  We discuss a method to construct hadronic scattering and decay amplitudes
from Euclidean correlators, by combining the approach of a regulated inverse
Laplace transform with the work of Maiani and Testa. Revisiting the original
result, we observe that the key observation, i.e. that only threshold
scattering information can be extracted at large separations, can be understood
by interpreting the correlator as a spectral function, $\rho(\omega)$,
convoluted with the Euclidean kernel, $e^{- \omega t}$, which is sharply peaked
at threshold. We therefore consider a modification in which a smooth step
function, equal to one above a target energy, is inserted in the spectral
decomposition. This can be achieved either through Backus-Gilbert-like methods
or more directly using the variational approach. The result is a shifted
resolution function, such that the large $t$ limit projects onto scattering or
decay amplitudes above threshold. The utility of this method is highlighted
through large $t$ expansions of both three- and four-point functions that
include leading terms proportional to the real and imaginary parts (separately)
of the target observable. This work also presents new results relevant for the
un-modified correlator at threshold, including expressions for extracting the
$N \pi$ scattering length from four-point functions and a new strategy to
organize the large $t$ expansion that exhibits better convergence than the
expansion in powers of $1/t$.
