Spectral Theory Of Differential Operators (STODO)

An undergraduate research program

Organizer: Wencai Liu

Description: I am running ongoing research projects. I have some projects about (inverse) spectral problems of (discrete)differential operators at hand, which are suitable to undergraduate students. It is expected to have graduate students and/or postdocs to work with me on projects.

Structure of each project: The project has two weekly meetings. In the first meeting (1-2 hours), we do lectures and student presentations. In the second meeting (1-2 hours), students study and work together on the project.

Acknowledgment: STODO is partially funded by a grant from College of Arts and Sciences Undergraduate Research Program

Announcement: I will be visiting UC Berkeley during Fall 2025 and will not be taking new students into the program during that semester.

Current project

Project 6

TBA

- Assistant: TBA
- Date: TBA
- Participated Student: TBA
- Outcome: TBA

Five Past Projects

I have run five projects about spectral problems of differential operators starting 2020.

Project 5

Extrema of spectral band functions of periodic Schrodinger operators

- Assistant: Matthew Faust
- Date: August 2023- June 2025
- Participated Student: Ethan Luo
- Outcome: A research paper, Extrema of spectral band functions of two-dimensional discrete periodic Schrodinger operators. J. Math. Phys. 2025. arXiv version

Project 4

Floquet isospectrality for periodic Schrodinger operators

- Assistants: Rodrigo Matos and Matthew Faust
- Date: August 2022-July 2024
- Weekly Schedule (including notes)
- Participated Students: Jenna Plute, Jonah Robinson (student leader), Yichen Tao, Ethan Tran, Cindy Zhuang
- Outcome: A research paper, Floquet isospectrality of the zero potential for discrete periodic Schrodinger operators. J. Math. Phys. 65 (2024), no. 7, Paper No. 073501, 8 pp. arXiv version

Blocker 628, Dec 7, 2022

Jenna Plute, Yichen Tao, Ethan Tran and Cindy Zhuang gave slide and poster presentations in the 6th annual TX-LA undergraduate mathematics conference, LSU, March 25-26 2023. Below are some pictures.

Project 3

Embedded eigenvalues of perturbed Dirac operators

- Assistant: Kang Lyu
- Date: January 2022-July 2022
- Weekly Schedule (including all notes)
- Participated students: Vishwam Khapre and Andrew Yu
- Outcome: A research paper, Sharp bound for embedded eigenvalues of Dirac operators with decaying potentials. New York J. Math. 28 (2022) 1317–1328.

The following two projects are part of inverse spectral problem of discrete Schrodinger operators.

Project 2

Ambarzumian type problems with mixed data.

- Assistant: Burak Hatinoglu
- Date: April 2020-August 2020
- Participated undergraduate students. Alejandra Puente, Sithija Manage, Jerik Eakins, Lucille Lamb, William Frendreiss
- Weekly Schedule (including all notes)
- Outcome: A research paper, Ambarzumian-type problems for discrete Schrödinger operators. Complex Anal. Oper. Theory 15 (2021), no. 8, Paper No. 118, 13 pp. Arxiv version

Project 1

Ambarzumian type problems of long range operators.

- Assistants: Burak Hatinoglu and Rodrigo Matos
- Date: August 2020-April 2021
- Weekly Tentative Schedule in Spring 2021 (including all notes)
- Weekly Tentative Schedule in Fall 2020 (including all notes)
- Participated undergraduate students: Alejandra Puente, Sithija Manage, Lucille Lamb, William Frendreiss