Quantum Mechanics and Geometry Seminar Winter 2018

Hofstadter's Butterfly, Magnetic fields in Crystals, and the Almost-Mathieu Equation

Harper's eqn is a difference eqn [discrete Laplacian plus potential on the integers, so an operator on little l_2 ] whose spectrum holds a lot of surprises. The eqn was derived as that governing energy levels for an electron travelling in a square lattice under the influence of a magnetic field. Hofstader, in numerical experiments, uncovered the fractal nature of its spectrum. By introducing a coupling parameter in front of  the potential one goes from Harper's eqn to the almost Mathieu eqn, a subject of intense study and a long-standing open problem [``The Ten Martinis Problem''] which was recently solved by Avila [recent Field's medal] and Jitomirskaya [UCI; sister of a friend of mine] using ideas from dynamical systems: KAM, ergodicity, Lyapanov exponents. In this first talk I will derive the eqn from the quantum mechanics of an electron on a lattice in a constant magnetic field.

References: articles of Avila, Jitomirskayaka and Zhou available on the arXiv, arxiv 1608.01799; Hofstader's thesis, a book called the Quantum and the Butterfly; survey article by Damilokov arxiv.1410.2445.