Inverse spectral problems for periodic Schrödinger operators

A periodic Schrödinger operator $LaTeX: H:L^2(\mathbb{R})\to L^2(\mathbb{R})$ $H:L^2(\mathbb{R})\to L^2(\mathbb{R})$ is defined by

$LaTeX: H=-\dfrac{d^2}{dx^2} + V$ $H=-\dfrac{d^2}{dx^2} + V$

where $LaTeX: V\in C(\mathbb{R})$ $V\in C(\mathbb{R})$ is periodic.

It is known that the spectrum of the operator LaTeX: H $H$ is purely continuous and consists of a union of at most countably many disjoint closed intervals.

In this project we consider inverse spectral problems for such operators. In other words, given the spectrum of an operator of the same form as LaTeX: H $H$ , can we reconstruct the potential LaTeX: V $V$ ? Is it unique?

The latter question has been answered in that if we can reconstruct a potential, then it is typically not unique. Several results on this problem exist and it is enough to know the edges of the intervals, as they also appear as eigenvalues of two related Sturm-Liouville problems.

This project can go in several directions depending on the interest of the student. One possibility is to consider the case where the operator $LaTeX: H:L^2(\mathbb{R};\mathbb{R}^m)\to L^2(\mathbb{R};\mathbb{R}^m)$ $H:L^2(\mathbb{R};\mathbb{R}^m)\to L^2(\mathbb{R};\mathbb{R}^m)$ and LaTeX: V $V$ is a real symmetric matrix-valued potential. This introduces new challenges, some of which occur already in the case where $V$ is diagonal. The main purpose of this MSc thesis is to do a literature review of the existing results and to identify new directions of research. The project will be done with me, Wilhelm Treschow, as the main supervisor, together with two collaborators as co-supervisors. One here in Lund and another from the University of Tennessee. The project relates to my own research on periodic Schrödinger operators.

Contact: wilhelm.treschow@math.lth.se