Spectral theory and resonances for Schrödinger operators

我正在研究无限图上的离散薛定谔算子，既有随机势，也有确定性势。这些随机势算符的谱理论是数学物理中的一个很大的领域，其中包括一些重要的未解决的问题，如关于粒子在无序介质中运动的扩展态猜想。我的工作引入了动力系统的思想，以证明在各种模型中绝对连续谱(离域)和点谱(局域)的存在。我想继续发展这些想法，例如绘制出树上横向周期势的相图。对于确定性薛定谔算子，还有一些有趣的开放问题。*共振是预解式在第二张薄片上的亚纯延拓中的极点。我正在估算共振态的时间演化。我还对涉及共振的最优化问题以及由波导中的散射理论引起的一些问题感兴趣。*****

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. *****