温州大学数理学院2025年第9期数理大讲堂

题目：Standing waves in the Schrodinger equation with Kato class potentials and $L^\infty$-bounded nonlinearities

报告人：Aleksander Cwiszewski

报告时间：2025年6月3日 14：00

报告地点：数理学院3号楼 3B205

Abstract: We consider the Schrödinger equation with potentials belonging to the Kato class and nonlinearities that are $L^\infty$-bounded. The associated parabolic equation is analyzed using Conley index techniques. To this end, we establish the admissibility of the parabolic semiflow and investigate the structure of its maximal invariant set -- namely, the set of all functions through which a full bounded solution of the semiflow passes. We show, in a sequence of steps, that this invariant set is bounded both in the space of bounded functions on Euclidean space and in the energy space. Finally, we present results demonstrating that, under the Landesman–Lazer conditions or the sign conditions, the Schrödinger equation admits standing wave solutions with frequencies lying within the spectrum of the Schrödinger operator below its essential spectrum.。

专家简介：Aleksander Cwiszewski ，托伦尼古拉哥白尼大学数学与计算机科学学院教授，是《Topological Methods in Nonlinear Analysis》(非线性分析中的拓扑方法)的执行主编。

研究领域：Aleksander Cwiszewski 教授的研究方向主要集中在“非线性分析与微分方程”，尤其是 “拓扑方法在动力系统中的应用”，涉及：微分方程解的稳定性与分岔理论 ；非局部微分方程的动力学行为 ；泛函分析在生物数学模型中的应用（如种群动力学、传染病模型）。学术成果：在 《Calculus of Variations and PDE》《Journal of Differential Equations》《Nonlinear Analysis》等国际知名期刊发表多篇论文。 参与波兰国家科学中心（NCN）资助的科研项目，研究方向与动力系统和应用数学相关。