In the context of solid-state physics the linear Schroedinger
equation is considered describing, typically, the motion of
electrons in potentials generated by ionic cores. The authors
consider the one-dimensional model with the two-scale WKB initial
condition and with potentials periodic on the lattice $\Gamma=2 \pi
\mathbb{Z}$.

In the dynamical regime where the direct numerical simulations are
prohibitively expensive and where the presence of many caustics
reduces the accuracy of semiclassical methods (i.e., e.g., of the
time-splitting spectral method of Ref. [3]) the authors generalize
the Gaussian beam method introduced in 2008 [23]. The idea is to
allow the phase function to be complex, still admitting that the
solution has a Gaussian profile for each of a few (non-crossing)
Bloch bands.

The necessary mesh size proves much coarser, allowed to be of the
square-root order in the small semiclassical parameter. This
favorably compares with the mere first-order mesh characterizing the
recent Bloch-decomposition-based techniques of Refs. [18, 19, 20].

Several (e.g., insulator) numerical examples illustrate the accuracy
and confirm the efficiency of the new method, with promising future
applications, especially in the higher spatial dimensions.


MR2643633 Jin, Shi; Wu, Hao; Yang, Xu; Huang, Zhongyi Bloch
decomposition-based Gaussian beam method for the Schrödinger
equation with periodic potentials. J. Comput. Phys. 229 (2010), no.
13, 4869--4883. 82B10 (65Mxx 81Q10)