We introduce a variation on Kac's question, "Can one hear the shape of a
drum?" Instead of trying to identify a compact manifold and its metric via its
Laplace--Beltrami spectrum, we ask if it is possible to uniquely identify a
point x on the manifold, up to symmetry, from its pointwise counting function
Nxβ(Ξ»)=Ξ»jββ€Ξ»βββ£ejβ(x)β£2, where here Ξgβejβ=βΞ»j2βejβ and ejβ form an orthonormal
basis for L2(M). This problem has a physical interpretation. You are placed
at an arbitrary location in a familiar room with your eyes closed. Can you
identify your location in the room by clapping your hands once and listening to
the resulting echos and reverberations?
The main result of this paper provides an affirmative answer to this question
for a generic class of metrics. We also probe the problem with a variety of
simple examples, highlighting along the way helpful geometric invariants that
can be pulled out of the pointwise counting function Nxβ.Comment: 26 pages, 1 figur