Abstract

The magnetic Schrödinger equation provides a probabilistic model of the motion of a charged particle in an electromagnetic field. The associated eigenvalue problem provides probability densities, via normalized eigenvectors, of the location of the charged particle at certain energies associated with the eigenvalues.  Properties of the magnetic and electric potentials can cause eigenvectors to be strongly spatially localized.  This phenomenon has been extensively studied in the case where the magnetic potential is absent, and a we will briefly illustrate some known results about localization and its driving mechanisms in that context.  Much less is known in the case where the behavior is dominated by the magnetic field.  Our talk will focus on that case, providing computational tools that exploit the notion of gauge invariance to dramatically reduce the cost of eigenvector computations, and providing practical predictors of where eigenvectors lower in the spectrum are likely to localize.    This is joint work with Hadrian Quan, Robin Reid, Stefan Steinerberger and Julie Zhu.