On the Structure of Eigenfunctions Corresponding to Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators
On the Structure of Eigenfunctions Corresponding to Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators
复制标题
局部扰动周期图算子嵌入特征值对应的特征函数结构
DOI：
10.1007/s00220-006-0105-2
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发表时间：
2005
影响因子：
2.4
通讯作者：
B. Vainberg
中科院分区：
文献类型：
--
作者：
P. Kuchment;B. Vainberg
本文主要探讨以下问题。考虑图（量子图）G上的周期自伴差分（微分）算子具有整数格的余紧自由作用，
$$\mathbb{Z}^{n}$$。众所周知，算子的局部扰动可能会将本征值嵌入连续谱中（这是二阶周期椭圆算子不常见的特征）。在这类例子的所有已知构造中，相应的本征函数都是紧支集的。人们不禁要问，情况是否总是如此。本文对此作了肯定的回答。更令人惊讶的是，人们可以估计本征模必须在离微扰不远的地方局域化（在微扰支持的邻域中，邻域的宽度仅取决于未微扰算子）。这一结果的有效性需要周期算子的费米（弗洛凯）曲面不可约的条件，这在某些情况下是已知的，并且预期周期薛定谔算子会满足。
AbstractThe article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co- compact free action of the integer lattice
$$\mathbb{Z}^{n}$$. It is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum (a feature uncommon for periodic elliptic operators of second order). In all known constructions of such examples, the corresponding eigenfunction is compactly supported. One wonders whether this must always be the case. The paper answers this question affirmatively. What is more surprising, one can estimate that the eigenmode must be localized not far away from the perturbation (in a neighborhood of the perturbation’s support, the width of the neighborhood dependent upon the unperturbed operator only). The validity of this result requires the condition of irreducibility of the Fermi (Floquet) surface of the periodic operator, which is known in some cases and is expected to be satisfied for periodic Schrödinger operators.