Quantum Mechanics and Geometry Seminar Winter 2018
McHenry Library Room 1270
For further information please contact Professor Richard Montgomery or call 831-459-2400
Friday, January 12, 2018
No Seminar
Friday, January 19, 2018
Richard Montgomery, University of California, Santa Cruz
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.
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.
Friday, January 26, 2018
TBA
Friday, February 2, 2018
TBA
Friday, February 9, 2018
TBA
Friday, February 16, 2018
TBA
Friday, February 23, 2018
TBA
Friday, March 2, 2018
TBA
Friday, March 9, 2018
TBA
Friday, March 16, 2018
TBA
Friday, March 23, 2018
TBA