Spectral determination of analytic axi-symmetric plane domains

Let ${\cal D}$ denote the class of bounded real analytic plane domains with the symmetry of an ellipse. We prove that if $Ω_1, Ω_2 \in {\cal D}$ and if the Dirichlet spectra coincide, $Spec(Ω_1) = Spec(Ω_2)$, then $Ω_1 = Ω_2$ up to rigid motion.

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