This chapter presents the basic mathematical facts concerning Schrödinger's equation with constant and time-dependent Hamiltonians: Stone's theorem on strongly continuous one-parameter groups of unitaries, the Kato–Rellich criterion for self-adjointness, and the Dyson expansion. In particular, Floquet's theory for periodic perturbations is outlined and illustrated by examples of kicked systems: the kicked top and the baker map. Then classical mechanics is introduced as a limit of quantum theory using coherent states and mean-field limits. The formalism of classical differentiable dynamics is briefly described and the classical and quantum aspects of the motion of a free particle on a compact Riemannian manifold are discussed including Weyl's theorem characterizing spectra of generalized Laplacians such as Beltrami–Laplace operators.
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