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报告题目：Spectral invariants for vector periodic NLS

报告人：Evgeny Korotyaev

报告时间：9月10日15：00-16：00

报告地点：立德楼407（线下）

zoom: 910 416 3215

报告摘要：

Firstly, we discuss the various properties of periodic Zkharov-Shabat operator, associated with scaler periodic NLS. We describe the main results and techniques. Secondly, we discuss first order operators with a periodic 3x3 matrix potential on the real line. This operator is the Lax operator for the periodic vector NLS equation. The spectrum of the operator covers the real line and it is union of the spectrum of multiplicity 3, separated by intervals (gaps) of multiplicity 1. We prove the following:

 · The corresponding 2 or 3-sheeted Riemann surface is described.

 · Necessary and sufficient conditions are given when the Riemann surface is 2-sheeted.

 In the case of the 2-sheeted Riemann surface the solution of vector NLS equation is determined in terms of solutions of scalar NLS equations.

 · One constructs an entire function, which is negative on the spectrum of multiplicity 3 and is positive on its gaps.

 · The conformal mapping of the upper half plane on the domain on the upper half plane is constructed and the main properties are This conformal mapping has asymptotics at high energy where coefficients are constants of motion. As a corollary we obtain the estimate of potentials in terms of gap lengths.

 · Finally the Borg type result is obtained.

报告人简介：Evgeny Korotyaev 教授是谱理论与可积系统、散射理论领域国际知名的数学家。他现任圣彼得堡国立大学数学-力学系教授，兼任俄罗斯高等经济研究大学教授，并受聘为东北师范大学前沿交叉研究院教授。Korotyaev 教授于 1982 年在圣彼得堡国立大学获博士学位，1996 年在圣彼得堡 Steklov 研究所获理学博士学位。三十余年来，他专注于逆谱理论、几何函数论、可积系统与周期介质上的Dirac算子与Schrödinger算子等方向的研究，在 Invent. Math., J. Reine Angew. Math., Math. Ann., Commun. Math. Phys.,Trans. Am. Math. Soc., Inverse Probl., JFA, JDE等顶级期刊发表论文 160 余篇，Google Scholar 引用量逾 3200 次。

报告题目：(Essential) numerical ranges for unbounded operators and pencils, with applications to spectral approximation

报告人：Marco Marletta

报告时间：9月10日16：00-17：00

报告地点：立德楼407 (线下)

zoom: 910 416 3215 密码：1123

报告摘要：This talk is based upon joint work with Boegli, Tretter and Ferraresso, and examines how a concept developed originally in the late 1960s to mid 1970s for bounded operators, can be generalised to unbounded operators, and used to explain how their spectra may (fail to be)approximated when the operators are approximated. The applications include Schrodinger, Dirac, Stokes and Maxwell operators.

报告人简介：Marco Marletta 是谱理论领域的专家，现任 Cardiff 大学 Mathematical Analysis Research Group 主任，是威尔士科学学会的会士、London Mathematical Society、European Mathematical Society 与 Society for Industrial and Applied Mathematics 的活跃成员。其研究兴趣涵盖 Spectral Theory of Differential Operators、Numerical Methods for PDEs、Inverse Problems 及 Maxwell Equations。Marletta 教授是 London Mathematical Society 理事会成员，并入选 UK Research and Innovation 机构 Future Leaders Fellowships 评估团队。他受邀在 2020 年为 Ricardo Weder 七十华诞会议、2022 年 Krakow 的 IWOTA、Oberwolfach 以及 Isaac Newton Institute 的重要项目中担任 plenary speaker。