Scattering using Euclidean Green functions

We show that it is possible to compute differential cross sections using matrix elements of polynomials in $e^{-\beta H}$ in normalizable states. These matrix elements can be calculated by quadrature using reflection-positive Euclidean Green functions. The the proposed method is based on an explicit ``time-dependent'' computation of the M{\o}ller wave operators using the Kato-Birman invariance principle to replace $H$ by $-e^{-\beta H}$ in the expression for the wave operators. The compact spectrum of $-e^{-\beta H}$ allows uniform polynomial approximations of continuous functions of $-e^{-\beta H}$. We tested the method using a solvable model with the range and strength of a typical nucleon-nucleon interaction and found convergence to the transition matrix elements for energies up to 1.5 GeV.