{"title":"PSLn(q) as operator group of isospectral drums","authors":"K. Thas","doi":"10.1088/0305-4470/39/50/L01","DOIUrl":null,"url":null,"abstract":"In a paper by M Kac (1966 Am. Math. Mon. 73 1–23), Kac asked his famous question ‘Can one hear the shape of a drum?’, which was answered negatively in Gordon et al (1992 Invent. Math. 110 1–22) by construction of planar isospectral pairs. In Buser et al (1994 Int. Math. Res. Not. 9), it is observed that all operator groups associated with the known counter examples are isomorphic to one of PSL3(2), PSL3(3), PSL4(2) and PSL3(4). We show that if (D1, D2) is a pair of non-congruent planar isospectral domains constructed from unfolding a polygonal base-tile and with associated operator group PSLn(q), then (n, q) belongs to this very restricted list.","PeriodicalId":87442,"journal":{"name":"Journal of physics A: Mathematical and general","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2006-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of physics A: Mathematical and general","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1088/0305-4470/39/50/L01","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 6
Abstract
In a paper by M Kac (1966 Am. Math. Mon. 73 1–23), Kac asked his famous question ‘Can one hear the shape of a drum?’, which was answered negatively in Gordon et al (1992 Invent. Math. 110 1–22) by construction of planar isospectral pairs. In Buser et al (1994 Int. Math. Res. Not. 9), it is observed that all operator groups associated with the known counter examples are isomorphic to one of PSL3(2), PSL3(3), PSL4(2) and PSL3(4). We show that if (D1, D2) is a pair of non-congruent planar isospectral domains constructed from unfolding a polygonal base-tile and with associated operator group PSLn(q), then (n, q) belongs to this very restricted list.
在卡茨(1966)的一篇论文中。数学。星期一(73 - 23),卡茨问了他著名的问题“人能听到鼓的形状吗?”, Gordon et al(1992)对此给出了否定的回答。数学。110 - 22)通过构造平面等谱对。In Buser et al . (1994);数学。Res. Not. 9),可以观察到与已知反例相关的所有算子群都与PSL3(2)、PSL3(3)、PSL4(2)和PSL3(4)中的一个同构。我们证明了如果(D1, D2)是由展开一个多边形基图构成的一对非全等平面等谱域,并且具有相应的算子群PSLn(q),则(n, q)属于这个极限表。