Abstract

To get around the difficulty that the singular nature of the potential function V(x) = V₀/|x|^α introduces into physics, the regularization methods are used. However, they affect the singular nature of the problem, and so we discuss here how quantum tunneling behaves if the original singular nature of the Schrodinger equation remains unperturbed. For this purpose, I am starting from the precondition that the singular terms are mutually compensated in the current probability density and the current can be considered continuous. As a result, it is obtained that the mild-singular potential (with 0 < α < 1) has finite but unusual tunnel transparency, in particular, a non-zero value at zero energy of the incident particle. The transparency of a Coulomb potential well oscillates infinitely at zero energy, and only the strongly singular potential (with α > 1) repeats the property of the regularized prototypes: to be impenetrable.

中文翻译：

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关于奇异势的量子隧道

摘要

为了解决势函数的奇异性质V ( x ) = V ₀ /|的困难 × | ^α引入物理学，使用正则化方法。然而，它们会影响问题的奇异性质，因此我们在这里讨论如果薛定谔方程的原始奇异性质保持不变，量子隧穿的表现如何。为此，我从奇异项在电流概率密度中相互补偿并且电流可以被认为是连续的前提出发。结果，获得了温和奇异势（0 < α < 1）具有有限但不寻常的隧道透明度，特别是在入射粒子的零能量处具有非零值。库仑势阱的透明度在零能量下无限振荡，只有强奇异势（α > 1）重复正则化原型的属性：不可穿透。