Vestigial effects of singular potentials in diffusion theory and quantum mechanics
Vestigial effects of singular potentials in diffusion theory and quantum mechanics
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扩散理论和量子力学中奇异势的残留效应
DOI：
10.1063/1.522632
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发表时间：
1975
影响因子：
1.3
通讯作者：
L. Shepp
中科院分区：
文献类型：
--
作者：
H. Ezawa;J. Klauder;L. Shepp
本文研究了Feynman-Kac积分中的形式为λV（x）=λ <$x−c <$−α，λ <$0的排斥奇异势作为α的函数。对于α <$2，这样的势完全抑制了到达奇点的路径对积分的贡献，因此，即使在系数λ↓0之后，势的某些残余仍然存在。对于2 <$α <$1，通过在奇点处适当的反项（在精神上类似于场论中的重整化反项）进行仔细定义，可以导致完全消除λ↓0势的影响。当α<1时，当λ↓0时，不存在电势的剩余效应。为了证明这些结果，我们依靠随机过程理论，特别是当地时间和随机微分方程。对Feynman-Kac积分建立的这些结果与微分方程理论中已知的结果一致。事实上，各种各样的残留效应可以从合适的选择的反条款中产生，这些对应于一个自然的过程。
Repulsive singular potentials of the form λV (x) =λ‖x−c‖−α, λ≳0, in the Feynman−Kac integral are studied as a function of α. For α≳2 such potentials completely suppress the contribution to the integral from paths that reach the singularity, and thus, unavoidably, certain vestiges of the potential remain even after the coefficient λ↓0. For 2⩾α⩾1 careful definition by means of suitable counterterms at the point of singularity (similar in spirit to renormalization counter terms in field theory) can lead to complete elimination of effects of the potential as λ↓0. For α<1 no residual effects of the potential exist as λ↓0. In order to prove these results we rely on the theory of stochastic processes using, in particular, local time and stochastic differential equations. These results established for the Feynman−Kac integral conform with those known in the theory of differential equations. In fact, a variety of vestigial effects can arise from suitable choices of counter terms, and these correspond in a natural ...
DOI：
10.1002/9781118231296.ch8
发表时间：
2018-11
期刊：
Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics
影响因子：
--
作者：
Dr. Gergely Záruba
通讯作者：
Dr. Gergely Záruba