A note on the persistence of multiplicity of eigenvalues of fractional Laplacian under perturbations

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-05-11 DOI:10.1016/j.na.2024.113558

Marco Ghimenti , Anna Maria Micheletti , Angela Pistoia

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引用次数: 0

Abstract

We consider the eigenvalue problem for the fractional Laplacian ${(- Δ)}^{s}$ , $s \in (0, 1)$ , in a bounded domain $Ω$ with Dirichlet boundary condition. A recent result (see Fall et al., 2023) states that, under generic small perturbations of the coefficient of the equation or of the domain $Ω$ , all the eigenvalues are simple. In this paper we give a condition for which a perturbation of the coefficient or of the domain preserves the multiplicity of a given eigenvalue. Also, in the case of an eigenvalue of multiplicity $ν = 2$ we prove that the set of perturbations of the coefficients which preserve the multiplicity is a smooth manifold of codimension 2 in $C^{1} (R^{n})$ .

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关于扰动下分数拉普拉斯特征值多重性持续性的说明

我们考虑的是分数拉普拉斯方程 (-Δ)s 的特征值问题，s∈(0,1)，在有界域 Ω 中，边界条件为 Dirichlet。最近的一个结果（见 Fall 等人，2023 年）指出，在方程系数或域 Ω 的一般小扰动下，所有特征值都是简单的。在本文中，我们给出了一个条件，即系数或域的扰动会保持给定特征值的多重性。此外，在特征值的多重性 ν=2 的情况下，我们证明了保持多重性的系数扰动集合是 C1(Rn) 中标度为 2 的光滑流形。

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

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.