We offer a consistent dynamical formulation of stationary scattering in two and three dimensions (2D and 3D) 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. 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 nonunitary quantum system. We study the application of our findings in the treatment of the scattering problem for $\delta$-function potentials in 2D and 3D 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.

中文翻译：

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二维和三维平稳散射的基本传递矩阵和动力学公式

我们提供了二维和三维（2D 和 3D）静态散射的一致动力学公式，该公式基于传递矩阵的适当多维泛化。这是一个作用于无限维函数空间的线性算子，我们可以将其表示为$2\times 2$带有运算符条目的矩阵。该算子对有关势的散射特性的信息进行编码，并享有与其一维祖先的组成特性类似的特性。我们的结果改进了在这个方向上的早期尝试 [ Phys. Rev. A 93 , 042707 (2016)] 通过阐明渐逝波的作用。我们表明，这种方法的正确表述需要引入一对相互交织的传递矩阵，每个矩阵都与有效非幺正量子系统的时间演化算子相关。我们研究了我们的发现在处理散射问题中的应用$\delta$-函数势在 2D 和 3D 中，并阐明其隐式正则化特性，该特性规避了出现在这些势的标准处理中的奇异项。我们还讨论了我们的方法在表征不可见（无散射）势和势的效用，第一波恩近似提供了散射幅度的精确表达式。