Orateur
Description
I consider Callias operators (that is, Dirac-Schrödinger operators perturbed by an operator-valued potential $A$) on odd, $d$-dimensional hyperbolic space. Without imposing uniform invertibility of the potential outside a compact region, the operators are no longer Fredholm, so that the Witten index, a type of regularized index defined via an iterated trace limit, becomes the natural substitute for the classical notion of index.
An explicit trace formula, which leads to a new index formula, has already been found in the Euclidean case, in dimension $d=1$ by Pushnitski, Gesztesy et al. and others, and recently, by Fürst, for $d \geq 3$.
My approach combines Fourier-Helgason analytic methods on symmetric spaces with Volterra-type constructions of heat kernels, adapted to the curved setting of hyperbolic space. In this talk, I present an explicit trace formula, which constitutes the first of its kind in a multi-dimensional, non-Euclidean setting.