Can you hear your location on a manifold?

Abstract

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 \[ N_x(\lambda) = \sum_{\lambda_j \leq \lambda} |e_j(x)|^2, \] where here $\Delta_g e_j = -\lambda_j^2 e_j$ and $e_j$ form an orthonormal basis for $L^2(M)$. This problem has several natural physical interpretations, two of which are acoustic: 1. 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 echoes and reverberations? 2. If a drum of a known shape is struck at some unknown point, can you determine this point by listening to the quality of the sound the drum produces? 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 $N_x$.