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. *****

我正在研究离散薛定谔运营商的无限图形，随机和确定性的潜力。随机势的这些算子的谱理论是数学物理学中的一个大领域，包括重要的未解决的问题，如关于无序介质中粒子运动的扩展态猜想。我的工作介绍了动力系统的想法，以显示存在的绝对连续谱（离域）和点谱（本地化）在各种模型。我想继续发展这些想法，例如绘制出树上横向周期势类的相图。确定性薛定谔算子也有一些有趣的开放性问题。** 共振是预解式在第二片上的亚纯延拓中的极点。我正在研究共振态的时间演化。我也有兴趣在优化问题涉及谐振，以及一些问题所产生的散射理论在波导。*****