薛定谔积分方程的分布理论
Distribution theory for Schrödinger's integral equation
摘要：
在量子力学中，大部分关于点相互作用的文献都集中在薛定谔方程的微分形式上。相反，本文研究薛定谔方程的积分形式。虽然已知这两种形式对于光滑势是等价的，但对于分布势却不是这样。在这里，我们假设的潜力是由一个分布上定义的不连续的测试功能的空间。首先，通过使用薛定谔的积分方程，我们证实了一个开创性的结果，库拉索夫，这是最初获得的薛定谔的微分方程的背景下。这暗示了两种形式的方程之间可能存在更深层次的联系。我们还勾画了一个推广的Kurasov的结果超曲面。其次，我们推导出一个新的封闭形式的解决方案，薛定谔的积分方程与一个δ素数的潜力。这一潜力引起了相当大的关注，包括一些争议。有趣的是，导出的传播子满足先前使用薛定谔微分方程导出的边界条件。第三，我们推导出边界条件的“超奇异”的潜力，由高阶导数的δ潜力。这些边界条件不能纳入自伴扩张的正规框架。我们表明，边界条件依赖于能量的解决方案，并保持概率。本文由此证实了几个开创性的结果，并导出了一些新的结果。总之，薛定谔积分方程是研究量子力学中奇异相互作用的有效工具。
英文摘要：
Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schrödinger's equation. This paper, in contrast, investigates the integral form of Schrödinger's equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schrödinger's integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schrödinger's differential equation. This hints at a possible deeper connection between both forms of the equation. We also sketch a generalisation of Kurasov's result to hypersurfaces. Second, we derive a new closed-form solution to Schrödinger's integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schrödinger's differential equation. Third, we derive boundary conditions for `super-singular' potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution, and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schrödinger's integral equation is viable tool for studying singular interactions in quantum mechanics.
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