Spectral theory and resonances for Schrödinger operators

薛定谔算子的谱理论和共振

• 批准号: RGPIN-2016-03748
• 负责人: Froese, Richard
• 金额: $ 1.09万
• 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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=651221
• 关键词: Spectral; theory; resonances; Schr; dinger

I am studying discrete Schrödinger operators on infinite graphs, for both random and deterministic potentials. The spectral theory of these operators for random potentials is a large field in mathematical physics and includes important unsolved problems such as the extended states conjecture about the motion of a particle in a disordered medium. My work has introduced dynamical systems ideas to show both existence of absolutely continuous spectrum (delocalization) and point spectrum (localization) in various models. I would like to continue to develop these ideas, for example to map out the phase diagram for the class of transversally periodic potentials on trees. There are also interesting open questions for deterministic Schrödinger operators. ******Resonances are poles in the meromorphic continuation of the resolvent onto a second sheet. I am working on estimates of the time evolution of resonant states. I am also interested in optimization problems involving resonances, as well as some questions arising from the scattering theory in waveguides. *****

我正在研究无限图上的离散Schrödinger算子，用于随机和确定性势。这些随机势算子的谱理论是数学物理中的一个大领域，包括一些重要的尚未解决的问题，如无序介质中粒子运动的扩展状态猜想。我的工作引入了动力系统的思想来展示各种模型中绝对连续谱（离域）和点谱（局部化）的存在。我想继续发展这些想法，例如，画出树上的一类横向周期势的相图。对于确定性Schrödinger操作符也有一些有趣的开放问题。******共振是溶剂亚纯延拓到第二薄片上的极点。我正在研究共振态的时间演化的估计。我也对涉及共振的优化问题，以及波导中散射理论产生的一些问题感兴趣。＊＊＊＊＊