Microlocal WKB Expansions
Microlocal WKB Expansions
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微局部 WKB 扩展
DOI：
10.1006/jfan.1999.3460
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发表时间：
1999
期刊：
影响因子：
--
通讯作者：
V. Sordoni
中科院分区：
文献类型：
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作者：
André Martinez;V. Sordoni
在微局部隧穿的基础上，研究了符号在R2 n的某点（x 0，x 0）处具有非退化极小值的半经典伪微分算子的本征函数的微局部WKB展开的存在性.更精确地说，在解析性的假设下，我们证明了当半经典参数h趋于0时，这样的算子的第一本征函数的FBI（或Bargman）变换在（x 0，<$0）附近允许形式为e−Φ（x，<$）/h ∑j <$0 hjaj（x，<$）的展开。这里Φ和aj是光滑（复）函数，并且其中展开有效的域可以根据变形性质来描述。我们的结果可以应用于例如磁薛定谔算子与电位承认一个非简并阱。
In relation with microlocal tunneling, we investigate the existence of micro-local WKB expansions for the eigenfunctions of semiclassical pseudodifferential operators whose symbol admits a non degenerate minimum at some point (x0, ξ0) of R2n. More precisely, under assumptions of analyticity, we prove that the FBI (or Bargman) transform of the first eigenfunction of such an operator admits near (x0, ξ0) an expansion of the form e−Φ(x, ξ)/h ∑j⩾0 hjaj(x, ξ) as the semiclassical parameter h tends to 0. Here Φ and the aj's are smooth (complex) functions, and the domain in which the expansion is valid can be described in terms of deformation properties. Our result can be applied e.g. to magnetic Schrodinger operators with electric potential admitting a non degenerate well.