Bremen-Hamburg²-Kiel-Oldenburg-Seminar 2024 – Funktionalanalysis und ihre Anwendungen

Vorläufiges Programm

• 14:00 - 14:45: Alden Waters (Uni Hannover): "Relative trace formulae for the Casimir effect"
• 15:00 - 15:30: Coffee Break
• 15:30 - 16:15: Badreddine Benhellal (Uni Oldenburg): "On the self-adjointness of two-dimensional relativistic shell interactions"
• 16:30 - 17:15: Dennis Schmeckpeper (TUHH): "Spectral theory of periodic quantum graphs via the Floquet transform"
• 18:00: Dinner (Restaurant Übersee)

Anreise zur Uni Bremen

• Vom Hauptbahnhof fährt die Straßenbahn 6 direkt zur Universität (Haltestelle „Zentralbereich“).
• Mit dem Auto färt man über die A27 bis zur Abfahrt Horn-Lehe, von dort ist die Universität ausgeschildert. Parken auf dem großen Parkplatz vorm Mathegebäude („MZH“) kostet 1€ pro Tag; den Parkplatz erreicht man über die Enrique-Schmidt-Straße.
• Die Vorträge finden im MZH (Gebäude der Mathematik und Informatik) statt, im Raum MZH 5600 im 5. Obergeschoss.
  Die Adresse: Bibliothekstraße 5, 28359 Bremen.

Abstracts

• Alden Waters: Consider a the complement of a bounded Lipschitz domain in three dimensional space. We consider Maxwell's equations with metallic boundary conditions, or equivalently, co-closed differential one forms with relative boundary conditions. I will give a relative trace formula for these operators which relates the trace of a certain linear combination of operators on space to the determinant of the Maxwell boundary layer operator on the boundary. This allows for a boundary layer expression for the Casimir energy.
  (Based on joint work with A. Strohmaier)
• Badreddine Benhellal: In this talk, we discuss the self-adjointness of the two-dimensional Dirac operator coupled with electrostatic and Lorentz scalar shell interactions of constant strength ε and μ supported on a closed Lipschitz curve. Namely, we present several new explicit ranges of ε and μ for which there is a unique self-adjoint realization with a domain included into $H^{1/2}$. A more precise analysis is carried out for curvilinear polygons, which allows one to take the corner openings into account. Compared to the preceding works on this topic, two new technical ingredients are employed: the explicit use of the Cauchy transform on non-smooth curves and an explicit characterization of the Fredholmness for singular integral operators.
  (Based on joint work with Konstantin Pankrashkin and Mahdi Zreik.)
• Dennis Schmeckpeper: Quantum graphs can be viewed as intervals glued together to form a graph-like structure. Each quantum graph possesses a distinctive Hamiltonian operator, i.e. the Laplacian plus some suitable potential, which acts on each interval and fulfills compatibility conditions at their boundaries. In this talk we use the so-called Floquet transform to exploit the inherent periodicities within the graphs to show that every such Hamiltonian allows a direct integral decomposition where the fibers are operators on a compact subgraph with only point spectrum.