The most commonly considered counterexamples to Kac’s 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.

Kac 的著名问题“人们能听到鼓的形状吗？”的最常被考虑的反例。– 即，域上两个拉普拉斯算子的等谱性是否意味着域是全等的？– 由成对的域组成，这些域由以不同方式排列的等距构建块的副本组成，因此交织拉普拉斯算子的酉算子充当重叠的“局部”等距的总和，将副本相互映射。

我们证明并探索了一个互补的积极陈述：如果在一对域上将拉普拉斯算子的两个适当实现交织在一起的算子保持不相交的支持，那么在通常远弱于单一性的附加假设下，域是全等的。我们特别针对连续函数空间和 $L^2$-空间上的 Dirichlet、Neumann 和 Robin Laplacians 展示了这一点。