APDE seminar @UPV/EHU: Fourier analytic methods in the spectral theory of Schrödinger operators

Abstract

Estimating the location and accumulation of eigenvalues for Schrödinger operators is a classical problem at the intersection of spectral theory and mathematical physics. In this talk, we illustrate how Fourier analysis, in particular Fourier restriction, yields sharp eigenvalue estimates for Schrödinger-type operators with short-range potentials. We highlight the limitations of such methods for long-range potentials and show how they can be overcome by introducing randomness. Our results shed new light on the critical temperature for superconductivity in BCS theory and on the energy and lifetime of resonances in nuclear physics.