Restrictions on Singular Spectrum for Trace Class Perturbations (26rit044)

The Banff International Research Station will host the "Restrictions on Singular Spectrum for Trace Class Perturbations" workshop in Banff from June 14 - 28, 2026.

Rank-one and finite rank self-adjoint perturbations emerge naturally when changing the boundary condition(s) of a self-adjoint differential operator, e.g.~from Neumann via mixed conditions towards Dirichlet conditions. The problem has over the years enjoyed much attention by mathematical physicists as well as operator and function theoreticians. Finite rank perturbations can be applied to obtain fast and cheap (more environmentally friendly) calculations in data science. This project concerns the singular spectrum under infinite rank perturbations such as trace class perturbations and beyond. Of particular interest is the development of an Aronszajn--Donoghue-type theory. Our goals feature a first analysis of the singular spectrum under trace class and more general perturbations.

To place this research into context, we recall that the main results of classical perturbation theory of self-adjoint operators --- such as the Weyl--von Neumann Theorem from the early 20th century, the Kato--Rosenblum Theorem 1950's, and the Weyl--von Neumann--Kuroda Theorem 1909/1935/1958 --- deal with the analysis of so-called essential and the absolutely continuous spectra under perturbations. We are interested in the behavior of the singular spectrum under trace class and more general perturbations, which is well-known to be a completely different beast and usually considered more difficult to capture.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), and Alberta's Advanced Education and Technology.