极球坐标系中拉普拉斯算子的奇异性及其对坐标原点处径向波函数的某些影响
Singular Behavior of the Laplace Operator in Polar Spherical Coordinates and Some of Its Consequences for the Radial Wave Function at the Origin of Coordinates
摘要：
研究了球坐标系中Laplace算子的奇异性。结果表明，在薛定谔方程向约化径向波函数过渡的过程中，出现了一个在整个物理学史和数学史上未曾被注意到的由狄拉克δ函数组成的附加项。避免这种贡献减少径向方程的可能性进行了讨论。证明了达到这一目的的充分必要条件是波函数在原点处的下降足够快。这个结果不依赖于势的性质：它是正则的还是奇异的。这一观察的各种表现和后果也被认为是。在我们的方法的基石是自然的要求，径向方程的解决方案，在同一时间必须服从完整的方程。
英文摘要：
Singular behavior of the Laplace operator in spherical coordinates is investigated. It is shown that in course of transition to the reduced radial wave function in the Schrodinger equation there appears additional term consisting the Dirac delta function, which was unnoted during the full history of physics and mathematics. The possibility of avoiding this contribution from the reduced radial equation is discussed. It is demonstrated that for this aim the necessary and sufficient condition is requirement the fast enough falling of the wave function at the origin. The result does not depend on character of potential:is it regular or singular. The various manifestations and consequences of this observation are considered as well. The cornerstone in our approach is the natural requirement that the solution of the radial equation at the same time must obey to the full equation.
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