Microlocal propagation near radial points and scattering for symbolic potentials of order zero
Microlocal propagation near radial points and scattering for symbolic potentials of order zero
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径向点附近的微局域传播和零阶符号势的散射
DOI：
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发表时间：
2005
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通讯作者：
A. Vasy
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作者：
Andrew Hassell;R. Melrose;A. Vasy
这推广了我们早先在二维情况下的结果。这类算子包括欧几里得空间上的拉普拉斯算子的扰动，它是由近无穷远的零次齐次位势引起的。几何散射理论的许多特殊结构可以追溯到基本经典系统的径向点的出现。在这种情况下，径向点精确地对应于V到X的约束的临界点V0，并且在附加的假设V0是Morse的情况下，得到了广义特征函数的泛函参数化。高维情况的主要微妙之处在于径向点的额外复杂性。扩展和改进了Guillumin和Schaeffer在这些点附近得到的范式，允许给出H的零空间的微局域描述，除了有限的能量“阈值”集合；在离散的“有效共振”能量集合上出现了额外的复杂情况。证明了V0值小于的每个临界点都是HU=u的解源。由此得到的广义特征空间的描述是一个相当精确的、分布的、渐近完备性的公式。我们还得到了L-2与时间相关的渐近完备性的形式，包括与非极小临界点相关的L-2通道的不存在。Herbst和Skibsted观察到的这种现象可以归因于这样一个事实：与非最小临界点相关的本征函数在无穷远处是“大的”；特别是它们太大而不能位于紧支集函数的预解式R(±i0)的范围内。
this extends our earlier results in the two-dimensional case. Included in this class of operators are perturbations of the Laplacian on Euclidean space by potentials homogeneous of degree zero near infinity. Much of the particular structure of geometric scattering theory can be traced to the occurrence of radial points for the underlying classical system. In this case the radial points correspond precisely to critical points of the restriction, V0, of V to@X and under the additional assumption that V0 is Morse a functional parameterization of the generalized eigenfunctions is obtained. The main subtlety of the higher dimensional case arises from additional complexity of the radial points. A normal form near such points obtained by Guillemin and Schaeffer is extended and refined, allowing a microlocal description of the null space of H to be given for all but a finite set of “threshold” values of the energy; additional complications arise at the discrete set of “effectively resonant” energies. It is shown that each critical point at which the value of V0 is less than is the source of solutions of Hu = u. The resulting description of the generalized eigenspaces is a rather precise, distributional, formulation of asymptotic completeness. We also derive the closely related L 2 and time-dependent forms of asymptotic completeness, including the absence of L 2 channels associated with the nonminimal critical points. This phenomenon, observed by Herbst and Skibsted, can be attributed to the fact that the eigenfunctions associated to the nonminimal critical points are “large” at infinity; in particular they are too large to lie in the range of the resolvent R( ±i0) applied to compactly supported functions.