标题： 时间微分和准周期势下的量子谐振子在$ R^d$上的可约性 显示英文标题

标题： Reducibility of quantum harmonic oscillator on $ R^d$ with differential and quasi-periodic in time potential

摘要： 我们改进了Grébert和Paturel在\cite{GP}中的结果，并证明在$R^d$上带有谐波势$|x|^2$和小$t$拟周期势作为$$ {\rm i}u_t - \Delta u+|x|^2u+\varepsilon V(\omega t,x)u=0, \ (t,x)\in R\times R^d $$的线性薛定谔方程，对于频率向量$\omega\in R^n$的大多数值，可以简化为一个自治系统。 新特点是势能$V(\theta,\cdot )$仅在${\mathcal{C}^{\beta}}(T^n, \mathcal{H}^{s}(R^d))$中，当$\beta$足够大时。 因此，此类线性 PDE 的任何解在时间上都是几乎周期性的，并且在某些合适的 Sobolev 范数中保持有界。 显示英文摘要

摘要： We improve the results by Gr\'ebert and Paturel in \cite{GP} and prove that a linear Schr\"odinger equation on $R^d$ with harmonic potential $|x|^2$ and small $t$-quasiperiodic potential as $$ {\rm i}u_t - \Delta u+|x|^2u+\varepsilon V(\omega t,x)u=0, \ (t,x)\in R\times R^d $$ reduces to an autonomous system for most values of the frequency vector $\omega\in R^n$. The new point is that the potential $V(\theta,\cdot )$ is only in ${\mathcal{C}^{\beta}}(T^n, \mathcal{H}^{s}(R^d))$ with $\beta$ large enough. As a consequence any solution of such a linear PDE is almost periodic in time and remains bounded in some suitable Sobolev norms.