WKB and Spectral Analysis¶of One-Dimensional Schrödinger Operators¶with Slowly Varying Potentials
WKB and Spectral Analysis¶of One-Dimensional Schrödinger Operators¶with Slowly Varying Potentials
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一维薛定谔算子的 WKB 和谱分析（具有缓慢变化的势）
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
A. Kiselev
中科院分区：
文献类型：
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作者：
Michael Christ;A. Kiselev
摘要：考虑具有适当边界条件的直线或半直线L2上的Schrödinger算子。如果势趋于零并且是有限项和，其中每项在L1+Lp中对某指数p<2有一定阶导数，则绝对连续谱等于1+的基本支持。几乎每一个广义特征函数都是有界的，并且在无穷远处满足一定的wkb型渐近性。此外，如果这些导数在权值|x|γ与γ >0下属于Lp，则谱测度的奇异分量的Hausdorff维数严格小于1。
Abstract: Consider a Schrödinger operator on L2 of the line, or of a half line with appropriate boundary conditions. If the potential tends to zero and is a finite sum of terms, each of which has a derivative of some order in L1+Lp for some exponent p<2, then an essential support of the the absolutely continuous spectrum equals ℝ+. Almost every generalized eigenfunction is bounded, and satisfies certain WKB-type asymptotics at infinity. If moreover these derivatives belong to Lp with respect to a weight |x|γ with γ >0, then the Hausdorff dimension of the singular component of the spectral measure is strictly less than one.