Semiclassical Resonances of Schr\"odinger operators as zeroes of regularized determinants

We prove that the resonances of long range perturbations of the (semiclassical) Laplacian are the zeroes of natural perturbation determinants. We more precisely obtain factorizations of these determinants of the form $\prod_{w = {\rm resonances}}(z-w) \exp (\varphi_p(z,h))$ and give semiclassical bounds on $\partial_z \varphi_p$ as well as a representation of Koplienko's regularized spectral shift function. Here the index $p \geq 1$ depends on the decay rate at infinity of the perturbation.