This is a question related to the "hearing the shape of a drum" problem. There are well-known examples of different shapes (non-isometric manifolds) which yield the same eigenvalue spectrum. However, has the following related problem been studied?
Supposing a perturbation in the shape of a vibrating drum (let's say, to be more precise, a small change in its metric tensor $g_{\mu\nu} \rightarrow (I+f)g_{\mu\nu}$, where $f$ is an arbitrary function), can we recover the function $f$ from studying the small change in its spectrum?
Any reference would be appreciated.