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 Schrödinger 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 $r\to \infty$ 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 $\mathbf{r}$ and its symmetry-related image $\overline{\mathbf{r}}$, 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 $\mathbf{S}$-matrix eigenvectors. The obtained properties are also valid at finite $\mathbf{r}$ and can be used as a tool to check upon and improve the efficiency of numerical approaches to the quantum scattering problem of interest.

中文翻译：

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二维量子散射中的对称诱导的非局部无散度电流

我们探索二维散射的势能，该势能相对于线性对称变换（例如旋转，反射和/或坐标交换）是不变的。通常，入射波的公共ansatz不考虑电势的对称性，因此，对应的Schrödinger方程的解不是表示各个对称变换的算子的本征态。这使得很难从概念上解释电势对称性在散射波场上的影响，除非无限距离的限制$\mathrm{[R}\to \infty$被采取。在后一种情况下，电势的对称性表示为散射幅度的条件。在当前的工作中，我们采用了最近导出的，对称感应的，非局部的和无散度的电流，这些电流是根据某一点处的波场构造的$\mathbf{[R}$ 及其对称性图像 $\overline{\mathbf{[R}}$，以推导散射溶液和相关散射矩阵的一般属性。这些特性源于对非局部电流的消失散度的要求，该散度与散射电势中存在对称性一一对应。实际上，它们以二维形式表示为相对于角动量为基础的波场扩展系数的条件。反过来，这限制了相关$\mathbf{小号}$-矩阵特征向量。所获得的属性在有限范围内也有效$\mathbf{[R}$ 可以用作检查和改善感兴趣的量子散射问题的数值方法的效率的工具。