Upper Bounds for the Energy Expectation inTime - Dependent Quantum Mechanics

We consider quantum systems driven by hamiltonians of the form H +W (t), where the spectrum of H consists in an innnite set of bands and W (t) depends arbitrarily on time. Let hHi ' (t) denote the expectation value of H with respect to the evolution at time t of an initial state '. We prove upper bounds of the type hHi ' (t) = O(t), > 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. Suucient conditions to have uniformly bounded expectation values are explicited and the consequences on other approaches of quantum stability are discussed.