论文标题
脱节的操作员和异谱拉普拉斯人
Disjointness-preserving operators and isospectral Laplacians
论文作者
论文摘要
All the known counterexamples to Kac' famous question "can one hear the shape of a drum", i.e., does isospectrality of two Laplacians on domains imply that the domains are congruent, consist of pairs of domains composed of copies of isometric building blocks arranged in different ways, such that the unitary operator intertwining the Laplacians acts as a sum of overlapping "local" isometries mapping the copies彼此。 我们证明并探索了一个互补的积极陈述：如果操作员在一对域上相互交织了两个适当的laplacian实现，则可以保留不相交的支持，那么在其上的其他假设下，通常比单位性弱弱了，则域是一致的。我们特别显示了Dirichlet，Neumann和Robin Laplacians在连续功能和$ l^2 $ - 空间上的内容。
All the known counterexamples to Kac' famous question "can one hear the shape of a drum", i.e., does isospectrality of two Laplacians on domains imply that the domains are congruent, consist of pairs of domains composed of copies of isometric building blocks arranged in different ways, such that the unitary operator intertwining the Laplacians acts as a sum of overlapping "local" isometries mapping the copies to each other. We prove and explore a complementary positive statement: if an operator intertwining two appropriate realisations of the Laplacian on a pair of domains preserves disjoint supports, then under additional assumptions on it generally far weaker than unitarity, the domains are congruent. We show this in particular for the Dirichlet, Neumann and Robin Laplacians on spaces of continuous functions and on $L^2$-spaces.
