Barbaroux J.M., Joye A.
Expectation Values of Observables in Time-Dependent Quantum Mechanics
(58K, LATeX)

ABSTRACT.  Let $U(t)$ be the evolution operator of the Schr\"odinger equation generated by 
an hamiltonian of the form $H_0(t)+W(t)$ where $H_0(t)$ commutes for all $t$ 
with a complete set of  time independent projectors
$\{P_j\}_{j=1}^{\infty}$.
Consider the observable $A=\sum_j  P_j\lambda_j$, where
$\lambda_j\simeq j^{\mu}$, $\mu >0$  for $j$ 
large. Assuming that the "matrix elements" of $W(t)$ behave as 
$\| P_jW(t)P_k\| \simeq 1/|j-k|^p$, $j\neq k$ for $p>0$ large enough, we 
prove 
estimates on the expectation value $\bra U(t)\varphi | AU(t)\varphi 
\ket\equiv \bra  A\ket_{\varphi}(t)$ for large times of the type 
$ \bra  
A\ket_{\varphi}(t)\leq c t^{\delta}$, where $\delta>0$ depends on 
$p$ and $\mu$. Typical applications 
concern the energy expectation $\bra  H_0\ket_{\varphi}(t)$ in case 
$H_0(t)\equiv H_0$ or the expectation of the position operator 
$\bra  x^2\ket_{\varphi}(t)$ on the lattice where $W(t)$ is the discrete 
laplacian or a variant of it and
$H_0(t)$ is a time dependent multiplicative potential.