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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=614599
• 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薛定谔算子，无论是随机势还是确定性势，这些随机势算子的谱理论是数学物理中的一个大领域，包括重要的未解决问题，例如关于粒子运动的扩展态猜想。我的工作引入了动力系统思想，以证明在各种模型中存在绝对连续谱（离域）和点谱（局部化），我想继续发展这些思想，例如绘制出地图。树上横向周期势类的相图 对于确定性薛定谔算子来说，还有一些有趣的开放问题。 共振是共振在第二片材上的亚纯延续中的极点，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的散射理论引起的一些问题感兴趣。 。