Applied/PDE Seminar: Qin Li (UW Wisconsin), 'Semi-classical limit for the Schroedinger equation with lattice potential, and band-crossing'

Event Date: 

Friday, February 1, 2013 -
3:00pm to 4:00pm

Event Contact: 

Carlos Garcia-Cervera
Email: cgarcia@math.ucsb.edu

In this talk, I am going to present the derivation of the semi-classical limit of the Schroedinger equation with lattice potential, where the lattice constant and the Planck constant are at the same order. Bloch theory is used to decompose the solution into eigenfunctions. Here we encounter two problems: 1. eigenvalues degenerate, when this happens one cannot distinguish the associated eigenfunctions, and the so called transition rate between energy bands should be introduced; 2. the evolution of the projection coefficients follow a coupled integro-differential equation, which can be hard to compute. By carrying out the Wigner transformation of all the Bloch bands, we find a complete basis on the phase space. The coefficients for them are controlled by a simple hyperbolic equation, and the transition rates at the point of the degeneracy are characterized explicitly. A domain decomposition method based on the distances between energy bands is designed associated to this newly developed model. This is a joint work with Lihui Chai and Shi Jin.