Summary
Let V Γ be a lattice periodic potential and A and Φ external electromagnetic potentials which vary slowly on the scale set by the lattice spacing. It is shown that the Wigner function of a solution of the Schrödinger equation with Hamiltonian operator \(H = \tfrac{1}{2}{{( - i{{\nabla }_{x}} - A(\varepsilon x))}^{2}} + {{V}_{\Gamma }}(x) + \phi (\varepsilon x)\) propagates along the flow of the semiclassical model of solid states physics up to an error of order ε. If ε-dependent corrections to the flow are taken into account, the error is improved to order ε 2. We also discuss the propagation of the Wigner measure. The results are obtained as corollaries of an Egorov type theorem proved in [PST3].
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Teufel, S., Panati, G. (2004). Propagation of Wigner Functions for the Schrödinger Equation with a Perturbed Periodic Potential. In: Blanchard, P., Dell’Antonio, G. (eds) Multiscale Methods in Quantum Mechanics. Trends in Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8202-6_17
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DOI: https://doi.org/10.1007/978-0-8176-8202-6_17
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