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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:

81Q99 General mathematical topics and methods in quantum theory
35J10 Schrödinger operator, Schrödinger equation
46N50 Applications of functional analysis in quantum physics
47N50 Applications of operator theory in the physical sciences
82B99 Equilibrium statistical mechanics
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