NUMERICAL EVIDENCE OF DYNAMICAL SPECTRAL RIGIDITY OF ELLIPSES AMONG SMOOTH Z 2 -SYMMETRIC DOMAINS SHANZA AYUB AND JACOPO DE SIMOI Abstract. We present numerical evidence for spectral rigidity among Z 2 - symmetric domains of ellipses of eccentricity smaller than 0.30. 1. Introduction The famous question “Can one hear the shape of a drum?“ posed by M. Kac in [7] has motivated over 50 years of research into what is now called the Inverse Spectral Problem. In this paper we present numerical evidence to support a conjecture that is closely related to this problem. Let us introduce the main concepts so that we can present our results. In this paper a domain will refer to a subset Ω ⊂ R 2 which is open, connected, bounded and whose boundary ∂ Ω is a sufficiently smooth curve; for simplicity 1 we consider here domains with C ∞ boundary. We denote with D the set of all such domains. Given a domain Ω ∈D, we denote its Laplace Spectrum with Sp(Ω) = {0 <λ 0 ≤ λ 1 ≤···≤ λ k ≤···} where the λ i are the eigenvalues of the Dirichlet 2 Boundary Problem, i.e. those λ for which there exists u ∈ L 2 (Ω) so that: Δu(x)+ λu(x) = 0 if x ∈ Ω u(x) = 0 if x ∈ ∂ Ω. Kac’s question can be then expressed, more formally, as “Does Sp(Ω) determine Ω?”. Clearly, domains that are isometric to each other (i.e. can be obtained by one another via a composition of rotations and translations) will have the same Laplace spectrum. From now on we will, in this paper, identify isometric domains, i.e. we consider two domains to be equal if they are isometric. Two domains Ω and Ω ′ are said to be Laplace isospectral if Sp(Ω) = Sp(Ω ′ ). We can thus further rephrase Kac’s question as: “Are isospectral domains necessarily isometric?” In full generality, this question has a negative answer: in [11] the authors con- struct an explicit example of a pair of isospectral domains that are not isometric, Date : June 12, 2020. 1 Our discussion can be actually applied to domains whose boundary is C 8 -smooth, but we do not want to insist on this point here. 2 Historically, the majority results in the field have been obtained with Dirichlet boundary conditions, although other type of boundary conditions can be treated and are equally relevant. In this paper we will follow this long established tradition and consider only Dirichlet boundary conditions. 1 arXiv:2006.06042v1 [math.DS] 10 Jun 2020