多瞬子和精确结果II：特殊情况、高阶效应和数值计算
Multi-Instantons and Exact Results II: Specific Cases, Higher-Order Effects, and Numerical Calculations
摘要：
在量子力学中瞬子处理的第二部分中，重点是与一些具有简并极小值的量子力学势有关的具体计算。我们使用论文第一部分中介绍的形式计算了主导多瞬子对配分函数的贡献。D. Jentschura，e-print quant-ph/0501136]。考虑以下势：（i）具有简并极小值的非对称势，（ii）周期余弦势，（iii）具有径向对称性的非简谐振子，以及（iv）与福克-普朗克方程类似的特定势。后一种势有一个特殊的性质，即基态能量的微扰级数在所有阶次都为零，因此形式上是收敛的（然而，基态能量是非零的正的）。对于势（ii），（iii）和（iv），我们计算微扰B函数以及瞬子A函数到四阶g。我们还考虑了双势阱的详细情况，并提出了一些高阶解析以及数值计算，以明确验证相关的acturtures到三瞬子的顺序。类似于这里概述的策略可能会导致我们目前的理解更加有限的问题的新的解决方案。
英文摘要：
In this second part of the treatment of instantons in quantum mechanics, the focus is on specific calculations related to a number of quantum mechanical potentials with degenerate minima. We calculate the leading multi-instanton constributions to the partition function, using the formalism introduced in the first part of the treatise [J. Zinn-Justin and U. D. Jentschura, e-print quant-ph/0501136]. The following potentials are considered: (i) asymmetric potentials with degenerate minima, (ii) the periodic cosine potential, (iii) anharmonic oscillators with radial symmetry, and (iv) a specific potential which bears an analogy with the Fokker-Planck equation. The latter potential has the peculiar property that the perturbation series for the ground-state energy vanishes to all orders and is thus formally convergent (the ground-state energy, however, is nonzero and positive). For the potentials (ii), (iii), and (iv), we calculate the perturbative B-function as well as the instanton A-function to fourth order in g. We also consider the double-well potential in detail, and present some higher-order analytic as well as numerical calculations to verify explicitly the related conjectures up to the order of three instantons. Strategies analogous to those outlined here could result in new conjectures for problems where our present understanding is more limited.
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