We explore two-dimensional scattering off a potential that is invariant with respect to linear symmetry transformations such as rotations, reflections, and/or coordinate exchange. The common ansatz of an incoming wave, in general, does not respect the symmetries of the potential and, therefore, the solution of the corresponding Schr\"odinger equation is not an eigenstate of the operator representing the respective symmetry transform. This renders it difficult to conceptually account for the imprint of the potentials' symmetry on the scattered wave field, unless the limit of an infinite distance $r\ensuremath{\rightarrow}\ensuremath{\infty}$ is taken. In the latter case, the symmetries of the potential are expressed as conditions on the scattering amplitude. In the present work, we employ the recently derived, symmetry-induced, nonlocal and divergence-free currents, which are constructed from the wave field at a point
r and its symmetry-related image
r, to derive general properties of the scattering solution and the associated scattering matrix. These properties originate from the requirement of a vanishing divergence for the nonlocal currents, which is in one-to-one correspondence with the presence of a symmetry in the scattering potential. In practice, they are expressed as conditions on the coefficients of the wave-field expansion with respect to the angular momentum basis in two dimensions. This, in turn, constrains the form of the related
S-matrix eigenvectors. The obtained properties are also valid at finite
r and can be used as a tool to check upon and improve the efficiency of numerical approaches to the quantum scattering problem of interest.
We study the partial-wave description of scattering of a particle from a one-dimensional potential in quantum mechanics. There are two partial waves, with even and odd parity. They are decoupled if the potential V(x) is symmetric, i.e., V(x)=V(−x). If V(x)≠V(−x), however, the two partial waves are coupled. This coupled-channel problem can be handled in terms of two eigenphases and a mixing parameter. This method enables us to shed new light on the transmission and reflection probabilities at threshold.