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Semiclassical limit for the Schrödinger equation with a short scale periodic potential. (English) Zbl 1052.81039
In this paper, the dynamics generated by the time Schrodinger equation with the potential consisting of a lattice periodic potential plus an external potential which varies slowly on the scale of the lattice spacing is investigated. Theorems proved in this paper show that for very slow changes of the external potential the time position operator and, more generally, semiclassical observables converge to a limit given by the semiclassical dynamics. Results are given for isolated bands only (no band crossing is considered).
MSC:
81Q99General mathematical topics and methods in quantum theory
35J10Schrödinger operator
46N50Applications of functional analysis in quantum physics
47N50Applications of operator theory in quantum physics
82B99Equilibrium statistical mechanics