I heard in a lecture that if you know the spectrum of the Laplacian operator in a domain, you actually know everything about the domain. What does this mean?
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Related quantum mechanical inverse scattering problem. – Qmechanic Mar 08 '21 at 05:07
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This problem is known as "hearing the shape of a drum". Is it possible to know the shape of a drum (a 2d membrane with a fixed boundary) only by hearing the sound it produces (its spectrum) ?
As nice as the idea sounds, the answer is negative. Gordon, Webb and Wolpert gave a counterexample, with two domains ("drums") that have the same spectrum under the laplacian operator:
(image from https://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum)
However, the answer can be made positive under some additional restrictions on the domain, such as only considering convex domains with well-behaved boundaries. We don't know how much these restrictions can be relaxed! It's an interesting problem.
