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

We realise the number of bound states of a Schrödinger operator on ${\mathbb{R}}^n$ as an index pairing in all dimensions. Expanding on ideas of Guillopé 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.