

In this talk we show how the Fredholm index and the top degree Chern character have to be regularized in the higher dimensional case in terms of higher order spectral shift functions. These functions were conjectured in the 80's by Koplienko, however their existence in all cases have only been shown in the relatively recent past in 2013 by D. Potapov, A. Skripka, and F. Sukochev.
We present a non-Fredholm version of the classical Callias index theorem and give some concrete examples, in which the regularized index may assume any real number.