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The semiclassical limit of quantum dynamics. I: Time evolution. (English) Zbl 0647.46060
The \(\hslash \to 0\) limit of the quantum dynamics determined by the Hamiltonian \(H(\hslash)=-(\hslash^ 2/2m)\Delta +V\) on \(L^ 2({\mathbb{R}}^ n)\) is studied for a large class of potentials. By convolving with certain Gaussian states, classically determined asymptotic behavior of the quantum evolution of states of compact support is obtained. For initial states of class \(C^ 1_ 0\) the error terms are shown to have \(L^ 2\) norms of order \(\hslash^{-\epsilon}\) for arbitrarily small positive \(\epsilon\).

MSC:
46N99 Miscellaneous applications of functional analysis
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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[1] DOI: 10.1007/BF01230088 · doi:10.1007/BF01230088
[2] DOI: 10.1016/0003-4916(81)90143-3 · doi:10.1016/0003-4916(81)90143-3
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[4] DOI: 10.1016/0003-4916(58)90032-0 · Zbl 0085.43103 · doi:10.1016/0003-4916(58)90032-0
[5] DOI: 10.1002/cpa.3160140303 · Zbl 0107.09102 · doi:10.1002/cpa.3160140303
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