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A Bloch Band Based Level Set Method for Computing
the Semiclassical limit of Schrodinger Equations
Zhongming Wang
Department of Mathematics
UC San Diego
Abstract
A novel Bloch band based level set method is proposed for computing
the semiclassical limit of Schrodinger equations in periodic media.
For the underlying equation subject to a highly oscillatory initial
data, a hybrid of the WKB approximation and homogenization leads to
the Bloch eigenvalue problem and an associated Hamilton-Jacobi system
for the phase in each Bloch band, with the Bloch eigenvalue be part
of the Hamiltonian. We formulate a level set description to capture
multi-valued solutions to the band WKB system, and then evaluate
total homogenized density over a sample set of bands. A superposition
of band densities is established over all bands and solution branches
when away from caustic points. The numerical approach splits the
solution process into several parts: i) initialize the level set
function from the band decomposition of the initial data; ii) solve
the Bloch eigenvalue problem to compute Bloch waves; iii) evolve the
band level set equation to compute multi-valued velocity and density
on each Bloch band; iv) evaluate the total position density over a
sample set of bands using Bloch waves and band densities obtained in
step ii) and iii), respectively. Numerical examples with different
number of bands are provided to demonstrate the good quality of the
method.
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