WKB Solutions Near an Unstable Equilibrium and Applications
WKB Solutions Near an Unstable Equilibrium and Applications
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接近不稳定平衡的 WKB 解及其应用
DOI：
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发表时间：
2014
期刊：
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通讯作者：
Maher Zerzeri
中科院分区：
文献类型：
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作者：
J. Bony;S. Fujiié;T. Ramond;Maher Zerzeri
在这篇综述中，我们提出了一些精确的结果，关于光谱和散射问题的薛定谔方程在半经典制度，我们已经得到了一系列的文件[BFRZ 1，BFRZ 2，BFRZ 3，ABR]。正如人们可以预料的那样，潜在的经典系统的属性在这个制度中起着至关重要的作用，我们已经研究了存在一个双曲不动点的情况下，相关的哈密顿流。例如，当电势具有局部最大值时，会发生这种情况。很多都被编码在我们称之为微局部柯西问题的不动点上，我们在这里描述一些细节。用物理学的语言来说，这个微局部柯西问题的研究就是双曲不动点上的n维隧道效应。
In this survey, we present some precise results concerning spectral and scattering problems for the Schrödinger equation in the semiclassical regime, that we have obtained in a series of papers [BFRZ1, BFRZ2, BFRZ3, ABR]. As one can expect, properties of the underlying classical system play a crucial role in this regime, and we have studied the case where there exists one hyperbolic fixed point for the associated Hamiltonian flow. This occurs for example when the potential has a local maximum. A lot is encoded in what we call a microlocal Cauchy problem at the fixed point, that we describe here with some details. In a physicist language, the study of this microlocal Cauchy problem is that of the n-dimensional tunneling effect at the hyperbolic fixed point.