Spectra of the Darboux–Treibich–Verdier operators
This course will study the spectrum of the Schrödinger operator with Darboux– Treibich–Verdier (DTV) potential. The DTV potential first discovered by Darboux in 1882 and rediscovered by Trebich and Verdier in 1988, represents a remarkable doubly periodic solution of stationary KdV hierarchy equations labeled by four integers.
Lecturer
Date
22nd September, 2023 ~ 19th January, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Friday | 09:50 - 12:15 | A3-2-303 | ZOOM 02 | 518 868 7656 | BIMSA |
Prerequisite
Calculus, Linear algebra
Syllabus
1. Introduction
2. Ordinary differential equations with regular singularities
3. Floquet Theory
4. KdV potentials
5. Spectral Theory of Hill’s equations
6. Spectra of the DTV operators
2. Ordinary differential equations with regular singularities
3. Floquet Theory
4. KdV potentials
5. Spectral Theory of Hill’s equations
6. Spectra of the DTV operators
Reference
[1] Z. Chen, E. Fu and C. S. Lin; A necessary and sufficient condition for the Darboux-TreibichVerdier potential with its spectrum contained in ℝ. Amer. J. Math. 144 (2022), no. 3, 851–87
[2] F. Gesztesy and H. Holden; Soliton equations and their algebro-geometric solutions. Vol. I. (1 + 1)-dimensional continuous models. Cambridge Studies in Advanced Mathematics, vol. 79, Cambridge University Press, Cambridge, 2003. xii+505 pp.
[3] Wilhelm Magnus and Stanley Winkler; Hill’s equation. (Corrected reprint of the 1966 edition) Dover Publications, Inc., New York (1979) viii+129.
[2] F. Gesztesy and H. Holden; Soliton equations and their algebro-geometric solutions. Vol. I. (1 + 1)-dimensional continuous models. Cambridge Studies in Advanced Mathematics, vol. 79, Cambridge University Press, Cambridge, 2003. xii+505 pp.
[3] Wilhelm Magnus and Stanley Winkler; Hill’s equation. (Corrected reprint of the 1966 edition) Dover Publications, Inc., New York (1979) viii+129.
Audience
Undergraduate
, Graduate
Video Public
Yes
Notes Public
Yes
Language
Chinese
Lecturer Intro
Erjuan Fu received her Ph.D. from University of Utah in 2019, under the guidance of Yuan-Pin Lee and Herb Clemens; after that, she was a postdoc in YMSC at Tsinghua University. Since Dec. 2022, she has been working at BIMSA as an assistant professor. Currently, her research is to study the spectrum for Darboux-Treibich-Verdier potential and related topics.