  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.
