  We make the first steps towards a generic theory for energy spreading and
quantum dissipation. The Wall formula for the calculation of friction in
nuclear physics and the Drude formula for the calculation of conductivity in
mesoscopic physics can be regarded as two special results of the general
formulation. We assume a time-dependent Hamiltonian $H(Q,P;x(t))$ with
$x(t)=Vt$, where $V$ is slow in a classical sense. The rate-of-change $V$ is
not necessarily slow in the quantum-mechanical sense. Dissipation means an
irreversible systematic growth of the (average) energy. It is associated with
the stochastic spreading of energy across levels. The latter can be
characterized by a transition probability kernel $P_t(n|m)$ where $n$ and $m$
are level indices. This kernel is the main object of the present study. In the
classical limit, due to the (assumed) chaotic nature of the dynamics, the
second moment of $P_t(n|m)$ exhibits a crossover from ballistic to diffusive
behavior. We define the $V$ regimes where either perturbation theory or
semiclassical considerations are applicable in order to establish this
crossover in the quantal case. In the limit $\hbar\to 0$ perturbation theory
does not apply but semiclassical considerations can be used in order to argue
that there is detailed correspondence, during the crossover time. In the
perturbative regime there is a lack of such correspondence. Namely, $P_t(n|m)$
is characterized by a perturbative core-tail structure that persists during the
crossover time. In spite of this lack of (detailed) correspondence there may be
still a restricted correspondence as far as the second-moment is concerned.
Such restricted correspondence is essential in order to establish the universal
fluctuation-dissipation relation.
