Alain Joye
Upper Bounds for the Energy Expectation in
Time-Dependent Quantum Mechanics
(61K, Latex)
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.