  We study the $2 \to 2$ scattering in the regime where the wavelength of the
scattered objects is comparable to their distance but is much larger than any
Compton wavelength in the quantum field theory. We observe that in this regime
- which differs from the eikonal - the Feynman diagram expansion takes the form
of a geometric series, akin to the Born series of quantum mechanics.
Conversely, we can define the Feynman diagram expansion as the Born series of a
relativistic effective-one-body (EOB) Schr\"odinger equation. For a
gravitational theory in this regime we observe that the EOB Schr\"odinger
equation reduces to the Regge-Wheeler or Teukolsky wave equations. We make use
of this understanding to study the tree-level Compton scattering off a Kerr
black hole. We compute the scalar and photon Compton amplitude up to
$O(a^{30})$ in the black hole spin $a$ and propose an all-order expression.
Remarkably, we find that boundary terms, which are typically neglected, give
non-zero contact pieces necessary for restoring crossing symmetry and gauge
invariance of the Kerr-Compton amplitude.
