• Lecturer: Angkana Rüland
• Date: Thursdays 09:15 - 10:45
• Room: Leipzig University, SG 3-14
• Language: English, German if desired
• Target audience: MSc students, PhD students, Postdocs
• Keywords: Partial Differential Equations, Inverse Spectral Theory
• Prerequisites: Analysis I-III and ODEs is required; basic knowledge of functional analysis and PDEs is helpful
• Remarks: if desired, this seminar could also be held as a block seminar.

Abstract:

In this seminar, we are going to study the equation $$- y'' + qy = \lambda y \qquad 0 \le x \le 1$$ subject to the boundary data \(y(0)=y(1)=0\). It is assumed that the function \(q : [0,1] \to \mathbb{R}\) is given. The real number \(\lambda\) is called an eigenvalue, if a non-trivial solution to the above problem exists.

In this seminar, we investigate the relation between the potential \(q\) and the set of eigenvalues. For example, typical questions are: For which sets of real numbers does there exist a potential which has this given set as eigenvalues? Which potentials are isospectral, i.e. which potentials give the same eigenvalues? Which additional pieces of information are determined by the potential?

The theory is surprisingly complete with rich relations to other fields of mathematics.