arXiv 2209.096722022-09-20math-phmath.APmath.MPmath.SP

大环面上薛定谔方程的代数解

Algebraic delocalization for the Schrödinger equation on large tori

Henrik Ueberschaer

备注：: 10 pages, no figures

摘要：

设$\mathcal{L}$是一个固定的$d$维格.本文研究了环面$\mathbb{R}^d/L\mathcal{L}$上具有正$L^\infty$势的定常薛定谔方程在极限下解的局部化性质，即$L\to\infty$，其中维数为$d \leq 3$ .我们表明，概率措施与$L^2$归一化的解决方案，与本征值$E$附近的底部的频谱，满足代数离域定理，该定理指出，这些概率措施不能本地化内的球半径$r = o(E^{-1/4+ε})$，除非本地化发生足够慢的代数衰减。特别地，我们将我们的结果应用于薛定谔算子建模无序系统，如d维连续Anderson- Bernoulli模型，其中本征函数的几乎必然指数局部化，在极限$E \to 0$中，Bourgain-Kenig在维度$d \geq 2$中证明了，并且表明我们的定理暗示了局部化长度在此极限中的代数爆破。

英文摘要：

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.

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