Inverse Spectral Problem for Analytic {{(mathbb{Z}/2 mathbb{Z})^{n}}} -Symmetric Domains in {{mathbb{R}^{n}}}

We prove that bounded real analytic domains in ${mathbb{R}^{n}}$ , with the symmetries of an ellipsoid and with one axis length fixed, are determined by their Dirichlet or Neumann eigenvalues among other bounded real analytic domains with the same symmetries and axis length. Some non-degeneracy conditions are also imposed on the class of domains. It follows that bounded, convex analytic domains are determined by their spectra among other such domains. This seems to be the first positive result for the well-known Kac problem, “Can one hear the shape of a drum?”, in higher dimensions.