A local perspective on manifolds with a Lie structures at infinity
Montag, 7.7.25, 16:15-17:45, Seminarraum 404
A Lie structure at infinity on a smooth manifold can be seen as the extension of the tangent bundle to a Lie algebroid on the closure of the respective manifold. This allows us to extend a Riemannian metric on the manifold up to infinity. In the talk, I will present the notion of a Riemannian manifold with Lie structure at infinity following Ammann, Lauter & Nistor (2004) and focus on the role of the Lie algebroid. I will use a local approach to show that the Riemannian metric blows up towards the boundary. Furthermore, I will show exemplarily how a Lie structure at infinity on a manifold yields a compactification when linking the Riemannian metric and the corresponding metric space.
Relativistic \(\delta\)-shell interactions
Montag, 28.7.25, 16:15-17:45, Seminarraum 404
In quantum mechanics, the practice of coupling Hamiltonians with singular potentials supported on lower-dimensional subsets of the ambient space is a well-established concept. For instance, the physicists Ralph Kronig and William Penney had already considered Schrödinger operators coupled with \(\delta\)-point potentials in their 1931 work, "Quantum mechanics of electrons in crystal lattices.“
However, one should remember that such \(\delta\)-potentials are merely idealized representations of real-world physical phenomena. Therefore, understanding how these operators can be rigorously approximated by more regular potentials is important for their physical interpretation and mathematical validity.
In this talk, we will present a result by Albert Mas and Fabio Pizzichillo which shows that the Dirac operator on \(\mathbb{R}^{3}\) coupled with suitable short-range potentials converges in the strong resolvent sense to the Dirac operator coupled with a \(\delta\)-potential supported on the boundary of a \(C^{2}\) domain. Following this, we'll discuss a possible generalization of this result to more general spinor bundles.