We show that non-obtuse trapezoids are uniquely determined by their Dirichlet Laplace spectrum. This extends our previous result [Hezari et al., Ann. Henri Poincare 18(12), 3759-3792 (2017)], which was only concerned with the Neumann Laplace spectrum. © 2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0036384 I. INTRODUCTION Kac popularized

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... he isospectral problem for planar domains with a paper 17 titled "Can one hear the shape of a drum?" For a bounded, connected domain Ω in R 2 , we define Δ B Ω to be the Laplace operator on Ω with the boundary condition B, where B is either Dirichlet or Neumann. We consider the Laplace eigenvalue equation B(u) = 0 on the boundary of Ω. The eigenvalues form a discrete subset of [0, ∞), 0 ≤ λ1 < λ2 ≤ λ3 ≤ ⋅ ⋅ ⋅. If one takes the Dirichlet boundary condition, requiring the function u to vanish at the boundary, then the set of eigenvalues, known as the spectrum of Ω, is in bijection with the resonant frequencies a drum would produce if Ω were its drumhead. With a perfect ear, one could hear all these frequencies and therefore know the spectrum. Kac's question mathematically means the following: If two such domains are isospectral, then are they isometric? Gordon, Webb, and Wolpert answered Kac's question in the negative 6,7 (see also Ref. 3 for an accessible presentation). All the known counterexamples to date consist of non-convex polygons. On the other hand, in certain settings, this isospectral question can have a positive answer. There are many types of positive results. One question is whether any domain is spectrally unique (up to rigid motions) among a very large class of domains. In this direction, Kac proved that disks can be heard among all domains. He used the heat trace invariants to prove that the area and perimeter of a domain are determined by its spectrum, so by the isoperimetric inequality, disks are spectrally determined. Watanabe 29 proved that there are certain nearly circular oval domains that are spectrally unique. Recently, Hezari and Zelditch 13 showed that one can hear the shape of nearly circular ellipses among all smooth domains. A weaker inverse spectral problem is to find domains that are locally spectrally unique, meaning that they can be heard among nearby domains in a certain topology. Marvizi and Melrose 21 constructed a two-parameter family of planar domains that are locally spectrally unique in the C ∞ topology. The two-parameter family consists of domains that are defined by elliptic integrals and that resemble ellipses but are not ellipses. For more on positive inverse spectral problems, we refer the readers to the surveys in Refs. 4 and 32. The notion of spectral rigidity of a domain Ω is even weaker than local spectral uniqueness. It means that any one-parameter family of isospectral domains containing Ω and staying within a limited class must be trivial, i.e., made out of rigid motions. In this setting, Popov and Topalov 24 recently showed that ellipses are spectrally rigid within the class of analytic domains with the two axial symmetries of an ellipse. In a recent article, 13 using a length spectral rigidity theorem of de Simoi, Kaloshin, and Wei, 27 Hezari and Zeldtich proved that nearly circular Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp where A(R) = hb is the area of the inner rectangle of T. In particular, the order of the wave trace singularity from this family is 1 2 . Remark 4. The same result holds for the C1,1 family in isosceles trapezoids, if the family is unobstructed as in Fig. 5 , but h must be replaced by hα and A(R) by half of the area that the C1,1 family sweeps. The proof is identical to the proof we provided in Ref. 14; hence, we omit it. The key point is that the geometrically diffractive orbits lying on the boundary of the C1,1 family each go through only one diffractive corner, and hence, the result of Ref. 16 regarding such families on ESCS can be used in the poof of Ref. 14. V. SPECTRAL UNIQUENESS OF A TRAPEZOID Before we present the proof of our main theorem, let us state some simple facts (the following five propositions) that will facilitate our argument. We begin by recalling a statement from our previous work 14 that specifies the length of the shortest closed geodesic in a trapezoid. Since the proof is quite short, we include it for the convenience of the reader.