The Neumann Isospectral Problem for Trapezoids

Journal article, 2017

We show that non-obtuse 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. We use the method of reflections to express the Dirichlet and Neumann wave kernels in terms of the wave kernel of the double polygon. Using Hillairet's trace formulas for isolated diffractive geodesics and one-parameter families of regular geodesics with geometrically diffractive boundaries for Euclidean surfaces with conical singularities (Hillairet in J Funct Anal 226(1):48-89, 2005), we obtain the new wave trace invariants for trapezoids. To handle the reflected term, we use another result of Hillairet (J Funct Anal 226(1):48-89, 2005), which gives a Fourier integral operator representation for the Keller and Friedlander parametrix (Keller in Proc Symp Appl Math 8:27-52, 1958; Friedlander in Math Proc Camb Philos Soc 90(2):335-341, 1981) of the wave propagator near regular diffractive geodesics. 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 of independent interest.