One can hear the shape of a trapezoid drum

What is it about?

If two bounded domains in the plane have the same Laplace spectrum, then are they the same shape? Kac popularized this question by rephrasing it as `can one hear the shape of a drum?' In general the answer is no, but if one restricts to certain classes of domains, the answer can be yes. Here we prove that if one considers domains that are trapezoids, then if they are isospectral, they are the same shape. Musically, this means that if you build two drums with trapezoidal drumheads, and the drums sound perfectly identical, then the trapezoids are the same shape!

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Photo by Case Hubbart on Unsplash

Photo by Case Hubbart on Unsplash

Why is it important?

An important ingredient in the proof is the study of closed geodesics in trapezoids and triangles. In order to connect them to the Laplace spectrum, we study the wave trace and its singularities. In many contexts one has a Poisson relation: the times at which the wave trace is singular is contained in the set of lengths of closed geodesics. Here due to the corners, we do not immediately have this relation. We are able to obtain it with help from Hillairet's wonderful works on the wave trace on euclidean surfaces with conical singularities. Closed geodesics in polygonal domains and the Poisson relation in singular geometric settings are areas which contain many open problems, and this work is a small contribution to those fields.

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