{"title":"关于奇阶微分算子的布洛赫特征值、带函数和带周期矩阵系数","authors":"O. A. Veliev","doi":"10.1007/s11005-024-01810-2","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we consider the Bloch eigenvalues, band functions and bands of the self-adjoint differential operator <i>L</i> generated by the differential expression of odd order <i>n</i> with the <span>\\(m\\times m\\)</span> periodic matrix coefficients, where <span>\\(n>1.\\)</span> We study the localizations of the Bloch eigenvalues and continuity of the band functions and prove that each point of the set <span>\\(\\left[ (2\\pi N)^{n},\\infty \\right) \\cup (-\\infty ,(-2\\pi N)^{n}]\\)</span> belongs to at least <i>m</i> bands, where <i>N</i> is the smallest integer satisfying <span>\\(N\\ge \\pi ^{-2}M+1\\)</span> and <i>M</i> is the sum of the norms of the coefficients. Moreover, we prove that if <span>\\(M\\le \\ \\pi ^{2}2^{-n+1/2}\\)</span>, then each point of the real line belong to at least <i>m</i> bands.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"114 3","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Bloch eigenvalues, band functions and bands of the differential operator of odd order with the periodic matrix coefficients\",\"authors\":\"O. A. Veliev\",\"doi\":\"10.1007/s11005-024-01810-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we consider the Bloch eigenvalues, band functions and bands of the self-adjoint differential operator <i>L</i> generated by the differential expression of odd order <i>n</i> with the <span>\\\\(m\\\\times m\\\\)</span> periodic matrix coefficients, where <span>\\\\(n>1.\\\\)</span> We study the localizations of the Bloch eigenvalues and continuity of the band functions and prove that each point of the set <span>\\\\(\\\\left[ (2\\\\pi N)^{n},\\\\infty \\\\right) \\\\cup (-\\\\infty ,(-2\\\\pi N)^{n}]\\\\)</span> belongs to at least <i>m</i> bands, where <i>N</i> is the smallest integer satisfying <span>\\\\(N\\\\ge \\\\pi ^{-2}M+1\\\\)</span> and <i>M</i> is the sum of the norms of the coefficients. Moreover, we prove that if <span>\\\\(M\\\\le \\\\ \\\\pi ^{2}2^{-n+1/2}\\\\)</span>, then each point of the real line belong to at least <i>m</i> bands.</p></div>\",\"PeriodicalId\":685,\"journal\":{\"name\":\"Letters in Mathematical Physics\",\"volume\":\"114 3\",\"pages\":\"\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-05-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Letters in Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s11005-024-01810-2\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Letters in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s11005-024-01810-2","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
摘要
在本文中,我们考虑了由奇数阶 n 的微分表达式与 \(m\times m\) 周期矩阵系数生成的自相关微分算子 L 的布洛赫特征值、带函数和带,其中 \(n>1.\我们研究了布洛赫特征值的定位和带状函数的连续性,并证明了集合 \(left[ (2\pi N)^{n},\infty \right) \cup (-\infty 、(-2\pi N)^{n}]\) 至少属于 m 个带,其中 N 是满足 \(N\ge \pi ^{-2}M+1\) 的最小整数,M 是系数的规范之和。此外,我们证明,如果 \(M\le \pi ^{2}2^{-n+1/2}\) ,那么实线上的每个点至少属于 m 个带。
On the Bloch eigenvalues, band functions and bands of the differential operator of odd order with the periodic matrix coefficients
In this paper, we consider the Bloch eigenvalues, band functions and bands of the self-adjoint differential operator L generated by the differential expression of odd order n with the \(m\times m\) periodic matrix coefficients, where \(n>1.\) We study the localizations of the Bloch eigenvalues and continuity of the band functions and prove that each point of the set \(\left[ (2\pi N)^{n},\infty \right) \cup (-\infty ,(-2\pi N)^{n}]\) belongs to at least m bands, where N is the smallest integer satisfying \(N\ge \pi ^{-2}M+1\) and M is the sum of the norms of the coefficients. Moreover, we prove that if \(M\le \ \pi ^{2}2^{-n+1/2}\), then each point of the real line belong to at least m bands.
期刊介绍:
The aim of Letters in Mathematical Physics is to attract the community''s attention on important and original developments in the area of mathematical physics and contemporary theoretical physics. The journal publishes letters and longer research articles, occasionally also articles containing topical reviews. We are committed to both fast publication and careful refereeing. In addition, the journal offers important contributions to modern mathematics in fields which have a potential physical application, and important developments in theoretical physics which have potential mathematical impact.