Knowing the potential, we can find the spectrum of the Schrödinger operator. The converse question is: Knowing the spectrum, can we reconstruct the potential? As an example, a harmonic potential has an equally spaced spectrum. But is the converse true?
This is, of course, similar to the 'hearing the shape of the drum' problem, which has a negative answer. But we also should notice that in classical mechanics, if the potential is symmetric, we can recover it from the oscillation period as a function of the energy of the particle. This is due to ingenious work by Abel.