Mathematical and Numerical Methods for Complex Quantum Systems

Semi-classical limit for the Schroedinger equation with lattice potential and band-crossing

Qin Li

University of Wisconsin-Madison

In this talk we derive and compute the semi-classical limit of the Schroedinger equation with lattice potential. In [Gerard-Markowich-Mauser-Poupaud, 1997], the limit is derived under the assumption that energy bands are O(1) separate, namely, the system is adiabatic. However, in reality, this assumption is generically invalid. We remove the assumption, and obtain a general model by performing multi-scale variable separation while using Bloch decomposition and the Wigner transformation. Asymptotically this new full system recovers to the old one in the adiabatic region. In the computation, we decompose the domain into regions depending on the distance to the energy band-crossing points, and apply associated schemes in different regions. This is a joint work with Lihui Chai and Shi Jin.