  We offer a consistent dynamical formulation of stationary scattering in two
and three dimensions that is based on a suitable multidimensional
generalization of the transfer matrix. This is a linear operator acting in an
infinite-dimensional function space which we can represent as a $2\times 2$
matrix with operator entries. This operator encodes the information about the
scattering properties of the potential and enjoys an analog of the composition
property of its one-dimensional ancestor. Our results improve an earlier
attempt in this direction [Phys. Rev. A 93, 042707 (2016)] by elucidating the
role of the evanescent waves. In particular, we show that a proper formulation
of this approach requires the introduction of a pair of intertwined transfer
matrices each related to the time-evolution operator for an effective
non-unitary quantum system. We study the application of our findings in the
treatment of the scattering problem for delta-function potentials in two and
three dimensions and clarify its implicit regularization property which
circumvents the singular terms appearing in the standard treatments of these
potentials. We also discuss the utility of our approach in characterizing
invisible (scattering-free) potentials and potentials for which the first Born
approximation provides the exact expression for the scattering amplitude.
