Spectral theory and resonance for Schrodinger operators

薛定谔算子的谱理论和共振

基本信息

批准号：
92997-2010
负责人：
Froese, Richard
金额：
$ 1.46万
依托单位：
University of British Columbia
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2015
资助国家：
加拿大
起止时间：
2015-01-01 至 2016-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569022
关键词：
Spectral theory resonance Schrodinger operators

项目摘要

My research is about spectral and scattering theory of Schrödinger Operators, and related topics in PDE and Geometry. In recent years I have focused on proving the existence of absolutely continuous spectrum for discrete Schrödinger operators whose behavior at infinity is irregular. The primary examples of such operators are Schrödinger operators with random potentials. One motivation for this work is the extended states conjecture, a big unsolved problem in the field of random Schrödinger operators. This conjecture asserts for the Anderson model on the integer lattice in dimensions three or higher there is some absolutely continuous spectrum at low disorder. The physical meaning of this conjecture is that a disordered solid should have some conducting energies provided the disorder is sufficiently low. The only situation where this conjecture has been proved is when the integer lattice is replaced with a tree. The original proof is due to Abel Klein. In my work with Hasler and Spitzer, I found a new proof of this result, and am proposing to use ideas from this proof to study a series of problems that share some of the difficulties of working on the integer lattice. In particular, I am now in a position to handle some graphs with loops of unbounded length. I also plan to combine our method with operator theory techniques to study operators on the lattice with slowly decaying random potentials. I am interested in some problems involving resonances, or scattering poles. I am proposing to study a problem involving the positions of resonances and one involving the long time behavior of resonant states.

我的研究是关于薛定谔算子的谱和散射理论，以及偏微分方程和几何中的相关主题。近年来，我专注于证明离散薛定谔算子的绝对连续谱的存在性，这些算子在无穷远处的行为是不规则的。算子是具有随机势的薛定谔算子。这项工作的动机之一是扩展态猜想，这是随机薛定谔算子领域中一个未解决的大问题，该猜想断言三维或更高维度的整数晶格上的安德森模型在低无序度下存在一些绝对连续的谱。这个猜想的意义是，如果无序度足够低，无序固体应该具有一些传导能量，这个猜想被证明的唯一情况是当整数晶格被替换时。最初的证明来自于 Abel Klein，在我与 Hasler 和 Spitzer 的合作中，我发现了这个结果的新证明，并提议使用这个证明中的想法来研究一系列具有一些困难的问题。特别是，我现在能够处理一些具有无界长度循环的图，我还计划将我们的方法与算子理论技术结合起来，研究具有缓慢衰减随机势的格子上的算子。我对共振或散射极的一些问题感兴趣，我建议研究一个涉及共振位置和共振态的长期行为的问题。