95-469 Alain Joye

Upper Bounds for the Energy Expectation in Time-Dependent Quantum Mechanics (61K, Latex) Oct 27, 95

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Abstract. We consider quantum systems driven by hamiltonians of the form $H+W(t)$, where the spectrum of $H$ consists in an infinite set of bands and $W(t)$ depends arbitrarily on time. Let $\bra H \ket_{\ffi}(t)$ denote the expectation value of $H$ with respect to the evolution at time $t$ of an initial state $\ffi$. We prove upper bounds of the type $\bra H \ket_{\ffi}(t) =O(t^{\delta})$, $\delta>0$, under conditions on the strength of $W(t)$ with respect to $H$. Neither growth of the gaps between the bands nor smoothness of $W(t)$ are required. Similar estimates are shown for the expectation value of functions of $H$. Sufficient conditions to have uniformly bounded expectation values are explicited and the consequences on other approaches of quantum stability are discussed.

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