Abstract:

The magnetic Schrödinger equation provides a (probabilistic) model of the motion of a charged particle in an electromagnetic field. The associated (time-independent) eigenvalue problem provides probability densities, via normalized eigenvectors, of the location of the charged particle at certain energies associated with the eigenvalues. Even for magnetic fields that have a seemingly simple structure, there appear to be non-trivial computational challenges for approximating eigenvalues and eigenvectors, particularly as the strength of the magnetic field increases. We will highlight some of these challenges, and introduce two independent approaches that aim to address these challenges in two different regimes. We will also provide some background concerning what is currently well-understood about the problem (not as much as one might hope), and hint at some progress toward improvements on the theoretical and computational fronts.

Bio:

Jeff Ovall (JO) is a Maseeh Professor of Mathematics at Portland State University, working on numerical analysis and scientific computing for partial differential equations and integral equations. Specific research topics of interest include operator eigenvalue problems, "exotic" discretization schemes, estimation of discretization error, and effective treatment of singular solutions. JO received a PhD in mathematics at the University of California, San Diego, in 2004. After postdoctoral positions at the Max Planck Institute for Mathematics in the Sciences (Leipzig, Germany) and the California Institute of Technology, and a faculty position at the University of Kentucky, JO joined the faculty at Portland State University in 2013. JO's favorite basic linear algebra result is the spectral theorem for normal matrices, and the favorite basic ODE result is that any solution of a first-order (scalar) autonomous equation must be monotone. JO's Erdős number is 4, with at least nine paths having no internal "links" in common and mathematical genealogy can be traced back as far as Niccolò Tartaglia (1499-1557).