# Is there an analogue of the LSZ reduction formula in quantum mechanics?

In quantum field theory the LSZ reduction formula gives us a method of calculating S-matrix elements. In order to understand better scattering in QFT, I will study scattering in non-relativistic quantum mechanics and that question ocurred to me.

In QFT, LSZ formula is a tool to obtain S-matrix from correlation function. In QFT correlation function is very easy to calculate(free and interacting theories). When outgoing particles became on-shell then we can relate S-matrix element to correlation function. I QM all the particles are always on-shell. In non-relativistic limit scattering matrix(amplitude) should be compared to Born-approximation. So if we Fourier transforms the $i \cal{M}$ (non-relativistic limit) back into position space, then we can see the behavior of potential.
$$S = \int_{xt} \psi^\dagger \left(i{\partial \over \partial t} + {\nabla^2\over 2m}\right)\psi - \psi^\dagger(x) \psi(x) V(x).$$ After that we can do the usual trick using generating functional. After taking the functional derivative with respect to current($J$), we can find correlation function. $${\displaystyle Z[J]=\int {\mathcal {D}}\phi e^{i(S[\phi ]+\int d^{d}xJ(x)\phi (x))}~,}$$ Ofcourse, propagator are different in QM as compared to QFT. In this way we can find S-matrix element using correlation function.