随机二聚体模型的动力学局部化
Dynamical Localization for the Random Dimer Model
摘要：
研究了一维随机二聚体模型，其中对所有的$H_ω=Δ+ V_ω$，$V_ω(2x)$是独立同分布的. Bernoulli随机变量取值为$\pm V, V>0$。我们证明了，对于$V$的所有值和概率为1的$ω$，$H$的谱是纯点。如果$V\leq1$和$V\neq 1/\sqrt{2}$，Lyapounov指数仅在$E=\pm V$给出的两个临界能量处为零。对于特定的值$V=1/\sqrt{2}$，分别$V=\sqrt{2}$，我们表明存在额外的临界能量在$E=\pm 3/\sqrt{2}$，分别。E=0。在任何不包含临界能量的紧致区间$I$上，本征函数被证明是半均匀指数局部化的，这意味着动力学局部化：对于所有$q>0$和所有$ψ\in\ell^2(\Z)$，都有足够快的下降：$$ \sup_t r^{(q)}_{ψ,I}(t) \equiv \sup_t < P_I(H_ω)ψ_t, |X|^q P_I(H_ω)ψ_t > <\infty. $$这里$ψ_t=e^{-iH_ωt} ψ$，$P_I(H_ω)$是$H_ω$在区间$I$上的谱投影。特别是如果$V>1$和$V\neq \sqrt{2}$，这些结果在整个光谱上都成立（因此可以使用$I=σ(H_ω)$）。
英文摘要：
We study the one-dimensional random dimer model, with Hamiltonian $H_ω=Δ+ V_ω$, where for all $x\in\Z, V_ω(2x)=V_ω(2x+1)$ and where the $V_ω(2x)$ are i.i.d. Bernoulli random variables taking the values $\pm V, V>0$. We show that, for all values of $V$ and with probability one in $ω$, the spectrum of $H$ is pure point. If $V\leq1$ and $V\neq 1/\sqrt{2}$, the Lyapounov exponent vanishes only at the two critical energies given by $E=\pm V$. For the particular value $V=1/\sqrt{2}$, respectively $V=\sqrt{2}$, we show the existence of additional critical energies at $E=\pm 3/\sqrt{2}$, resp. E=0. On any compact interval $I$ not containing the critical energies, the eigenfunctions are then shown to be semi-uniformly exponentially localized, and this implies dynamical localization: for all $q>0$ and for all $ψ\in\ell^2(\Z)$ with sufficiently rapid decrease: $$ \sup_t r^{(q)}_{ψ,I}(t) \equiv \sup_t < P_I(H_ω)ψ_t, |X|^q P_I(H_ω)ψ_t > <\infty. $$ Here $ψ_t=e^{-iH_ωt} ψ$, and $P_I(H_ω)$ is the spectral projector of $H_ω$ onto the interval $I$. In particular if $V>1$ and $V\neq \sqrt{2}$, these results hold on the entire spectrum (so that one can take $I=σ(H_ω)$).
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