Realizations of Elliptic Operators on Manifolds with Singularities

by Thomas Krainer

Thursday 25 Apr 2024, 14:15 → 15:15 Europe/Zurich

Sitzungszimmer (MI)

The development of the mathematical tools for analyzing boundary value problems for elliptic operators, such as the Dirichlet, Neumann, or Robin problems for the Laplacian, is a major achievement of the 20th century. Mathematics in Göttingen, with Hilbert giving birth to what we now know as functional analysis through his work on integral equations, and Weyl’s early work on Sturm-Liouville and spectral theory, played a pivotal role in that achievement and shaped our contemporary point of view. The analysis of classical boundary value problems for elliptic operators on smooth domains came to a culmination in the 1960s, and continues to be under active investigation today for operators with non-smooth coefficients and in domains with non-smooth boundaries. Seemingly unrelated at first are problems arising in mathematical physics that involve the Schrödinger equation with Hamiltonians that are singular, such as the quantum N-body or the Aharonov-Bohm Hamiltonian. Eigenfunction behavior where the potentials are singular reveals insight about the physics of such systems.

A unifying mathematical perspective on these problems is that an elliptic operator is considered acting in L^2 that initially is defined only on functions supported away from all boundaries or singularities. This operator has in general many different extensions to a closed operator, and the domain of an extension determines the admissible behavior of functions at the boundary or the singularities. In this talk I plan to review and elaborate on this viewpoint, and explain how it can be useful for studying elliptic operators that exhibit singular behavior that can be cast in a geometric language of manifolds with certain singularities, in particular those of cone or edge type and iterated versions thereof. I plan to focus on model operators associated to this type of singularities as they determine the principal behavior, and they are accessible with functional analytic methods that do not require advanced tools from Fourier analysis and operator theory.