Imbedded singular continuous spectrum for Schrodinger operators
Imbedded singular continuous spectrum for Schrodinger operators
复制标题
薛定谔算子的嵌入式奇异连续谱
DOI：
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发表时间：
2001
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影响因子：
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通讯作者：
A. Kiselev
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作者：
A. Kiselev
(1.1)Hv = -^ + V（x）和零点处的一些自伴边界条件。算符（1.1）描述了电场V（x）中的带电粒子，如电子。当V（x）快速衰减时，人们期望Hy的谱和动力学性质保持接近自由算子Ho的谱和动力学性质。回想一下，我们可以用正则的方式将算子（1.1）与谱测度p联系起来（例如，参见[6，2]），它包含了关于量子系统的许多信息。经典的结果在这个方向上，可以追溯到世纪初，是，如果V G L 1，频谱措施的正半轴保持纯粹绝对连续。在一维中，绝对连续的光谱对应于粒子的弹道传播速率。一个自然的问题是什么是衰变的临界速率，在这种临界速率下，Hy的光谱和动力学性质可能发生变化。只要V（x）有任何衰减，基本谱与[0，oo）重合，但谱的质量和动力学可能会改变。1928年，Wigner和von Neumann [27]证明了存在势V（x）满足V（x）< ^j使得Hy有正的嵌入本征值E = 1。这是一个纯粹的量子共振现象，因为V可以选择任意小，并且束缚态起源于势中的长程关联，而不是通常的限制效应。Naboko [17]和Simon [25]提供的结构表明，如果势的衰减任意慢于库仑，则可能发生更剧烈的变化。即，对于任何无穷大的正函数h（x），存在势V，使得|V（x）|Hy具有正特征值的稠密集。由于最近的一些结果[3，21]，已知正半轴上的绝对连续谱对于|V（o;）|< C（1 + x）~ 1/2~e，实际上对于VGL ~ 2 [8]。更精确地说，对于这样的势，谱测度的绝对连续部分/iac给正勒贝格测度的（0，oo）的任何子集以正权重。因此，Naboko
(1.1) Hv = -^ + V{x) and some self-adjoint boundary condition at zero. The operator (1.1) describes a charged particle, such as an electron, in the electric field V(x). When V(x) is decaying quickly, one expects the spectral and dynamical properties of Hy to remain close to those of the free operator Ho. Recall that with the operator (1.1) one can associate in a canonical way a spectral measure p, (see, for example, [6, 2]) which contains much information about the quantum system. The classical result in this direction, going back to the beginning of the century, is that if V G L1, the spectral measure on the positive semi-axis remains purely absolutely continuous. In one dimension, an absolutely continuous spectrum corresponds to the ballistic rate of propagation of the particle. A natural question to ask is what are the critical rates of decay at which some changes in the spectral and dynamical properties of Hy may happen. As far as V(x) has any decay at all, the essential spectrum coincides with [0, oo), but the quality of the spectrum and dynamics may change. In 1928, Wigner and von Neumann [27] showed that there exist potentials V(x) satisfying V(x) < ^jsuch that Hy has positive imbedded eigenvalue E = 1. This is a purely quantum resonance phenomenon, since V can be chosen arbitrarily small and the bound state originates from long-range correlations in the potential rather than the usual confining effect. Naboko [17] and Simon [25] provided constructions which show that much more drastic changes are possible if the potential decays arbitrarily slower than Coulomb. Namely, for any positive function h(x) which grows at infinity, there exist potentials V such that |V(x)| < yjgand Hy has a dense set of positive eigenvalues. Due to some recent results [3, 21], it is known that the absolutely continuous spectrum on the positive semi-axis is preserved for |V(o;)| < C(l + x)~l/2~e and in fact for V G L2 [8]. More precisely, for such potentials the absolutely continuous part of the spectral measure, /iac, gives positive weight to any subset of (0, oo) of positive Lebesgue measure. Therefore, the Naboko