A note on measure-geometric Laplacians

详细信息    查看全文

• 作者：M. Kesseböhmer ; T. Samuel ; H. Weyer
• 关键词：Measure ; geometric Laplacians ; Spectral asymptotics ; Singular measures ; Chebyshev polynomials ; Salem measures
• 刊名：Monatshefte f¨¹r Mathematik
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：181
• 期：3
• 页码：643-655
• 全文大小：707 KB
• 刊物主题：Mathematics, general;
• 出版者：Springer Vienna
• ISSN：1436-5081
• 卷排序：181

文摘

We consider the measure-geometric Laplacians \(\Delta ^{\mu }\) with respect to atomless compactly supported Borel probability measures \(\mu \) as introduced by Freiberg and Zähle (Potential Anal. 16(1):265–277, 2002) and show that the harmonic calculus of \(\Delta ^{\mu }\) can be deduced from the classical (weak) Laplacian. We explicitly calculate the eigenvalues and eigenfunctions of \(\Delta ^{\mu }\). Further, it is shown that there exists a measure-geometric Laplacian whose eigenfunctions are the Chebyshev polynomials and illustrate our results through specific examples of fractal measures, namely inhomogeneous self-similar Cantor measures and Salem measures.