Upper bounds for the energy expectation in time-dependent quantum mechanics |
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Authors: | Alain Joye |
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Institution: | (1) Department of Mathematics and Center for Transport Theory and Mathematical Physics, Virginia Polytechnic Institute and State University, 24061-0123 Blacksburg, Virginia;(2) Present address: Centre de Physique Théorique, CNRS Marseille, Luminy Case 907, F-13288 Marseille Cedex 9, France;(3) PHYMAT, Université de Toulon et du Var, B.P. 132, F-83957 La Garde Cedex, France |
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Abstract: | We consider quantum systems driven by Hamiltonians of the formH+W(t), where the spectrum ofH consists of an infinite set of bands andW(t) depends arbitrarily on time. Let H![lang](/content/k70804491r679q3j/xxlarge9001.gif)
(t) denote the expectation value ofH with respect to the evolution at timet of an initial state . We prove upper bounds of the type H![rang](/content/k70804491r679q3j/xxlarge9002.gif)
(t)=O(t
), >0, under conditions on the strength ofW(t) with respect toH. Neither growth of the gaps between the bands nor smoothness ofW(t) is required. Similar estimates are shown for the expectation value of functions ofH. Sufficient conditions to have uniformly bounded expectation values are made explicit and the consequences on other approaches to quantum stability are discussed. |
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Keywords: | Quantum stability energy expectations quantum diffusion |
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