Lecture Preview: "Frontiers in Numbers" Forum Session III
Publication Time:2025-11-12
Lecture Title
Spectral Invariants for Vector Periodic NLS
Speaker
Evgeny Korotyaev
Date and Time
September 10, 15:00–16:00
Venue
Room 407, Lide Building (In-person)
Zoom
ID: 910 416 3215 Password: 1123
Abstract
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.
About the Speaker
Professor Evgeny Korotyaev is an internationally renowned mathematician in the fields of spectral theory, integrable systems, and scattering theory. He is currently a Professor in the Department of Mathematics and Mechanics at Saint Petersburg State University, a Professor at the Higher School of Economics in Russia, and a Professor at the Institute of Frontier Interdisciplinary Research, Northeast Normal University. Professor Korotyaev received his Ph.D. from Saint Petersburg State University in 1982 and a Doctor of Science degree from the Steklov Institute of Mathematics, Saint Petersburg, in 1996. Over the past thirty years, he has focused on research in inverse spectral theory, geometric function theory, integrable systems, and Dirac and Schrödinger operators on periodic media. He has published over 160 papers in top journals such as Inventiones Mathematicae, Journal für die reine und angewandte Mathematik, Mathematische Annalen, Communications in Mathematical Physics, Transactions of the American Mathematical Society, Inverse Problems, Journal of Functional Analysis, and Journal of Differential Equations, with more than 3,200 citations on Google Scholar.
Lecture Title
(Essential) Numerical Ranges for Unbounded Operators and Pencils, with Applications to Spectral Approximation
Speaker
Marco Marletta
Date and Time
September 10, 16:00–17:00
Venue
Room 407, Lide Building (Online)
Zoom
ID: 910 416 3215 Password: 1123
Abstract
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.
About the Speaker
Marco Marletta is an expert in spectral theory and currently serves as the Head of the Mathematical Analysis Research Group at Cardiff University. He is a Fellow of the Learned Society of Wales and an active member of the London Mathematical Society, the European Mathematical Society, and the Society for Industrial and Applied Mathematics. His research interests include the spectral theory of differential operators, numerical methods for PDEs, inverse problems, and Maxwell’s equations. Professor Marletta is a member of the Council of the London Mathematical Society and has been selected to serve on the Future Leaders Fellowships review panel of UK Research and Innovation. He has been invited as a plenary speaker at significant events, including the Ricardo Weder 70th Birthday Conference (2020), IWOTA 2022 in Krakow, Oberwolfach, and major programs at the Isaac Newton Institute.
