  We introduce a nonperturbative approximation scheme for performing scattering
calculations in two dimensions that involves neglecting the contribution of the
evanescent waves to the scattering amplitude. This corresponds to replacing the
interaction potential $v$ with an associated energy-dependent nonlocal
potential ${\mathscr{V}}_k$ that does not couple to the evanescent waves. The
scattering solutions $\psi(\mathbf{r})$ of the Schr\"odinger equation,
$(-\nabla^2+{\mathscr{V}}_k)\psi(\mathbf{r})=k^2\psi(\mathbf{r})$, has the
remarkable property that their Fourier transform $\tilde\psi(\mathbf{p})$
vanishes unless $\mathbf{p}$ corresponds to the momentum of a classical
particle whose magnitude equals $k$. We construct a transfer matrix for this
class of nonlocal potentials and explore its representation in terms of the
evolution operator for an effective non-unitary quantum system. We show that
the above approximation reduces to the first Born approximation for weak
potentials, and similarly to the semiclassical approximation, becomes valid at
high energies. Furthermore, we identify an infinite class of complex potentials
for which this approximation scheme is exact. We also discuss the appealing
practical and mathematical aspects of this scheme.
