  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.
