$1$ -d中连续Anderson Hamilton算子的局部化交叉
Localization crossover for the continuous Anderson Hamiltonian in $1$-d
摘要：
我们研究了具有白噪声势的连续Anderson哈密顿量$\mathcal{H}_L$在其大小为$L$的线段上的谱的行为。我们放大能级$E$的顺序$1$（散装制度）或顺序$1\ll E \ll L$（交叉制度）。我们表明，（适当重新标度）特征值和质心的点过程收敛到泊松点过程。我们还证明了指数本地化的本征函数在一个明确的速度。此外，我们表明，本征函数收敛到明确的限制：在交叉制度，这些限制是普遍的。结合我们的姊妹论文arXiv：2102.05393的结果，这完全识别了$\mathcal{H}_L$光谱的局域和离域相位之间的转变。两个主要的技术挑战是两点或Minami估计的证明，以及对亚椭圆扩散收敛到平衡的估计，其证明依赖于Malliavin演算和hypocompativity理论。
英文摘要：
We investigate the behavior of the spectrum of the continuous Anderson Hamiltonian $\mathcal{H}_L$, with white noise potential, on a segment whose size $L$ is sent to infinity. We zoom around energy levels $E$ either of order $1$ (Bulk regime) or of order $1\ll E \ll L$ (Crossover regime). We show that the point process of (appropriately rescaled) eigenvalues and centers of mass converge to a Poisson point process. We also prove exponential localization of the eigenfunctions at an explicit rate. In addition, we show that the eigenfunctions converge to well-identified limits: in the Crossover regime, these limits are universal. Combined with the results of our companion paper arXiv:2102.05393, this identifies completely the transition between the localized and delocalized phases of the spectrum of $\mathcal{H}_L$. The two main technical challenges are the proof of a two-points or Minami estimate, as well as an estimate on the convergence to equilibrium of a hypoelliptic diffusion, the proof of which relies on Malliavin calculus and the theory of hypocoercivity.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
