In this talk, we discuss qualitative spectral properties of self-adjoint Schrödinger and Dirac operators. We first briefly review some of the standard results for regular potentials from the literature and turn to more recent developments afterwards. Our main objective in this lecture is to discuss differential operators with singular potentials supported on curves or hyperplanes, where in the case of Dirac operators it is necessary to distinguish the so-called non-critical and critical cases for the strength of the singular perturbation. In particular, it turns out that Dirac operators with singular potentials in the critical case have some unexpected spectral properties. This talk is based on joint some recent works with P. Exner, M. Holzmann, V. Lotoreichik, T. Ourmieres-Bonafos, and K. Pankrashkin.

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The exhibiting artists are Marsha Borden, Helena Chastel, Kathryn Frund, Phoebe Godfrey, Hugh MacDonald, Bob Pavlik, Dan Potter, and R. Douglass Rice. Open Air 2025 is open daily and will remain on view through October 6, 2025. June 19, 2025 iis the last day to visit indoor art exhibitions. Exhibitions inside the AVS Gallery will resume on September 11, 2025

The exhibiting artists are Marsha Borden, Helena Chastel, Kathryn Frund, Phoebe Godfrey, Hugh MacDonald, Bob Pavlik, Dan Potter, and R. Douglass Rice. Open Air 2025 is open daily and will remain on view through October 6, 2025. June 19, 2025 iis the last day to visit indoor art exhibitions. Exhibitions inside the AVS Gallery will resume on September 11, 2025

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