论文标题
schrödinger方程的代数离域化
Algebraic delocalization for the Schrödinger equation on large tori
论文作者
论文摘要
令$ \ Mathcal {l} $为固定的$ D $ - 维晶格。我们研究了固定schrödinger方程的解决方案的本地化属性，其在限制中具有正$ l^\ infty $在tori $ \ mathbb {r}^d/l \ mathcal {l} $上，如$ l \ to \ in \ infty $，用于dimension $ d \ d \ leq 3 $。我们表明，与$ l^2 $一量的解决方案相关的概率度量，并在频谱底部附近具有特征值$ e $，满足一个代数的离域定理，该定理指出这些概率度量不能定位在半径$ r = o（e^{e^{e^{ - 1/4+ε}）$的情况下，与定位均可定位。 In particular, we apply our result to Schrödinger operators modeling disordered systems, such as the d-dimensional continuous Anderson- Bernoulli model, where almost sure exponential localization of eigenfunctions, in the limit as $E \to 0$, was proved by Bourgain-Kenig in dimension $d \geq 2$, and show that our theorem implies an algebraic blow-up of localization length in this limit.
Let $\mathcal{L}$ be a fixed $d$-dimensional lattice. We study the localization properties of solutions of the stationary Schrödinger equation with a positive $L^\infty$ potential on tori $\mathbb{R}^d/L\mathcal{L}$ in the limit, as $L\to\infty$, for dimension $d \leq 3$. We show that the probability measures associated with $L^2$-normalized solutions, with eigenvalue $E$ near the bottom of the spectrum, satisfy an algebraic delocalization theorem which states that these probability measures cannot be localized inside a ball of radius $r = o(E^{-1/4+ε})$, unless localization occurs with a sufficiently slow algebraic decay. In particular, we apply our result to Schrödinger operators modeling disordered systems, such as the d-dimensional continuous Anderson- Bernoulli model, where almost sure exponential localization of eigenfunctions, in the limit as $E \to 0$, was proved by Bourgain-Kenig in dimension $d \geq 2$, and show that our theorem implies an algebraic blow-up of localization length in this limit.
