Math 521: Advanced Topics in Real Analysis
In this course we will discuss some recent developments in the theory of Schrödinger operators. The main emphasis will be on half-line operators with decaying potentials. We will first discuss the discrete spectrum and address topics such as the Hardy inequality and Bargmann and Lieb-Thirring estimates. Then we turn to the essential spectrum and relate the decay rate of the potential to the spectral type inside the essential spectrum. The highlights here include the Wigner-von Neumann example, the Kiselev-Last-Simon treatment of random decaying potentials, the Christ-Kiselev-Remling proof of the 1/2-conjecture, Remling's bounds on the size of the embedded singular spectrum, Kiselev's construction of examples with embedded singular continuous spectrum (which solved one of Simon's 21st-century problems), and the Deift-Killip approach to square-integrable potentials. Most of these results were major events in spectral theory during the 1990's. If time permits, we will also discuss even more recent results linking the discrete spectrum and the essential spectrum directly, without prior knowledge of the potential.
Prerequisites: Previous exposure to measure theory and functional analysis, especially the spectral theory of self-adjoint operators, will be assumed.
Meeting Time: MWF 11:00 AM -- 11:50 AM
Location: Herman Brown Hall (Room 453)
Email: [my last name] at rice dot edu
Office: Herman Brown 434
Office Hour: M 3:00 PM -- 4:00 PM and by appointment