The Neumann isospectral problem for trapezoids

We show that trapezoids with identical Neumann spectra are congruent up to rigid motions of the plane. The proof is based on heat trace invariants and some new wave trace invariants associated to certain diffractive billiard trajectories. The reason we can only treat the Neumann case is that the wave trace is “more singular” for the Neumann case compared to the Dirichlet case. This is a new observation which is interesting on its own.