Global eigenfamilies on closed manifolds

IF 1.2 2区数学 Q1 MATHEMATICS

Journal of the London Mathematical Society-Second Series Pub Date : 2025-07-25 DOI:10.1112/jlms.70228

Oskar Riedler, Anna Siffert

{"title":"Global eigenfamilies on closed manifolds","authors":"Oskar Riedler, Anna Siffert","doi":"10.1112/jlms.70228","DOIUrl":null,"url":null,"abstract":"We study globally defined <math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>λ</mi>\n <mo>,</mo>\n <mi>μ</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$(\\lambda,\\mu)$</annotation>\n </semantics></math>-eigenfamilies on closed Riemannian manifolds. Among others, we provide (non-)existence results for such eigenfamilies, examine topological consequences of the existence of eigenfamilies and classify <math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>λ</mi>\n <mo>,</mo>\n <mi>μ</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$(\\lambda,\\mu)$</annotation>\n </semantics></math>-eigenfamilies on flat tori. It is further shown that for <math>\n <semantics>\n <mrow>\n <mi>f</mi>\n <mo>=</mo>\n <msub>\n <mi>f</mi>\n <mn>1</mn>\n </msub>\n <mo>+</mo>\n <mi>i</mi>\n <msub>\n <mi>f</mi>\n <mn>2</mn>\n </msub>\n </mrow>\n <annotation>$f=f_1+i f_2$</annotation>\n </semantics></math> being an eigenfunction decomposed into its real and its imaginary part, the powers <math>\n <semantics>\n <mrow>\n <mo>{</mo>\n <msubsup>\n <mi>f</mi>\n <mn>1</mn>\n <mi>a</mi>\n </msubsup>\n <msubsup>\n <mi>f</mi>\n <mn>2</mn>\n <mi>b</mi>\n </msubsup>\n <mo>∣</mo>\n <mi>a</mi>\n <mo>,</mo>\n <mi>b</mi>\n <mo>∈</mo>\n <mi>N</mi>\n <mo>}</mo>\n </mrow>\n <annotation>$\\lbrace f_1^a f_2^b\\mid a,b\\in \\mathbb N\\rbrace$</annotation>\n </semantics></math> satisfy highly rigid orthogonality relations in <math>\n <semantics>\n <mrow>\n <msup>\n <mi>L</mi>\n <mn>2</mn>\n </msup>\n <mrow>\n <mo>(</mo>\n <mi>M</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$L^2(M)$</annotation>\n </semantics></math>. In establishing these orthogonality relations, one is led to combinatorial identities involving determinants of products of binomials, which we view as being of independent interest.","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"112 2","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2025-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70228","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70228","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

We study globally defined $(λ, μ)$ -eigenfamilies on closed Riemannian manifolds. Among others, we provide (non-)existence results for such eigenfamilies, examine topological consequences of the existence of eigenfamilies and classify $(λ, μ)$ -eigenfamilies on flat tori. It is further shown that for $f = f_{1} + i f_{2}$ being an eigenfunction decomposed into its real and its imaginary part, the powers ${f_{1}^{a} f_{2}^{b} ∣ a, b \in N}$ satisfy highly rigid orthogonality relations in $L^{2} (M)$ . In establishing these orthogonality relations, one is led to combinatorial identities involving determinants of products of binomials, which we view as being of independent interest.

Abstract Image

查看原文本刊更多论文

闭流形上的全局特征族

研究了闭黎曼流形上全局定义的（λ, μ） $(\lambda,\mu)$ -特征族。其中，我们提供了这些特征族的（不）存在性结果，检查了特征族存在的拓扑结果，并在平坦环面上分类了（λ, μ） $(\lambda,\mu)$ -特征族。进一步证明，当f = f 1 + i f 2 $f=f_1+i f_2$是一个分解为实部和虚部的特征函数时，幂{ f 1a f 2b∣a, b∈N }$\lbrace f_1^a f_2^b\mid a,b\in \mathbb N\rbrace$在l2 (M) $L^2(M)$中满足高度刚性正交关系。在建立这些正交关系的过程中，我们得到了包含二项乘积的行列式的组合恒等式，我们认为这是一个独立的问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.