Weyl’s asymptotic formula for fractal Laplacians defined by a class of self-similar measures with overlaps

IF 0.6 3区数学 Q3 MATHEMATICS

Analysis Mathematica Pub Date : 2023-06-08 DOI:10.1007/s10476-023-0222-6

W. Tang, Z. Y. Wang

引用次数: 0

Abstract

We observe that some self-similar measures that we call essentially of finite type satisfy countable measure type condition. We make use of this condition to set up a framework to obtain a precise analog of Weyl’s asymptotic formula for the eigenvalue counting function of Laplacians defined by measures, emphasizing on one-dimensional self-similar measures with overlaps. As an application of our result, we obtain an analog of a semi-classical asymptotic formula for the number of negative eigenvalues of fractal Schrödinger operators as the parameter tends to infinity.

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由一类有重叠的自相似测度定义的分形拉普拉斯算子的Weyl渐近公式

我们观察到一些我们称之为本质上有限型的自相似测度满足可数测度型条件。利用这一条件，我们建立了一个框架，获得了由测度定义的拉普拉斯算子特征值计数函数的Weyl渐近公式的精确模拟，强调了具有重叠的一维自相似测度。作为我们结果的一个应用，我们得到了分形Schrödinger算子负本征值个数的半经典渐近公式的一个模拟，当参数趋于无穷大时。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Analysis Mathematica MATHEMATICS-

CiteScore

1.00

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： Traditionally the emphasis of Analysis Mathematica is classical analysis, including real functions (MSC 2010: 26xx), measure and integration (28xx), functions of a complex variable (30xx), special functions (33xx), sequences, series, summability (40xx), approximations and expansions (41xx). The scope also includes potential theory (31xx), several complex variables and analytic spaces (32xx), harmonic analysis on Euclidean spaces (42xx), abstract harmonic analysis (43xx). The journal willingly considers papers in difference and functional equations (39xx), functional analysis (46xx), operator theory (47xx), analysis on topological groups and metric spaces, matrix analysis, discrete versions of topics in analysis, convex and geometric analysis and the interplay between geometry and analysis.