Abstract or Additional Information
In two papers in the 90's, Zhongwei Shen studied non-asymptotic bounds for the eigenvalue counting function of the magnetic Schrödinger operator, as well as the localization of eigenfunctions. But in dimensions 3 or above, his methods required a strong quantitative assumption on the gradient of the magnetic field; in particular, this excludes many singular or irregular magnetic fields, and the questions of treating these later cases had remained open, giving rise to a problem and a conjecture in this area at the intersection of harmonic analysis, mathematical physics, and partial differential equations. In this talk, we present our solutions to these questions, and other new results on the exponential decay of solutions (eigenfunctions, integral kernels, resolvents) to Schrödinger operators. We will introduce the Filoche-Mayboroda landscape function for the (non-magnetic) Schrödinger operator, present its pointwise equivalence to the classical Fefferman-Phong-Shen maximal function (also known as the critical radius function in harmonic analysis literature), and then show how one may use directionality assumptions on the magnetic field to construct a new landscape function in the magnetic case. We resolve the problem and the conjecture of Z. Shen (and recover other results in the irregular setting) by putting all these observations together.