Absolutely continuous spectrum of perturbed Stark operators
Absolutely continuous spectrum of perturbed Stark operators
复制标题
扰动斯塔克算子的绝对连续谱
DOI：
10.1090/s0002-9947-99-02450-2
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发表时间：
1997
影响因子：
1.3
通讯作者：
A. Kiselev
中科院分区：
文献类型：
--
作者：
A. Kiselev
本文证明了具有衰减扰动或满足一定光滑性假设扰动的Stark算子的绝对连续谱的稳定性的新结果。我们证明了Stark算子的绝对连续谱是稳定的，如果扰动势以（1 + x）~ 3 -e的速率衰减，或者如果它是从H_6 lder空间C_1（R）连续可微的导数，且任何ae > 0. 0.引言在本文中，我们研究了绝对连续的spect的稳定性;rum的一维斯塔克运营商在各类扰动。斯塔克-薛定谔算符描述了带电粒子在恒定电场中的行为。绝对连续的谱是一个事实的表现，即由算子描述的粒子以相当快的速度传播到无穷远（参见，例如[2]，[12]）。因此，有趣的是描述类的扰动，保持绝对连续谱的斯塔克运营商。在第一部分中，我们研究了Stark算子在衰减势作用下的扰动。这一部分的灵感来自Naboko和Pushnitski最近的工作[14]。我们所证明的一般图象与自由薛定谔算子的微扰情形非常相似。然而，根据物理直觉，绝对连续谱在比自由情况更强的扰动下是稳定的。如果在自由情况下，保持自由算符纯绝对连续谱的短程势由条件lq（x）l < C（1 + xK）-2-E给出（在幂标上），则在Stark算符情况下，相应的条件为lq（x）l < C（1 + xK）-2-E.如果c在上述范围内允许为零，则嵌入特征值可能在两种情况下都出现（参见例如[14]，[151]）。此外，在这两种情况下，如果我们允许势通过任意函数增长到无穷大而衰减得更慢，则可能出现非常丰富的奇异谱，例如密集的本征值集，-（参见[13]自由情况和[141]斯塔克情况，以获得这些结果的精确公式和证明）。本文的第一部分进一步进行了类比，表明斯塔克算子的绝对连续谱在满足lq（x）l < C（1 + Jxl）-3-E的扰动下保持不变，特别是即使在本征值密集的区域中也是如此;因此在这种情况下，这些本征值是真正嵌入的。自由情况下的类似结果在[9]中得到了证明，1997年4月14日由编辑收到。1991年数学学科分类。初级34 L40，81 Q10。?) 1999 American Mathematical Society 243本内容于2016年9月10日星期六04：57：39 UTC从 207.46.13.156下载所有使用受 http://about.jstor.org/terms约束244亚历山大KISELEV [10].本文的主要证明策略与文献[9]和[10]类似：先研究广义本征函数的渐近性，然后应用Gilbert-Pearson理论[7]导出谱结果。虽然我们在处理Stark算子时引入的主要新工具与自由情况相同，即a.e.收敛的傅里叶型积分算子，有一些重大的差异。首先，光谱参数进入我们以不同方式研究的最终方程，这使得分析更加复杂。其次，我们采用不同的方法来分析渐近性。代替Harris-Lutz渐近方法，我们研究了适当的Prilfer变换变量，简化了整体考虑。在第二部分的工作，我们讨论扰动的潜力有一些额外的光滑性，但没有衰减。事实证明，对于斯塔克算子的衰减或额外的光滑性的潜力对光谱性质的影响是有点类似。很久以前就知道，如果一个势扰动Stark算子有两个有界导数，则谱保持纯绝对连续（实际上，导数的某些增长也是允许的，详见第2节或Walter [21]的原始结果）。我们注意到，在[4]中，通过应用不同类型的技巧（用Brackre方法代替研究解的渐近性），也得到了类似于Walter关于绝对连续谱保持性的结果。另一方面，如果扰动势是6个函数在R上整数点上的导数序列，且具有一定的耦合，则谱可能转向纯点[3]，[5]，[1]。从某种意义上说，6'相互作用是所有可用的一维薛定谔算子自然扰动中最奇异和最不可微的。因此，我们在平滑度尺度的非常相反的两侧具有非常不同的谱特性。这项工作关闭了部分差距。我们改进了Walter [21]关于保持绝对连续谱所需的最小光滑性的著名结果，并证明了实际上一阶导数的存在性和最小光滑性足以暗示谱的绝对连续性.在提交这篇论文后，作者了解了J. Sahbani [17]的工作，其中特别证明了与本工作的定理2.1有关的结果。Sahbani的结果比定理2.1稍强：势V '（x）的导数必须是有界的和Dini连续的，以保持绝对连续谱。此外，他还指出，在这种情况下，嵌入奇异谱可能只由孤立的本征值组成。文[17]所用的方法是共轭算子方法的推广。1.考虑一个自伴算子Hq，定义为L2（-oo，oo）上的微分表达式Hqu =-u”-xu + q（x）u.让我们介绍一些符号。对于函数f C L2，我们用bf表示其傅里叶变换：
We prove new results on the stability of the absolutely cont;inuous spectrum for perturbed Stark operators with decaying or satisfying certain smoothness assumption perturbation. We show that the absolutely continuous spectrum of the Stark operator is stable if the perturbing potential decays at the rate (1 + x) 3 -e or if it is continuously differentiable with derivative from the H6lder space C, (R), with any ae > 0. 0. INTRODUCTION In this paper, we study the stability of the absolutely continuous spect;rum of one-dimensional Stark operators under various classes of perturbations. Stark Schr6dinger operators describe behavior of the charged particle in the constant electric field. The absolutely continuous spectrum is a manifestation of the fact that the particle described by the operator propagates to infinity at a rather fast rate (see, e.g. [2], [12]). It is therefore interesting to describe the classes of perturbations which preserve the absolutely continuous spectrum of the Stark operators. In the first part of this work, we study perturbations of Stark operators by decaying potentials. This part is inspired by the recent work of Naboko and Pushnitski [14]. The general picture that we prove is very similar to the case of perturbatiorns of free Schr6dinger operators [9]. In accordance with physical intuition, however, the absolutely continuous spectrum is stable under stronger perturbations than in the free case. If in the free case the short range potentials preserving purely absolutely continuous spectrum of the free operator are given by condition (on the power scale) lq(x)l < C(1 + xKl)'E, in the Stark operator case the corresponding condition reads lq(x)l < C(1 + IXD)-2-E. If c is allowed to be zero in the above bounds, imbedded eigenvalues may occur in both cases (see, e.g. [14], [151). Moreover, in both cases if we allow potential to decay slower by an arbitrary function growing to infinity, very rich singular spectrum, such as a dense set of eigenvalues, -may occur (see [13] for the free case and [141 for the Stark case for precise formulation and proofs of these results). The first part of this work draws the parallel further, showing that the absolutely continuous spectrum of Stark operators is preserved under perturbations satisfying lq(x)l < C(1 + Jxl)-3-E, in particular even in the regimes where a dense set of eigenvalues occurs; hence in such cases these eigenvalues are genuinely imbedded. Similar results for the free case were proven in [9], Received by the editors April 14, 1997. 1991 Mathematics Subject Classification. Primary 34L40, 81Q10. ?)1999 American Mathematical Society 243 This content downloaded from 207.46.13.156 on Sat, 10 Sep 2016 04:57:39 UTC All use subject to http://about.jstor.org/terms 244 ALEXANDER KISELEV [10]. Our main strategy of the proof here is similar to that in [9] and [10]: we study the asymptotics of the generalized eigenfunctions and then apply Gilbert-Pearson theory [7] to derive spectral consequences. While the main new tool we introduce in our treatment of Stark operators is the same as in the free case, namely the a.e. convergence of the Fourier-type integral operators, there are some major differences. First of all, the spectral parameter enters the final equations that we study in a different way and this makes analysis more complicated. Secondly, we employ a different method to analyze the asymptotics. Instead of Harris-Lutz asymptotic method we study appropriate Prilfer transform variables, simplifying the overall consideration. In the second part of the work we discuss perturbations by potentials having some additional smoothness properties, but without decay. It turns out that for Stark operators the effects of decay or of additional smoothness of potential on the spectral properties are somewhat similar. It was known for a long time that if a potential perturbing Stark operator has two bounded derivatives the spectrum remains purely absolutely continuous (actually, certain growth of derivatives is also allowed, see Section 2 for details or Walter [21] for the original result). We note that the results similar to Walter's on the preservation of absolutely continuous spectrum were also obtained in [4] by applying different types of technique (Mourre method instead of studying asymptotics of solutions). On the other hand, if the perturbing potential is a sequence of derivatives of 6 functions in integer points on R with certain couplings, the spectrum may turn pure point [3], [5], [1]. In some sense, the 6' interaction is the most singular and least "differentiable" among all available natural perturbations of one-dimensional Schrbdinger operators [11]. Hence we have very different spectral properties on the very opposite sides of the smoothness scale. This work closes part of the gap. We improve the well-known results of Walter [21] concerning the minimal smoothness required for the preservation of the absolutely continuous spectrum and show that in fact existence and minimal smoothness of the first derivative is sufficient to imply absolute continuity of the spectrum. After submitting this paper, the author learned about the work of J. Sahbani [17], where, in particular, the results related to Theorem 2.1 of the present work are proven. Sahbani's results are slightly stronger than Theorem 2.1: the derivative of potential V'(x) is required to be bounded and Dini continuous in order for the absolutely continuous spectrum to be preserved. In addition, he shows that the imbedded singular spectrum in this case may only consist of isolated eigenvalues. The approach employed in [17] is an extension of conjugate operator method. 1. DECAYING PERTURBATIONS Consider a self-adjoint operator Hq defined by the differential expression Hqu =-u" -xu + q(x)u on the L2(-oo, oo). Let us introduce some notation. For the function f C L2 we denote by bf its Fourier transform: