Dr. Chris Marx, Asst. Professor of Mathematics at Oberlin College will present “What's the cage size of an electron?” on Wednesday, April 6 at 4:15 in Olin 114.
In physics, the quantum Hall effect captures how the electrons of a crystal respond to an external magnetic field. Depending on the properties of the material, the magnetic field may enhance or reduce the movability of electrons, thus making the crystal behave more like a metal or an insulator.
To describe this phenomenon mathematically, we will introduce quasi-periodic Schrödinger operators: infinite band matrices whose elements are generated by an irrational rotation on the circle. Insulator-like behavior amounts to exponentially decaying eigenvectors, representing electrons “trapped'' in some finite region of the crystal. The goal of this talk will be to derive bounds on the cage-size(“localization length’’) of such trapped electrons. We will thereby be able to predict metal- or insulator-like behavior of the crystal.
The results presented in this colloquium are joint work with two former students, Laura Shou and Jake Wellens. The talk will be of expository nature, and only uses calculus and some rudiments of analysis and linear algebra.