"Can one hear the shape of a drum?" is a well known problem, originating from Kac, 1966, that questions whether an (idealized) drum head is completely specified by its spectrum. That is: is the Dirichlet boundary $D$ uniquely determined given the complete eigenspectrum $\lambda_i$ for the Helmholtz equation?:
\begin{cases} \Delta u + \lambda u = 0\\ u|_{\partial D} = 0 \end{cases}
While there are exceptions, apparently the spectrum is unique for a 2D convex drumhead (Zelditch, page above), so relatively strong statements can be made without considering high dimensional systems.
I'd like to know: "Can you hear the material of a drumhead?" I'm interested in a similar equation with an additional term, $V(\mathbf x)u(\mathbf x)$. For purposes of the analogy, Kacs problem asks whether the boundary of an ideal drum is determined by the sound; I'm asking to what extent the drum head itself can be characterized if there is some varation in material or thickness.
Specifically, I'm interested in how well $V(\mathbf x)$ is determined based on all eigenvalues $\lambda$ and eigenfunctions $u(\mathbf x)$ for the following:
\begin{cases} \left[\Delta_{\mathbf x} + \lambda + V(\mathbf x)\right]u(\mathbf x) = 0\\ u|_{\mathbf x \rightarrow \infty} = 0 \end{cases}
As the original problem is a particular case of this one ($V(\mathbf x) = 0$ over $D$, $V(\mathbf x = D) = \infty$), so strictly speaking the answer is 'not completely'.
As a side note, this question arose from research in non relativistic quantum mechanics, and can be stated as: how similar are two potentials if they share the same eigenspectrum and eigenfunctions? If the discrete spectrum is matched, does this imply the continuous spectrum is also? I've already posted a similar question in Physics SE. Thanks!!