Feynman-Kac Functional and the Schrödinger Equation
Feynman-Kac Functional and the Schrödinger Equation
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DOI：
10.1007/978-1-4612-3938-3_1
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发表时间：
1981
期刊：
影响因子：
--
通讯作者：
K. Chung;K. M. Rao
中科院分区：
文献类型：
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作者：
K. Chung;K. M. Rao
2 KL Chung和Km Rao正连续边界函数f，只要这个量是有限的(至少是D中的一点x)，就可以用下面L的(2)中给出的显式公式得到解。因此，费曼-卡克公式为问题提供了自然的格林算子。对于具有有限测度的区域，其结果是最好的，因为它包含了Dirichlet问题的概率方法的经典解。其他结果对于任意域也是有效的，并且似乎其中一些结果在这里被证明的条件比通常在非概率处理中给出的条件要宽松一些。例如，除了狄利克雷问题意义上的正则性外，不假定边界的光滑性条件，基本结果在没有这种正则性的情况下也成立。当然，薛定谔方程是椭圆型偏微分方程组的一种情况，关于它的文献很多，但我们没有求助于后一种理论。这些方法之间的比较应该被证明是值得的，并将在另一份出版物中讨论。众所周知，薛定谔方程与拉普拉斯方程的本质区别在于，区域大小的条件是保证解的唯一性所必需的。在我们的上下文中，一开始就很明显这一点的关键是量ud(X)
2 KL CHUNG and KM RAO positive continuous boundary function f, a solution is obtained in the explicit formula given in (2) of § l below, provided that this quantity is finite (at least at one point x in D). Thus the Feynman-Kac formula supplies the natural Green's operator for the problem. For a domain with finite measure, the result is the best possible as it includes the already classical solution of the Dirichlet problem by probability methods. Other results are valid for an arbitrary domain and it seems that some of them are proved here under less stringent conditions than usually given in non-probabilistic treatments. For instance, no condition on the smoothness of the boundary is assumed beyond that of regularity in the sense of the Dirichlet problem, and the basic results hold without this regularity. Of course, the Schrodinger equation is a case of elliptic partial differential equations on which there exists a huge literature, but we make no recourse to the latter theory. Comparisons between the methods should prove worthwhile and will be discussed in a separate publication. It is well known that the Schrodinger equation differs essentially from the Laplace equation in that a condition on the size of the domain is necessary to guarantee the uniqueness of solution. In our context it is evident at the outset that the key to this is the quantity uD (x)