Shi Jin, University of Wisconsin-Madison

"Semiclassical computational methods for quantum dynamics with band-crossing"

Band-crossing is a quantum dynamical behavior that contributes
to important physics and chemistry phenomena such as quantum tunneling, Berry connection, chemical reaction etc. As in a typical quantum mechanics or high frequency wave simulation, one needs to numerically resolve the small wave length, as required by the Shannon-Nyquist Theorem, which is prohibitively expensive.

For semiclassical models for surface hopping, we introduce a nonlinear geometric optics method based numerical strategy which is able to capture accurately pointwise solutions without numerically resolving the high frequency oscillations in both space and time. Our first main idea is to use the nonlinear geometric optics ansatz, which builds the oscillatory phase into an independent variable. We then choose suitable initial data, based on the Chapman-Enskog expansion, for the new model. We establish uniform (in the wave length) numerical convergence for a scalar model, and give numerical examples to demonstrate that the method has the desired properties, defying the Shanon-Nyquist Theorem in the sense of the maximum norm.