Analysis and PDE Seminar Homepage, Spring 2014

Location: 383N

Time: Thursday, 2:15pm.

• Wednesday, April 2, 4-5pm, 383N (geometry seminar slot, unusual date/time/room):

  Setsuro Fujiie (Ritsumeikan University)

  ``Semiclassical distribution of resonances created by homoclinic trajectories''

  Abstract: We consider the Schrödinger operator −h^2 ∆ + V (x) in the multidimensional Euclidean space with a semiclassical parameter h and a smooth potential decaying at infinity. Assuming that the underlying classical mechanics on the energy surface of a fixed positive energy has a trapped set consisting of homoclinic trajectories, we describe the precise asymptotic distribution of resonances in a complex neighborhood of this energy. The method is based on the microlocal study of solutions near the trapped set, especially near the hyperbolic fixed point. This is a joint work with Jean-Francois Bony, Thierry Ramond and Maher Zerzeri.

• Monday, May 5, Bay Area Microlocal Analysis Seminar at Stanford:

  • 2:15-3:15pm, Room 381T:

    Jeff Galkowski (UC Berkeley)

    ``Distribution of Resonances for Quantum Corrals''

    Abstract: We consider resonances for the operator $-\Delta +V\otimes \delta_{\partial\Omega}$ where $\Omega\subset\re^d$ is a bounded domain. This operator is a model for quantum corrals as well as other lossy systems. We give a bound on the size of the resonance free region for very general $\Omega$ and in the case that $\partial \Omega$ is strictly convex, we give a dynamical characterization of the resonance free region that is generically sharp. We describe how this characterization can be thought of as a Sabine Law in certain cases.

  • 4-5pm, Room 381U:

    Guy David (Université Paris-Sud)

    ``A variant of the Alt, Caffarelli, and Friedman free boundary problem motivated by the localization of eigenfunctions''

    Abstract: I will try to describe joint work with M. Filoche, D. Jerison, and S. Mayboroda. The initial motivation (Filoche-Mayboroda) concerns the localization of eigenfunctions, say, for a Schrödinger operator with a complicated bounded potential, or the Laplacian on a complicated domain. What we do is try to find an automatic decomposition of the domain into small pieces, related to the given operator, and for this we minimize a variant of the Alt, Caffarelli, and Friedman free boundary problem, where we authorize a large number of phases (instead of 2). The results concern the regularity of the minimizers.

• Monday, May 12, 2:15-3:15pm, Room 383N (unusual date/time):

  Graham Cox (University of North Carolina)

  ``A Morse index theorem for elliptic operators on bounded domains''

  Abstract: The Maslov index is a symplecto-geometric invariant that counts signed intersections of Lagrangian subspaces. It was recently shown that the Maslov index can be used to compute Morse indices of Schrödinger operators on star-shaped domains. We extend these results to general selfadjoint, elliptic operators on domains with arbitrary boundary geometry, and discuss some applications. (Joint work with C. Jones and J. Marzuola.)

• Schedule for Winter 2014.
• Schedule for Autumn 2013.
• Schedule for Spring 2013.
• Schedule for Winter 2013.
• Schedule for Autumn 2012.
• Schedule for Winter/Spring 2012.
• Schedule for Autumn/Winter 2010-2011.
• Schedule for Winter-Spring 2010.
• Schedule for Autumn 2009.
• Schedule for Spring 2009.
• Schedule for Winter 2009.
• Schedule for Spring 2008.
• Schedule for Winter 2008.
• Schedule for Autumn 2007.