一维局部化的数论问题：具有Lifshitz尾的“爆米花函数”及其Dedekind $η$的连续逼近
Number-theoretic aspects of 1D localization: "popcorn function" with Lifshitz tails and its continuous approximation by the Dedekind $η$
摘要：
我们讨论了当一个观测量是两个独立的指数加权整数的商时，物理系统中出现的分布的数论性质。以规则$\sim f^L$（$0<f<1$）分布的线性聚合物链的系综的谱密度用作具体实例，其中$L$是链长。在$f\to 1$中，频谱密度可以通过在所有有理点处不连续的“爆米花”（“popcorn”）函数来表示。我们建议一个连续的爆米花功能的基础上，Dedekind $η$ -功能的实轴附近。此外，我们提供了简单的参数，基于“欧几里德果园”建设，这表明存在的Lifshitz尾巴，典型的1D安德森本地化，在光谱边缘。我们强调，超度量结构的谱密度最终与数论关系的渐近模函数。我们还注意到在实轴附近的Dedekind $η$ -函数与Borwein和Borwein在1993年研究的某些连分数的不变测度之间的联系。
英文摘要：
We discuss the number-theoretic properties of distributions appearing in physical systems when an observable is a quotient of two independent exponentially weighted integers. The spectral density of ensemble of linear polymer chains distributed with the law $\sim f^L$ ($0<f<1$), where $L$ is the chain length, serves as a particular example. At $f\to 1$, the spectral density can be expressed through the discontinuous at all rational points, Thomae ("popcorn") function. We suggest a continuous approximation of the popcorn function, based on the Dedekind $η$-function near the real axis. Moreover, we provide simple arguments, based on the "Euclid orchard" construction, that demonstrate the presence of Lifshitz tails, typical for the 1D Anderson localization, at the spectral edges. We emphasize that the ultrametric structure of the spectral density is ultimately connected with number-theoretic relations on asymptotic modular functions. We also pay attention to connection of the Dedekind $η$-function near the real axis to invariant measures of some continued fractions studied by Borwein and Borwein in 1993.
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。
