97-405 Barbaroux J.M., Joye A.

Expectation Values of Observables in Time-Dependent Quantum Mechanics (58K, LATeX) Jul 16, 97

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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.

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