Trace formula and Levinson's theorem as an index pairing in the presence of resonances

Abstract

We realise the number of bound states of a Schr\"{o}dinger operator on $\mathbb{R}^n$ as an index pairing in all dimensions. Expanding on ideas of Guillop\'{e} and others, we use high-energy corrections to find representatives of the $K$-theory class of the scattering operator. These representatives allow us to compute the number of bound states using an integral formula involving heat kernel coefficients.